morfismos, vol 8, no 1, 2004

VOLUMEN 8NÚMERO 1

ENERO A JUNIO DE 2004 ISSN: 1870-6525

MorfismosComunicaciones EstudiantilesDepartamento de Matematicas

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VOLUMEN 8NÚMERO 1

ENERO A JUNIO DE 2004ISSN: 1870-6525

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Contenido

Multiobjective Markov Control Processes: a Linear Programming Approach

Onesimo Hernandez-Lerma and Rosario Romera . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Tutte uniqueness of locally grid graphs

D. Garijo, A. Marquez and M.P. Revuelta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

No-inmersion de espacios lente

Enrique Torres Giese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Morfismos, Vol. 8, No. 1, 2004, pp. 1–33

Multiobjective Markov Control Processes: aLinear Programming Approach ∗

Onesimo Hernandez-Lerma Rosario Romera

Abstract

This paper studies discrete-time multiobjective Markov controlprocesses (MCPs) on Borel spaces and unbounded costs. Un-der mild assumptions, it shows the existence of Pareto policies,which, as in multiobjective optimization problems, are also char-acterized as optimal policies for a certain class of single-objective(or “scalar”) MCPs. A similar result is obtained for strong Paretopolicies, which are Pareto policies whose cost vector is the closest,in the Euclidean norm, to the virtual minimum. To obtain theseresults, the basic idea is to transform the multiobjective MCP intoan equivalent multiobjective measure problem (MMP). In addition,MMP is restated as a primal multiobjective linear program and itis shown that solving the dual program is in fact the same as solv-ing the scalarized MCPs. A multiobjective LQ example illustratesthe main results.

2000 Mathematics Subject Classification: 93E20, 90C40, 90C29.Keywords and phrases: Markov control processes, multiobjective controlproblems, Pareto optimality, (infinite–dimensional) multiobjective linearprogramming.

1 Introduction

In a standard optimal control problem there is a decision–maker or con-troller that wishes to optimize a single objective function. Thus, forinstance, in a production control problem it is tacitly assumed thatthe given objective function somehow aggregates several different costs

∗Invited Article. Research partially supported by CONACYT (Mexico) Grant37355-E for OHL, and DGES (Spain) Grant PB96-0111 for RR.

1

2 O. Hernandez-Lerma and R. Romera

(manufacturing costs, holding costs, distribution costs, etc.) and possi-bly several income sources (for example, sales, investments, and so on).However, there are situations in which it is convenient, or perhaps evennecessary, to optimize separately these functions and the controller isthen led to consider a multiobjective problem of the form (say): “mini-mize” the cost vector

V (π) := (V1(π), . . . , Vq(π)) ∈ IRq

over the class of all admissible policies π (see Section 2 for details). Inparticular, if π∗ minimizes V (π) in the sense of Pareto, then π∗ is saidto be a Pareto policy. On the other hand, letting

(1.1) V ∗i := inf

πVi(π) for i = 1, . . . , q,

and defining the virtual minimum V ∗ := (V ∗1 , . . . , V ∗

q ), an importantissue is to find strong Pareto policies, namely, Pareto policies π∗ whosecost vector V (π∗) is the “closest” (e.g. in the usual Euclidean norm) toV ∗. This is the control–theoretic analogue of a goal programming prob-lem [36] in which the goal or target is V ∗. (We might of course considerother “goals”, but V ∗ is the most common.) Still another key problemoccurs when the individual costs V1(π), . . . , Vq(π) are ranked in order of“importance”. In this case, a lexicographically (or hierarchically) opti-mal policy turns out to be a particular Pareto policy.

Contributions of this paper. In this paper we study discrete-time multiobjective Markov control processes (MCPs) on Borel spacesand unbounded costs. The main problems we are concerned with are theexistence and characterization of both Pareto and strong Pareto policies,and also of weak and proper Pareto policies (Definition 2.5). Actually,the existence of Pareto, weak Pareto, and proper Pareto policies is veryeasy because it can be obtained via the usual scalarization approach,in which the multiobjective MCP is reduced to a single–objective (orstandard or scalar) MCP with a “weighted” objective function of theform

(1.2) λ·V (π) := λ1V1(π) + · · · + λqVq(π)

for some vectors λ in the nonnegative orthant IRq+. However, the exis-

tence of strong Pareto policies as well as the characterization problemare more complicated, and, to the best of our knowledge, this is the

Multiobjective Markov Control Processes 3

first paper dealing with these issues for general MCPs. (See below forrelated literature.)

To study the latter problems we propose here to use so–called occu-pation measures to transform the multiobjective MCP into an equivalentmultiobjective measure problem (MMP) on a suitable space of measures.The original multiobjective control problem is thus greatly simplified be-cause the MMP turns out to have a linear objective (vector) functiondefined on a convex set of measures. This implies that, for instance, theexistence of strong Pareto policies essentially reduces to find the distancefrom the virtual minimum V ∗ to a convex set. Similarly, the charac-terization of Pareto policies (known as the “theorem of equivalence” inPareto optimality [4]) can be obtained by standard convex–analytic ar-guments. Moreover, introducing suitable vector spaces, we reformulatethe MMP as a primal multiobjective linear program and this allows usto show that solving the dual linear program is in fact the same as solv-ing the scalar problem (1.2) using dynamic programming. As far as wecan tell, this interpretation of the scalarization approach for multiob-jective control problems as the dual of a multiobjective linear programhas never been reported before in the literature. We should also notethat to obtain the latter duality result, as well as the characterizationof Pareto policies (Theorem 3.4, below) without our MMP approachwould be extremely difficult — perhaps impossible — to obtain.

Related literature. Vector optimization problems can be tracedback to (at least) the late 19th century; see e.g. [4, 31] for earlierreferences. However, according to the excellent survey by Salukvadze[35, Chapter 1], in control theory they were first introduced by Zadeh[46] in 1963. The scalarization and the hierarchical (or lexicographical)approaches were introduced by Reid and Citron [34] and Waltz [43],respectively.

Concerning multiobjective MCPs, the existence and characteriza-tion of Pareto policies have been studied by many authors but for par-ticular classes of MCPs, for instance with a countable state space [10-12,14,23,24,28,40-42,44,45], or in Borel spaces but with bounded costs[29, 37, 39]. It should be noted that for some of these classes of MCPsone can obtain very interesting results. For example, if the state andaction spaces are both finite, the set of Pareto policies can be completelycharacterized using Theorem 1 of Arrow et al. [3], as in [14]. Moreover,


for finite state spaces, there are multiobjective versions of value itera-tion [23, 24, 45] and of policy iteration [12, 41, 42], which, as they arecomputationally appealing, it would be interesting to investigate if theycan be extended to MCPs in uncountable spaces. On the other hand,some papers [12, 23, 24, 29] deal with a vector–minimization problemmore general than ours in the sense that instead of the convex cone IRq

+,they work with the partial order induced by an arbitrary pointed convexcone in IRq. But it turns out that they restrict the control problem tosome subclass of policies (e.g. deterministic stationary), whereas herewe work with the set of all (randomized, history–dependent) policies. Atany rate, extending our MMP approach to the case of a general pointedconvex cone seems to be a purely notational problem.

Organization of the paper. The remainder of the paper is orga-nized as follows. In Section 2 we introduce the multiobjective MCP weare concerned with, as well as the precise notions of Pareto optimality.We consider a vector of discounted cost criteria but in Section 8 webriefly explain, among other things, how our results can be translatedto average costs. In Section 3 we state our hypotheses (Assumption3.1) and the so-called “theorem of equivalence” in Pareto optimality[4]. In fact, we state this theorem in two parts, Theorem 3.2(a) and(the converse) Theorem 3.4, because the proof of the latter requires theMMP, which is introduced until Section 4. On the other hand, Theorem3.2(a) is the easy part of the “theorem of equivalence” and it directlyyields the existence of Pareto policies. Section 3 also includes Exam-ple 3.5 on a multiobjective LQ (Linear system with Quadratic costs)MCP in which explicit Pareto policies can be calculated. In Section 5we introduce the virtual minimum V ∗ for our multiobjective MCP, andshow the existence of strong Pareto policies. We also extend a result ofTanaka [37] that can be very useful to compute strong Pareto policies;see Theorem 5.2(b). This fact is illustrated in Example 5.7, which isa continuation of the LQ Example 3.5. Section 6 presents the multi-objective Linear Programming (LP) formulation of the multiobjectiveMCP. The idea (as for scalar and constrained MCPs [16,17,20,21]) is tointroduce suitable dual pairs of vector spaces in which the MMP (4.7)can be formulated as a multiobjective linear program. The multiobjec-tive LP formulation is borrowed from Balbas and Heras [7]. Section 7contains the proof of Theorem 3.4, and, finally, in Section 8 we brieflymention some connections between our multiobjective MCPand con-strained MCPs, multiobjective problems with average cost criteria, and


multiobjective problems with “mixed” average and discounted criteria.

Remark 1.1.(Notation) If S is a Borel space (that is, a Borel subsetof a complete and separable metric space), we denote its Borel σ-algebraby B(S). If S and T are Borel spaces, then a stochastic kernel on S givenT is a function (t, B) → q(B|t) from T ×B(S) to the interval [0, 1] suchthat q(B|· ) is a measurable function on T for each fixed B ∈ B(S), andq(· |t) is a probability measure on B(S) for each fixed t.

2 Multiobjective MCPs

A multiobjective Markov control model can be represented as

(2.1) (X, A, IK, Q, (c1, . . . , cq), δ, γ0),

where X and A are Borel spaces that stand for the state space and thecontrol (or action) set, respectively. We also have the constraint set IK,a Borel subset of X×A, and which is assumed to contain the graph of ameasurable map from X to A (this ensures that the set IF in Definition2.1, below, is nonempty). For each x ∈ X, the x-section in IK, namely

A(x) := a ∈ A|(x, a) ∈ IK,

is a (nonempty) Borel subset of A whose elements are the admissiblecontrol actions in the state x. The transition law Q is a stochastic kernelon X given IK, whereas

(2.2) c := (c1, . . . , cq) : IK → IRq

is a vector function whose components are used to define the differentcost criteria. Finally, δ ∈ (0, 1) is a given discount factor, and γ0 is theinitial distribution, a probability measure on X.

If q = 1, then (2.1) will be referred to as a “scalar” (or “standard”)Markov control model.

Definition 2.1.Φ denotes the family of stochastic kernels ϕ on A givenX that satisfy the constraint ϕ(A(x)|x) = 1 for all x ∈ X, and IF standsfor the class of measurable functions f from X to A such that f(x) ∈A(x) for all x ∈ X.


Let H0 := X, and Hn := IKn × X for n = 1, 2, . . .. A control policyis a sequence π = πn, n = 0, 1, . . . of stochastic kernels πn on A givenHn that satisfy the condition

(2.3) πn(A(xn)|hn) = 1

for each “history” hn = (x0, a0, . . . , xn−1, an−1, xn) in Hn and n =0, 1, . . .. We denote by Π the set of control policies. A control pol-icy π = πn is said to be randomized stationary if there exists ϕ ∈ Φsuch that πn(· |hn) = ϕ(· |xn) for every history hn ∈ Hn and n = 0, 1, . . ..The set of such policies will be identified with the family Φ in Definition2.1. On the other hand, π = πn is called deterministic stationary ifthere exists f ∈ IF such that πn(· |hn) is the Dirac measure concentratedat f(xn) for all hn ∈ Hn and n = 0, 1, . . .. We shall identify IF with thecollection of deterministic stationary policies.

The multiobjective MCP. Consider the control model (2.1) andlet (Ω,F) be the (canonical) measurable space consisting of the sam-ple space Ω := (X × A)∞, and the corresponding product σ-algebra F .Then, for each policy π ∈ Π, there is a probability measure P π

γ0and

a stochastic process (xt, at), t = 0, 1, . . . defined on Ω in a canonicalway, where xt and at represent the state and the control variables attime t (t = 0, 1, . . .) when using the policy π. The expectation operatorwith respect to P π

γ0is denoted by Eπ

γ0.

For each i = 1, . . . , q and π ∈ Π, consider the δ-discounted cost

(2.4) Vi(π, γ0) := (1 − δ)Eπγ0

∞

t=0

δtci(xt, at) ,

which will be well defined under our Assumption 3.1 below. Now letV (π, γ0) ∈ IRq be the cost vector

(2.5) V (π, γ0) := (V1(π, γ0), . . . , Vq(π, γ0)).

The multiobjective control problem we are concerned with is to find apolicy π∗ that “minimizes” V (· , γ0) in the sense of Pareto. To state thisin a precise form we first introduce some notation and terminology.

Pareto optimality. We consider IRq with the usual partial order;that is, for q-vectors u and v, the inequality u ≤ v means that ui ≤ vi


for all i = 1, . . . , q. We also have

u < v ⇔ u ≤ v and u = v;

u v ⇔ ui < vi for all i = 1, . . . , q.

A sequence uk ⊂ IRq converging to u is said to converge in the di-rection v ∈ IRq if there is a sequence of positive numbers tk such thattk → 0 and

(2.6) limk→∞

(uk − u)/tk = v.

Let Γ be a subset of IRq. The tangent cone to Γ at u ∈ Γ, denotedT(Γ, u), is the set of all the directions v ∈ IRq in which some sequencein Γ converges to u. There are several equivalent definitions of tangentcone; see e.g. [5]. In particular, if Γ is a convex set, then ([5],p. 64)

(2.7) T(Γ, u) = closure [t>0

1

t(Γ − u) ].

Note that Γ − u is contained in T(Γ, u).

Definition 2.2. Let Γ be a subset of IRq. A vector u∗ in Γ is said tobe

(a) a Pareto point of Γ if there is no u ∈ Γ such that u < u∗;

(b) a weak Pareto point of Γ if there is no u ∈ Γ such that u << u∗;

(c) a proper Pareto point of Γ if u∗ is a Pareto point and, in addition,the tangent cone to Γ at u∗ does not contain vectors v < 0.

Let Par(Γ), WPar(Γ) and PPar(Γ) denote, respectively, the set ofPareto points of Γ, the set of weak Pareto points, and the set of properPareto points. Then

(2.8) PPar(Γ) ⊂ Par(Γ) ⊂ WPar(Γ).

Moreover, if Γ is a closed convex set, then (by Theorem 1 in [3]) Par(Γ)is contained in the closure of PPar(Γ).


m2

m1u1

u2*

*

Γ

Figure 1.

m2

m1u1

u2*

*

Γ

Figure 1.

Example 2.3.Let Γ ⊂ IR2 be as in Figure 1. Then WPar(Γ) coincideswith the boundary of Γ, whereas Par(Γ) is the subset of the boundaryconsisting of the vector (u∗

1,m2) and the vectors whose first coordinateis in the half–closed interval (u∗

1,m1]. Finally, the proper Pareto pointsof Γ are the vectors in Par(Γ) with first coordinate in the open inter-val (u∗

1,m1). Also note that the vector (u∗1,m2) is the lexicographical

minimum of Γ in the sense of the following definition.

Definition 2.4. If u and v are vectors in IRq, u is said to be lexicogra-phically smaller than v (in symbols: u ≤L v) if the first nonzero termof the sequence v1 − u1, . . . , vq − uq is positive. Moreover, a vector uin Γ ⊂ IRq is called the lexicographical minimum of Γ if u ≤L u for allu ∈ Γ.

A direct application of Definitions 2.2 and 2.4 shows that the lexi-cographical minimum is a Pareto point.

Pareto policies. The above concepts can be extended to multi-objective MCPs in the same way as it is done for vector optimizationproblems [27, 31, 36]. First, as the initial distribution γ0 is fixed, we shallsimplify the notation by dropping γ0 from expressions such as (2.4) and(2.5). For instance, we shall write Vi(π, γ0) simply as Vi(π).


Definition 2.5.Let Γ(Π) be the set of cost vectors in (2.5), i.e.

(2.9) Γ(Π) := V (π)|π ∈ Π.

A policy π∗ ∈ Π is said to be

(a) a Pareto policy (respectively, a weak Pareto policy or a properPareto policy) if its corresponding cost vector V (π∗) is in Par(Γ(Π))(respectively, in WPar(Γ(Π)) or in PPar(Γ(Π)));

(b) lexicographically optimal if V (π∗) is the lexicographical minimumof Γ(Π).

In other words, π∗ ∈ Π is a Pareto policy (or Pareto optimal) ifthere is no policy π such that V (π) < V (π∗), and similarly for weak orproper Pareto policies.

The set Γ(Π) in (2.9) is called the performance set (also known as theobjective or achievable set) of the multiobjective MCP. An example inwhich Γ(Π) is similar to the set Γ in Figure 1 is given in [18], where it isshown that the so–called cµ–rule for priority queues is lexicographicallyoptimal — hence a nonproper Pareto policy. In fact, there are manyexamples of lexicographically optimal policies, including Blackwell opti-mal policies [26], bias optimal policies [21, 25], and average cost optimalpolicies that in addition minimize the cost variance [22].

Remark 2.6.(a) To find lexicographically optimal policies we may pro-ceed as follows. Let Π0 := Π, and for i = 1, . . . , q let

(2.10) Vi := infVi(π)|π ∈ Πi−1,

and, finally, let Πi be the set of policies in Πi−1 that attain the minimumin (2.10). Then, assuming that the sets Πi are nonempty, Πq consistsof the lexicographically optimal policies. Moreover, if π is in Πq, then

its cost vector V (π) = (V1, . . . , Vq) is of course the lexicographical min-imum of Γ(Π).

(b) The procedure in (2.10) is also valid for q = ∞, that is, for infi-nite cost vectors V (π), as in Blackwell optimality [26], for instance.

(c) If for some i = 1, . . . , q the set Πi in (a) consists of a single policyπi, then πi is the unique lexicographically optimal policy.

(d) As in (2.8), PPar(Γ(Π)) ⊂ Par(Γ(Π)) ⊂ WPar(Γ(Π)).


3 Pareto optimal policies

To study the existence and characterization of Pareto policies, in theremainder of the paper we impose the following assumption.

Assumption 3.1.The multiobjective Markov control model (2.1) sat-isfies that:

(a) The constraint set IK ⊂ X × A is closed.

(b) The functions ci are nonnegative and lower semicontinuous and,moreover, at least one of them, say c1, is inf-compact, which meansthat for each r ∈ IR, the level set

(3.1) Kr := (x, a) ∈ IK|c1(x, a) ≤ r

is compact.

(c) The transition law Q is weakly continuous; that is, denoting byCb(S) the space of continuous bounded functions on a topologicalspace S, the map

(3.2) (x, a) →X

h(y)Q(dy|x, a) is in Cb(IK) for each h ∈ Cb(X).

(d) There exists a policy π ∈ Π such that Vi(π) < ∞ for all i =1, . . . , q. (Recall that Vi(π, γ0) ≡ Vi(π).)

Observe that Assumption 3.1 is not restrictive at all. In fact, it holdsin most applications to queueing systems, productions models, etc. Inparticular, Assumption 3.1(c) holds if the state process xt evolvesaccording to a discrete-time equation of the form

xt+1 = G(xt, at, ξt), t = 0, 1, . . . ,

where the ξt are i.i.d. disturbances independent of the initial state x0,and G(x, a, s) is a given measurable function, continuous in (x, a) ∈ IKfor each s. This class of systems includes the LQ problem in Examples3.5 and 5.7, below.

The existence problem. To study the existence of Pareto policieswe shall first follow the well-known “scalarization” approach. Thus,


given a q–vector λ > 0 we consider the scalar (or real-valued) cost-per-stage function

(3.3) cλ(x, a) := λ· c(x, a) =q

i=1

λici(x, a),

and, as in (2.4), we consider a δ-discounted cost V λ(π) ≡ V λ(π, γ0) with

(3.4) V λ(π) := (1 − δ)Eπγ0

∞

t=0

δtcλ(xt, at) .

Using (3.3) and (2.5) we may write V λ(π) as

(3.5) V λ(π) = λ·V (π) =q

i=1

λiVi(π).

It is clear that minimizing V λ(· ) over Π is equivalent to minimize V λ(· )multiplied by a positive constant. Hence, occasionally we shall assumethat the vector λ in (3.3)-(3.5) belongs to the set

(3.6) Λ := λ ∈ IRq++|

q

i=1

λi = 1,

where IRq++ is the set of vectors λ 0. We may then state an existence

result as follows. (Observe that part (d) in Theorem 3.2 gives a littlemore than the existence of Pareto policies because, in fact, it ensuresthe existence of deterministic stationary Pareto policies.)

Theorem 3.2. Suppose that for some q–vector λ = (λ1, . . . , λq) > 0there is a policy π∗ ∈ Π that is optimal for the scalar criterion (3.4),i.e.

(3.7) V λ(π∗) ≤ V λ(π) ∀ π ∈ Π.

Then:

(a) π∗ is a weak Pareto policy.

(b) If in addition

(3.8) V λ(π∗) < V λ(π) ∀ π ∈ Π with V (π) = V (π∗),

then π∗ is a Pareto policy.


(c) If λ 0 (in particular if λ is in Λ), then π∗ is a proper Paretopolicy.

(d) If λ1 > 0, then there exists a deterministic stationary policy fλ ∈IF that is a weak Pareto policy; if, moreover, π∗ ≡ fλ satisfies(3.8), then fλ is a Pareto policy. Finally, if λ 0, then fλ is aproper Pareto policy.

Proof: (a) Suppose that π∗ is not a weak Pareto policy. Then thereexists a policy π ∈ Π such that V (π) V (π∗) and, therefore, as λ > 0,we get V λ(π) < V λ(π∗), which contradicts (3.7).

(b) Similarly, if π∗ is not a Pareto policy, there exists π ∈ Π suchthat V (π) < V (π∗). Hence V λ(π) ≤ V λ(π∗), which contradicts (3.8).

(c) If π∗ is not a proper Pareto policy, then the tangent cone to Γ(Π)at V (π∗), i.e. T (Γ(Π), V (π∗)), contains a vector v < 0. Therefore, thereexists a sequence πk in Π and a sequence tk of positive numberssuch that

limk→∞

(V (πk) − V (π∗))/tk = v.

As λ 0, we have λ · v < 0. It follows that for all k sufficiently large

λ · (V (πk) − V (π∗)) = V λ(πk) − V λ(π∗) < 0,

which contradicts (3.7).

(d) Suppose that λ1 > 0. Then, by Assumption 3.1(b), λ1 · c1(x, a)is nonnegative and inf–compact, and, therefore (by (3.3) and the firstpart of Assumption 3.1(b)), so is cλ. The latter fact together withAssumption 3.1(a),(c),(d) implies the existence of a deterministic sta-tionary policy π∗ ≡ fλ that satisfies (3.7); see e.g. [15] or Theorem 4.2.3in [20]. Hence, by part (a), fλ is a weak Pareto policy. The remainingstatements in (d) are proved similarly.

To obtain the converse of parts (a),(b),(c) in Theorem 3.2 we willuse a special reformulation (introduced in Section 4) of the originalmultiobjective MCP. This requires to restrict the “admissible” policiesto the following set.

Definition 3.3.Π0 denotes the set of policies π ∈ Π for which Vi(π) <∞ for all i = 1, . . . , q.


By our Assumption 3.1(d), the set Π0 is nonempty. The followingtheorem is proved in Section 7.

Theorem 3.4. Let π∗ be a policy in Π0. If π∗ is a weak Pareto policy,then there exists a q–vector λ > 0 for which (3.7) holds. If π∗ is aproper Pareto policy, then λ 0.

As a Pareto policy is weak Pareto (recall (2.8)), Theorem 3.4 tacitlyincludes the case in which π∗ is a nonproper Pareto policy. Thus Theo-rem 3.4 is a (slight) extension of the so–called “theorem of equivalence”in Pareto optimality [4]. Finally, observe that Theorems 3.4 and 3.2indeed characterize weak and proper Pareto policies because they yieldthat, for instance, π∗ ∈ Π0 is a proper Pareto policy if and only if π∗

minimizes the scalar criterion (3.4) for some q–vector λ 0.

The following example illustrates Theorem 3.2.

Example 3.5. Let α and β be nonzero real numbers and consider thescalar linear system

(3.9) xt+1 = αxt + βat + ξt for t = 0, 1, . . . ,

with state and control spaces X = A = IR. The disturbances ξt are i.i.d.random variables, independent of the initial state x0, and such that

(3.10) E(ξ0) = 0 and E(ξ20) =: σ2 < ∞.

For i = 1, . . . , q, let si and ri be strictly positive numbers, and let ci(x, a)be the quadratic cost

(3.11) ci(x, a) := six2 + ria

2.

Then, for each q–vector λ > 0, the scalar problem (3.3)-(3.5) corre-sponds to the linear system (3.9) with quadratic cost

(3.12) cλ(x, a) = (λ· s)x2 + (λ· r)a2

with s := (s1, . . . , sq) and r := (r1, . . . , rq). Moreover, for each i =1, . . . , q, let zi be the unique positive solution of the Riccati equation

(3.13) δβ2z2 + (ri − riα2δ − siβ

2δ)z − siri = 0.


Now replace si and ri with the coefficients λ· s and λ· r in (3.12), re-spectively, and let z(λ) be the corresponding unique positive solutionof (3.13). Then, as is well-known (see, for instance, p. 72 in [20]), theoptimal control policy fλ ∈ IF for the scalar problem is

(3.14) fλ(x) = − λ· r + δβ2z(λ)−1

αβδz(λ)x ∀x ∈ X,

and, moreover, for each initial state x0 = x, the optimal cost functionis

(3.15) V λ(fλ, x) = z(λ) (1 − δ)x2 + δσ2 ∀x ∈ X,

with σ2 as in (3.10). Therefore, assuming that the initial distributionγ0 satisfies that

(3.16) γ0 := x2γ0(dx) < ∞,

the optimal cost V λ(fλ) ≡ V λ(fλ, γ0) in the left-hand side of (3.7) isobtained by integrating both sides of (3.15) with respect to γ0. Thisyields

(3.17) V λ(fλ) = k(γ0)z(λ), with k(γ0) := (1 − δ)γ0 + δσ2.

By Theorem 3.2, fλ is a proper Pareto policy if λ 0, and a weakPareto policy if λ > 0. In particular, let e(i) be the unit vector withcoordinates ei(i) = 1 and ej(i) = 0 for j = i. Then replacing λ in (3.17)with e(i) we obtain the “partial” minimum cost in (1.1), i.e.

(3.18) V ∗i := inf

πVi(π) = Vi(fe(i)) = k(γ0)zi ∀ i = 1, . . . , q.

This gives the virtual minimum V ∗ = (V ∗1 , . . . , V ∗

q ), which is illustratedin Figure 2 for the case q = 2. In that figure, the Pareto set Par(Γ(Π)) isthe part of the boundary of Γ(Π) with first coordinate in [V ∗

1 , V1(fe(2))].On the other hand, by the uniqueness of optimal policies for LQ (linear–quadratic) systems, it follows from Remark 2.6(a),(c) that fe(1) is the

lexicographically optimal policy, whose corresponding cost vector V :=V (fe(1)) has coordinates Vi = Vi(fe(1)) for all i = 1, . . . , q, i.e.

(3.19) V = (V ∗1 , V2(fe(1)), . . . , Vq(fe(1))).


Γ(Π))

V2*

V *1 V1 (f V1 (π)

V2(π)

e

V2(fe(1 )

( 2 ) )

Figure 2. See (3.18), (3.19).

Remark 3.6.Consider a single, or scalar, LQ system with cost c(x, a) =sx2 + ra2; see (3.11). If the coefficients s and r are both positive,then an optimal policy for this problem can be interpreted as a properPareto policy for a two–dimensional multiobjective control problem withindividual costs c1(x, a) := x2 and c2(x, a) := a2. In fact, a similarinterpretation is valid for any scalar control problem with additive costs,say of the form

c(x, a) = r1c1(x, a) + · · · + rqcq(x, a)

with positive coefficients r1, . . . , rq. See [19] for details.

4 A multiobjective measure problem

In this section we reformulate the multiobjective MCP as an equiva-lent multiobjective measure problem (MMP) on a suitable vector spaceof measures. This reformulation greatly simplifies the proofs of someresults and, in addition, it can be used to write the multiobjective MCPas a multiobjective linear program (see Section 6).

Occupation measures. For each policy π ∈ Π, let µπ ≡ µπγ0

be thecorresponding δ-discount expected occupation measure, which is defined


as

(4.1) µπ(D) := (1 − δ)∞

t=0

δtP πγ0

(xt, at) ∈ D ∀D ∈ B(X × A).

This is a probability measure on X × A that, by (2.3), is concentratedon IK. Moreover, if π is in Π0 (see Definition 3.3), then a standardargument (see, for instance, Remark 9.4.2(b) in [21, p. 85]) yields thatVi(π) in (2.4) can be written as

(4.2) Vi(π) = µπ, ci :=IK

ci dµπ (i = 1, . . . , q).

To state other properties of occupation measures we shall use the fol-lowing notation: if µ is a finite signed measure on X ×A, we denote itsvariation by |µ| = µ+ + µ−, and its marginal (or projection) on X by µ,that is,

µ(B) := µ(B × A) ∀B ∈ B(X).

We also introduce the following sets of measures.

Definition 4.1.M(IK) denotes the vector space of finite signed mea-sures on X × A, concentrated on IK, and such that

(4.3) |µ|, ci = ci d|µ| < ∞ ∀i = 1, . . . , q.

Further, M+(IK) ⊂ M(IK) stands for the convex cone of nonnegativemeasures in M(IK), and Mδ(IK) ⊂ M+(IK) is the subfamily of nonneg-ative measures for which

(4.4) µ(B) = (1 − δ)γ0(B) + δIK

Q(B|x, a)µ(d(x, a)) ∀B ∈ B(X).

As µ(X) = µ(X × A), it is evident from (4.4) that

(4.5) Mδ(IK) is a convex set of probability measures.

It also turns out that Mδ(IK) coincides with the family of occupationmeasures in (4.1). More precisely (as in [15, pp. 386-387] or [20, Theo-rem 6.3.7], for instance), we have the following result in which Π0 is asin Definition 3.3.

Lemma 4.2. If π is a policy in Π0, then its occupation measure µπ is inMδ(IK). Conversely, if µ is in Mδ(IK), then µ is the occupation measureof a policy in Π0 (that is, there exists π ∈ Π0 such that µπ = µ).


For µ ∈ Mδ(IK) and c as in (2.2), let

(4.6) µ, c := ( µ, c1 , . . . , µ, cq ).

Now consider the following multiobjective measure problem (MMP):

(4.7) minimize µ, c |µ ∈ Mδ(IK).

By (4.2) and Lemma 4.2, MMP is equivalent to our original multiob-jective MCP if we restrict ourselves — which we do in the rest of thispaper — to the set

(4.8) Γ(Π0) := V (π)|π ∈ Π0

in lieu of the set Γ(Π) in (2.9). On the other hand, from (4.2), (4.5) andLemma 4.2 we may immediately conclude the following.

Lemma 4.3. Γ(Π0) can be expressed as

(4.9) Γ(Π0) = µ, c |µ ∈ Mδ(IK),

which is a convex subset of IRq+.

Actually, the convexity of Γ(Π0) is a well–known fact (see e.g. [10,33, 38]). However, we wish to emphasize here that this convexity is astraightforward, trivial, consequence of the MMP formulation: see (4.6)and (4.5). This illustrates the advantage of using the MMP instead ofthe original multiobjective MCP.

In the following section we use the MMP (4.7) to show the existenceof “strong” Pareto policies, and in Section 7 we use it to prove Theorem3.4.

5 Strong Pareto optimality

For each i = 1, . . . , q, let V ∗i ≡ V ∗

i (γ0) be the optimal δ-discounted costof the scalar MCP with cost-per-stage ci(x, a), that is,

V ∗i := inf

πVi(π) (with Vi(π) as in (2.4)).

The q-vector V ∗ := (V ∗1 , . . . , V ∗

q ) is called the virtual minimum for themultiobjective MCP. (V ∗ is also known as the utopian or the ideal or


the shadow minimum.) Let · be the Euclidean norm in IRq, and letρ : Π0 → IR+ be the map defined as

(5.1) ρ(π) := V (π) − V ∗ for π ∈ Π0.

This is a utility function (or a strongly monotonically increasing function[27]) for the multiobjective MCP in the sense that if π and π are suchthat V (π) < V (π ), then ρ(π) < ρ(π ). (In (5.1) we took the Euclideannorm to fix ideas, but in fact we may take any norm in IRq. See Remark5.6.)

Definition 5.1.A policy π∗ ∈ Π0 is said to be strong Pareto optimal(or a strong Pareto policy) if it minimizes the function ρ, that is,

(5.2) ρ(π∗) = infρ(π)|π ∈ Π0 =: ρ∗.

As ρ is a utility function, it is clear that a strong Pareto policy isPareto optimal, but of course the converse is not true.

Let Γ(Π0) be as in (4.8). For each λ ∈ IRq, let

(5.3) ∆(λ) := infλ· (V (π) − V ∗)|π ∈ Π0

be the so-called support function of Γ(Π0) − V ∗ at λ. Moreover, letS ⊂ IRq be the closed unit sphere centered at the origin, and let S1 beits boundary, i.e.,

S := λ | λ 1 and S1 := λ | λ = 1

Theorem 5.2. Suppose that ρ∗ > 0. Then:

(a) There exists a strong Pareto policy;

(b) There exists a vector λ∗ ∈ S1 ∩ IRq++ such that

(5.4) ρ∗ = ∆(λ∗) = maxλ∈S

∆(λ)

and, moreover, for any strong Pareto policy π∗, the vector λ∗ is“aligned” with V (π∗) − V ∗, i.e.

(5.5) λ∗· (V (π∗) − V ∗) = λ∗ V (π∗) − V ∗ = ρ∗.


For completeness and ease of reference, before proving Theorem 5.2we state some well-known technical facts. The following lemma can beobtained from the definition of inf–compactness and Prohorov’s Theo-rem [8].

Lemma 5.3. Let Y be a metric space and M a family of probabilitymeasures on Y . If there exists a nonnegative and inf-compact functionv on Y such that

sup µ, v |µ ∈ M < ∞,

then M is relatively compact, that is, for each sequence µn in M thereis a probability measure µ on Y and a subsequence µm of µn suchthat µm converges weakly to µ in the sense that

(5.6) µm, u µ, u u ∈ Cb(Y ).

Lemma 5.4. Let Y be a metric space, and v : Y → IR lower semicon-tinuous and bounded below. If µm and µ are probability measures on Yand µm converges weakly to µ (that is, as in (5.6)), then

(5.7) lim infm→∞

µm, v µ, v .

Lemma 5.4 is well known: see, for instance, statement (12.3.37) in[21, p. 225].

Lemma 5.5. The set Mδ(IK) (in Definition 4.1) is closed with respectto the topology of weak convergence.

Proof: Let µm be a sequence in Mδ(IK) such that µm converges weaklyto µ. Choose an arbitrary function h in Cb(X). By (3.2), h(y)Q(dy|· )is in Cb(IK), and, therefore, by the weak convergence of µm to µ, we get

h(y)Q(dy|x, a)µm(d(x, a)) → h(y)Q(dy|x, a)µ(d(x, a)).

Similarly, the marginals µm converge weakly to the marginal µ. Hence,as each µm satisfies (4.4), so does the limiting probability measure µ.Thus, to complete the proof that µ is in Mδ(IK), it only remains to showthat (4.3) holds for µ. This, however, follows from Assumption 3.1(b)and Lemma 5.4, which together yield

lim infm→∞

µm, ci µ, ci i = 1, . . . , q.


This implies that µ satisfies (4.3).

Proof of Theorem 5.2. (a) By (4.6) and Lemma 4.2, we may expressρ∗ in (5.2) as

ρ∗ = inf µ, c V ∗ |µ ∈ Mδ(IK).

Now let µn be a sequence in Mδ(IK) such that, as n → ∞,

(5.8) µn, c V ∗ ρ∗.

Choose an arbitrary ε > 0 and let n(ε) be such that

µn, c V ∗ ρ∗ + ε ∀ n ≥ n(ε).

This implies the existence of a constant k such that µn, ci k for alln ≥ n( ) and i = 1, . . . , q. In particular,

(5.9) µn, c1 k ∀n ≥ n( ).

Thus, as c1 is inf-compact (Assumption 3.1(b)), (5.9) and Lemma 5.3imply the existence of a subsequence µm of µn and a probabilitymeasure µ∗ on X×A, concentrated on IK (by Assumption 3.1(a)), suchthat µm converges weakly to µ∗. By Lemma 5.5, µ∗ is in Mδ(IK), and,by (5.7) and (5.8),

(5.10) µ∗, c V ∗ = ρ∗.

Finally, let π∗ ∈ Π0 be the policy associated to µ∗, and use (4.2) torewrite (5.10) as V (π∗) − V ∗ = ρ∗. This completes the proof of part(a).

(b) If π∗ ∈ Π0 is strong Pareto optimal, then the support functionin (5.3) becomes

∆(λ) = λ· (V (π∗) − V ∗),

and the vector λ∗ := (V (π∗) − V ∗)/ V (π∗) − V ∗ satisfies (5.4) and(5.5).

Remark 5.6.By the convexity of Γ(Π0) (Lemma 4.3), finding a strongPareto policy essentially reduces to the problem of finding the distancefrom the virtual minimum V ∗ to the convex set Γ(Π0). This yields, inparticular, that part (b) in Theorem 5.2 can be seen as a special case


of the “Minimum Norm Duality” in Luenberger [32, p. 136, Theorem1]. Hence, as the latter result is true for an arbitrary normed linearspace, in (5.1) we may take any norm instead of the Euclidean one. Forinstance, one could take a weighted p–norm, with 1 ≤ p ≤ ∞, which isvery common in vector optimization [27, 31, 36].

Example 5.7. (Example 3.5 continued). Consider again the LQ prob-lem (3.9)–(3.11). For each i = 1, . . . , q, let V ∗

i = k(γ0)zi be the partialminimum in (3.18), where zi is the unique positive solution of (3.13).Thus, letting z∗ := (z1, . . . , zq), the LQ problem’s virtual minimumV ∗ = (V ∗

1 , . . . , V ∗q ) becomes

(5.11) V ∗ = k(γ0)z∗.

Moreover, to find a strong Pareto policy we may proceed as follows.From (5.11) and (3.17), the support function in (5.3) is given by

∆(λ) = k(γ0)[z(λ) − λ· z∗] ∀λ ∈ IRq.

Now let λ∗ ∈ S1 ∩ IRq++ be as in Theorem 5.2(b). Then a strong Pareto

policy is obtained from (3.14) taking λ = λ∗, and the cost vector “clos-est” to V ∗ is given by (3.17) with λ = λ∗.

6 The multiobjective LP approach

In this section we follow Balbas and Heras [7] to formulate our mul-tiobjective MCP as a multiobjective linear program. This requires tointroduce two dual pairs (M(IK), F (IK)) and (M(X), F (X)) of vectorspaces, which are essentially the same as those defined in [20, §6.3] or[21, §12.3]. (The reader may consult the latter references or [2] for gen-eral facts on infinite-dimensional scalar linear programming (LP).)

Define w : IK → IR++ as

(6.1) w(x, a) := 1 + c1(x, a) + · · · + cq(x, a).

(More generally, our approach may use any nonnegative “weight” func-tion w(x, a) provided that it is bounded away from zero and that itmajorizes all of the functions ci(x, a). Thus, instead of w in (6.1) wecould use, for instance, w := +max(c1, . . . , cq) for any > 0.) Observethat (4.3) is equivalent to

(6.2) w d|µ| < ∞.


Therefore, the vector space M(IK) can be described as the space of finitesigned measures µ on X × A, concentrated on IK, and for which (6.2)holds.

Now let F (IK) be the vector space of real-valued measurable func-tions v on IK such that

(6.3) v w := sup(x,a)

|v(x, a)|/w(x, a) < ∞.

From (6.1) it follows that each of the cost functions ci belongs to F (IK),and, on the other hand, (M(IK), F (IK)) is a dual pair of vector spaceswith respect to the bilinear form

(6.4) µ, v := v dµ for µ ∈ M(IK), v ∈ F (IK).

We also consider another dual pair (M(X), F (X)) defined exactly asabove but replacing IK and w with X and

w0(x) := infa∈A(x)

w(x, a) ∀x ∈ X,

respectively.

Weak topologies. Henceforth we consider M(IK) to be endowedwith the weak toplogy σ(M(IK), F (IK)), which will be referred to as theσ-weak topology. Thus a sequence (or a net) µn σ-converges to µ if

(6.5) µn, v µ, v v ∈ F (IK).

This should not be confused with the “weak convergence” (5.6), which isrestricted to continuous and bounded functions. (Note that, of course,Cb(IK) ⊂ F (IK).) The vector spaces F (IK),M(IK), and F (X) are alsoendowed with the corresponding σ-weak topologies.

In the remainder of this section we suppose that Assumption 3.1 andthe following Assumption 6.1 are both satisfied.

Assumption 6.1. X w0(y)Q(dy|· ) is in F (IK); that is, for some con-stant k,

Xw0(y)Q(dy|x, a) ≤ kw(x, a) ∀(x, a) ∈ IK.


Assumptions 6.1 and 3.1(d) ensure, in particular, that the initialdistribution γ0 is in the space M(X).

Let L : M(IK) → M(X) be the linear map µ → Lµ defined as

(6.6) (Lµ)(B) := µ(B) − δIK

Q(B|x, a)µ(d(x, a)).

The adjoint L∗ : F (X) → F (IK) of L, that is, the linear map L∗ forwhich

(6.7) Lµ, u = µ,L∗u µ ∈ M(IK), u ∈ F (X),

is given by

(6.8) (L∗u)(x, a) = u(x) − δX

u(y)Q(dy|x, a) ∀(x, a) ∈ IK.

By Assumption 6.1, L∗ indeed maps F (X) into F (IK), which is equiva-lent to say that L is σ–weakly continuous.

Multiobjective LP. For each µ in M(IK), let µ, c be as in (4.6)and consider the primal program (PP):

minimize µ, c

subject to: Lµ = (1 − δ)γ0, µ ∈ M+(IK).(6.9)

Comparing (PP) with the MMP (4.7) we can see that they are essentiallythe same but the former has a little more “structure”: the constraint(4.4) has been rewritten in (6.9) using the σ-weakly continuous map L.

A feasible solution µ∗ for (PP) is said to be optimal if there is nofeasible µ such that µ, c < µ∗, c . If such an optimal solution ex-ists, then (PP) is said to be solvable. Thus, from Theorem 3.2(d) andthe equivalence of (4.7) and the multiobjective MCP, we conclude thefollowing.

Corollary 6.2. (PP) is solvable.

To state the dual program we need some notation. Let F (X)q be thevector space of IRq-valued functions u = (u1, . . . , uq) with ui ∈ F (X)


for all i = 1, . . . , q. For u ∈ F (X)q and λ ∈ IRq, let uλ ∈ F (X) andL∗u ∈ F (IK)q be the functions given by

(6.10) uλ := λ·u =q

i=1

λiui, and L∗u := (L∗u1, . . . , L∗uq),

respectively. Moreover, if ν is in M(X), we write

ν, u := ( ν, u1 , . . . , ν, uq ).

Then, from [7, p. 380], we can see that the dual program (DP) of (PP)is as follows:

(DP) maximize (1 − δ)γ0, u

subject to: λ·L∗u ≤ λ· c with u ∈ F (X)q, for some λ ∈ IRq++.(6.11)

In fact, if we let

Fλ := u ∈ F (X)q|λ· Lµ, u λ· µ, c µ ∈ M+(X)

and use (6.7), it then follows that the dual constraint (6.11) can beexpressed as in [7], namely:

u is in Fλ for some λ ∈ IRq++.

On the other hand, using (6.10) and (6.8) we can write (6.11) in themore explicit form

(6.12) uλ(x) ≤ cλ(x, a) + δX

uλ(y)Q(dy|x, a) ∀(x, a) ∈ IK,

for some λ ∈ IRq++. The latter inequality yields

(6.13) uλ(x) ≤ mina∈A(x)

cλ(x, a) + δX

uλ(y)Q(dy|x, a) ∀x ∈ X,

which, when the equality holds, that is,

(6.14) uλ(x) = mina∈A(x)

cλ(x, a) + δX

uλ(y)Q(dy|x, a) ∀x ∈ X.

becomes the dynamic programming equation (d.p.e.) for the scalar MCPwith cost function (1 − δ)−1V λ(π, x), where V λ(π, x) is the function in(3.5) when the initial state is x0 = x.


Remark 6.3.Let V λ∗ (x) := infπ V λ(π, x) for all x ∈ X. Then (1 −

δ)−1V λ∗ (x) is the (pointwise) minimal solution of the d.p.e. (6.14).

Moreover, if V λ∗ is in F (X) and uλ satisfies (6.12)-(6.13), then well-

known arguments (see [20, Lemma 4.2.7], for instance) give that

(6.15) uλ(x) ≤ (1 − δ)−1V λ∗ (x) ∀x ∈ X,

and for this reason uλ is said to be a subsolution of the d.p.e. (6.14).Note that (6.15) yields

(6.16) (1 − δ)γ0, uλ γ0, V

λ∗ .

Therefore (by the equivalence of (6.11) and (6.12)), we can see the dualprogram (DP) as the problem of maximizing integrals as in the left-hand side of (6.16) over the family of subsolutions uλ of the d.p.e. fora class of scalar MCPs parameterized by λ ∈ IRq

++. Thus, the mul-tiobjective LP formulation gives us a “primal-dual” interpretation ofthe relation between our original multiobjective MCP and the scalarMCPs in (3.3)-(3.5). This interpretation can also be obtained from the“complementary slackness” property in the following proposition from[7] adapted to our current situation.

Proposition 6.4. Let µ be a feasible solution for (PP) and u a feasiblesolution for (DP). Then

(a) (Weak duality.) We never have (1 − δ)γ0, u > µ, c .

(b) (Complementary slackness.) If in addition

(6.17) µ, c − L∗u = 0,

then µ is optimal for (PP) and u is optimal for (DP).

Proof: Part (a) is straightforward, and in turn (a) implies (b) because,by (6.7) and (6.9), we can write (6.17) as

(1 − δ)γ0, u = µ, c .

Now, to obtain the primal-dual interpretation mentioned in the lastpart of Remark 6.3, it suffices to note that (6.17) is equivalent to

(6.18) µ, cλ − L∗uλ = 0 ∀λ ∈ IRq++.


In fact, by (6.8), we can recognize the integrand cλ − L∗uλ in (6.18) asthe difference between the two sides of (6.12). Therefore, we can obtaina solution (µ, uλ) for (6.18) in the obvious manner: choose an arbitraryλ ∈ IRq

++ and let V λ∗ be as in Remark 6.3. Let

uλ∗(x) := (1 − δ)−1V λ

∗ (x) ∀x ∈ X.

Furthermore (as in the proof of Theorem 3.2(d)), let f∗ ∈ IF be a sta-tionary policy such that f∗(x) ∈ A(x) attains the minimum in the d.p.e.(6.14) for all x ∈ X, and, finally, let µ∗ be the occupation measure as-sociated with f∗. Then, by their very definitions, it follows that µ∗ isfeasible for (PP), uλ

∗ is feasible for (DP), and

µ∗, cλ − L∗uλ

∗ = 0.

7 Proof of Theorem 3.4

Let us first suppose that π∗ ∈ Π0 is a proper Pareto policy. Let µ∗ ∈Mδ(IK) be the occupation measure corresponding to π∗ (see (4.1)). By(4.2), (4.6) and Lemma 4.2, to prove Theorem 3.4 it suffices to show theexistence of a q–vector λ 0 such that

λ · µ∗, c λ · µ, c µ ∈ Mδ(IK)

(cf. (3.7)) or, equivalently,

(7.1) µ − µ∗, cλ 0 ∀ µ ∈ Mδ(IK),

with cλ as in (3.3). With this in mind, consider the set Γ(Π0) in (4.9),and let T0 := T (Γ(Π0), µ∗, c ) be the tangent cone to Γ(Π0) at µ∗, c .As Γ(Π0) is convex (Lemma 4.3), we have

(7.2) Γ(Π0) µ∗, c T0.

(Recall (2.7).) Let B be the set of q–vectors u < 0 such that

(7.3)q

i=1

ui = −1,

and note that T0 − B is a convex set that does not contain the vectorzero. Therefore, by a well–known separation theorem (e.g. [5, p. 30],[32, p. 133], there exists a vector λ = 0 such that

λ · (v − u) > 0 ∀ v ∈ T0, u ∈ B.


In particular, by (7.2),

(7.4) λ · ( µ, c µ∗, c ) > λ · u ∀ µ ∈ Mδ(IK), u ∈ B,

and taking µ = µ∗ we obtain that λ · u < 0 for all u ∈ B. Therefore,choosing an arbitrary i ∈ 1, . . . , q and letting u ∈ B be the vector withcomponents ui = −1 and uj = 0 for j = i, we conclude that λi > 0;hence, as i ∈ 1, . . . , q was arbitrary, λ 0. Thus to complete theproof it only remains to verify that µ∗ and λ satisfy (7.1), so that µ∗

indeed minimizes λ · µ, c = µ, cλ . Suppose that this is not the caseand let µ ∈ Mδ(IK) be such that µ, cλ < µ∗, cλ , i.e.

(7.5) µ − µ∗, cλ < 0.

For each r ≥ 0, let vr be the vector in T0 defined as

vr := r( µ, c µ∗, c ) = r µ − µ∗, c .

Then, by (7.5), λ · vr = r µ − µ∗, cλ as r → ∞, which con-tradicts (7.4). This completes the proof of Theorem 3.4 when π∗ is aproper Pareto policy.

Let us now suppose that π∗ is a weak Pareto policy and let µ∗ bethe corresponding occupation measure. Then µ∗, c is a weak Paretopoint of Γ(Π0), i.e. there is no µ ∈ Mδ(IK) such that µ, c µ∗, c .Let

C1 := u ∈ IRq|u µ∗, c ,

C2 := u ∈ IRq|u µ, c for some µ ∈ Mδ(IK).

Then C1 and C2 are disjoint convex sets, and in addition C1 is open.Therefore, by the separation theorem in [32, p. 133, Theorem 3], thereis a q–vector λ = 0 and a real number α such that

(7.6) λ · u < α ≤ λ · v ∀ u ∈ C1, v ∈ C2.

Moreover, the vector µ∗, c is in the intersection of C2 and the closureof C1, which yields that α = λ · µ∗, c . Hence the first inequality in(7.6) gives

λ · ( µ∗, c w) ≤ λ · µ∗, c w ∈ IRq+,

which implies that λ · w ≥ 0 for all w ≥ 0, and so λ ≥ 0. Thus λ > 0because λ = 0. Finally, by the definition of C2 and the second inequalityin (7.6), we obtain that λ · µ∗, c λ · µ, c for all µ ∈ Mδ(IK), whichconcludes the proof of Theorem 3.4.


8 Further remarks

In this final section we briefly discuss some connections between ourresults and other problems for MCPs.

Constrained MCPs. For each i = 1, . . . , q, let Vi(π) = Vi(π, γ0)be as in (2.4), and let k2, . . . , kq be q − 1 nonnegative given numbers.Then the problem

minimize V1(π)

subject to: Vi(π) ≤ ki for i = 2, . . . , q; π ∈ Π,(8.1)

is called a constrained MCP. In this case, a policy π for which (8.1)holds and, in addition, V1(π) < ∞ is said to be feasible for the con-strained MCP. Let us suppose that the set Πco ⊂ Π of feasible policiesis nonempty. Then, under Assumption 3.1, there is an optimal pol-icy π∗ ∈ Πco for the constrained MCP (see e.g. [16]), and under anadditional Slater–like condition, π∗ is also a Pareto policy for the mul-tiobjective MCP in Section 2 above; see [30], for instance.

For additional results on constrained MCPs or for MCPs with weightedcriteria, see, for instance, [1, 10, 11, 14, 16, 17, 28, 30, 33, 38].

Average cost. Let us rewrite (2.4) as

(8.2) Vi(π, γ0) = lim supn→∞

Eπγ0

n−1

t=0

δtci(xt, at) /n−1

t=0

δt.

This is, of course, the same as (2.4) if 0 < δ < 1, whereas if δ = 1 weget the average cost (AC) criterion

(8.3) Ji(π, γ0) = lim supn→∞

1

nEπ

γ0

n−1

t=0

ci(xt, at) .

It is easily verified that all of the results in Sections 3, 4 and 5 remainvalid when δ = 1, with some obvious changes. For example, the setM1(IK) in Definition 4.1 (and (4.5)) is the set of probability measures µon X × A, concentrated on IK, and such that (as in (4.4))

(8.4) µ(B) =IK

Q(B|x, a)µ(d(x, a)).


Similarly, by (8.4), the constraint equation (6.9) in the multiobjectiveLP formulation becomes

(8.5) L1µ = 0, µ ∈ M+(IK),

where L1 is given by (6.6) with δ = 1. Finally, as in the discountedcase (8.1), we can also consider constrained MCPs with the AC crite-rion and obtain an optimal policy for the constrained problem, which isa Pareto policy for the multiobjective MCP. For details see [17], wherea probability measure µ for which (8.5) holds is called stable.

Mixed average-discounted criteria. The average cost case in(8.3)-(8.5) can be used to study multiobjective MCPs with cost vectorsof the form

(J1(π, γ0), . . . , Jr(π, γ0), Vr+1(π, γ0), . . . , Vq(π, γ0))

in which the Ji(π, γ0) are ACs as in (8.3), and the Vj(π, γ0) are dis-counted costs as in (8.2) with possibly different discount factors δj (j =r + 1, . . . , q). The key fact that allows us to do this is that the originalmultiobjective MCP is reduced to solving a Pareto problem of the form(4.7) but on the set M1(IK) of stable probability measures. The corre-sponding technical details are essentially the same as in Remarks 2.2(c)and 3.7(b) of [17].

Further research: the balance space approach. In this pa-per we used two main approaches to analyze a multiobjective MCP: thescalarization approach (to study the problem of existence of Pareto poli-cies) and the MMP approach (to study the characterization of Paretopolicies). In fact, the former approach is the “dual” of the latter in aprecise sense (see Section 6). On the other hand, there is a nonscalar-ized approach called the balance space approach introduced by Galperin[13] for vector optimization problems. This approach, in addition toallowing an interesting economic interpretation of the so–called “bal-ance points”, has proved to be very effective from the computationalviewpoint and also to study key issues, such as the sensitivity of vec-tor minimization problems [6]. It might be worth investigating if thiseffectiveness also holds for multiobjective control problems.


Onesimo Hernandez-LermaDepartamento de Matematicas,CINVESTAV-IPN,A.P. 14-470,Mexico D.F. 07000,[email protected]

Rosario RomeraDepartamento de Estadıstica,Universidad Carlos III de MadridCalle Madrid 126,Getafe 28903,Madrid Espana,[email protected]

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Tutte uniqueness of locally grid graphs ∗

D. Garijo A. Marquez M.P. Revuelta

Abstract

A graph is said to be locally grid if the structure around each ofits vertices is a 3× 3 grid. As a follow up of the research initiatedin [4] and [3] we prove that most locally grid graphs are uniquelydetermined by their Tutte polynomial.

2000 Mathematics Subject Classification: 05C75, 05C10.Keywords and phrases: Locally grid graph; Tutte polynomial.

1 Introduction

Given a graph G, the Tutte polynomial of G is a two-variable polynomialT (G;x, y), which contains considerable information on G [1]. A graphG is said to be Tutte unique if T (G;x, y) = T (H;x, y) implies G ∼= Hfor every other graph H. In Section 2 we prove that, locally grid graphsare Tutte unique.

Given a fixed graph H, a connected graph G is said to be locallyH if for every vertex x the subgraph induced on the set of neighborsof x is isomorphic to H. For example, if P is the Petersen graph, thenthere are three locally P graphs [2]. The locally grid condition is slightlydifferent since it involves not only a vertex and its neighbors, but alsofour vertices at distance two. From now on, all graphs considered haveno isolated vertices.

We first recall some definitions and results about locally grid graphsfrom [4].

∗Invited article. Partially supported by projects BFM2001-2474-ORI and PAIFQM-164. This work is part of the first author’s Ph.D. thesis writen at the Universityof Sevilla.

35

36 D. Garijo, A. Marquez and M.P. Revuelta

Let N(x) be the set of neighbors of a vertex x. We say that a4−regular connected graph G is a locally grid graph if for each vertex xthere exists an ordering x1, x2, x3, x4 of N(x) and four different verticesy1, y2, y3, y4, such that, taking the indices modulo 4,

N(xi) ∩ N(xi+1) = x, yi

N(xi) ∩ N(xi+2) = x

and there are no more adjacencies among x, x1, . . . , x4, y1, . . . , y4 thanthose required by these conditions (Figure 1).

Figure 1: Locally Grid Structure

Locally grid graphs are simple, two-connected, triangle-free, andeach vertex belongs to exactly four cycles of length 4.

Let H = Pp×Pq be the p×q grid, where Pl is a path with l vertices.Label the vertices of H with the elements of the abelian group Zp × Zq

in the natural way. Vertices of degree four already have the locally gridproperty, hence we have to add edges between vertices of degree two andthree in order to obtain a locally grid graph. A complete classificationof locally grid graphs is given in [4], and they fall into the followingfamilies. In all the Figures, the vertices of the graph are represented bydots and two points with the same label correspond to points that areidentified in the surface.

The Torus T δp,q with p ≥ 5, 0 ≤ δ ≤ p/2, δ + q ≥ 5 if q ≥ 4,

δ + q ≥ 6 if q = 2, 3 or 4 ≤ δ < p/2 with δ = p/3, p/4 if q = 1.(Figure 2a)

E(T δp,q) = E(H) ∪ (i, 0), (i + δ, q − 1), 0 ≤ i ≤ p − 1

∪ (0, j), (p − 1, j), 0 ≤ j ≤ q − 1.

For δ = 0 we obtain the toroidal grid Cp ×Cq, in this case we will writeTp,q. We can assume that δ ≤ p/2.

The Klein Bottle K1p,q with p ≥ 5, p odd, q ≥ 5. (Figure 2b)

E(K1p,q) = E(H) ∪ (j, 0), (p − j − 1, q − 1), 0 ≤ j ≤ p − 1

∪ (0, j), (p − 1, j), 0 ≤ j ≤ q − 1.

Tutte uniqueness of locally grid graphs 37

1

2

3

4

5

6 7 8 9 10 11 12

6 7 8 9 10 11 12

5

4

3

2

1

a)

1

2

3

4

5

6 7 8 9 10 11 12

6 7 8 9 10 12 11

5

4

3

2

1

b)

Figure 2: a) T 27,5 b) K1

7,5

The Klein Bottle K0p,q with p ≥ 6, p even, q ≥ 4 (Figure 3a).

E(K0p,q) = E(H) ∪ (j, 0), (p − j − 1, q − 1), 0 ≤ j ≤ p − 1

∪ (0, j), (p − 1, j), 0 ≤ j ≤ q − 1.

The Klein Bottle K2p,q with p ≥ 6, p even, q ≥ 5 (Figure 3b).

E(K2p,q) = E(H) ∪ (j, 0), (p − j, q − 1), 0 ≤ j ≤ p − 1

∪ (0, j), (p − 1, j), 0 ≤ j ≤ q − 1.

1

2

3

4

5

6 7 8 9 10 11

9 8 7 6 11 10

5

4

3

2

1

a)

1

2

3

4

5

6 7 8 9 10 11

10 9 8 7 6 11

5

4

3

2

1

b)

Figure 3: a) K06,5 b) K2

6,5

The graphs Sp,q with p ≥ 3 and q ≥ 6. (Figure 4). If p ≤ q


E(Sp,q) = E(H) ∪ (j, 0), (p − j, q − p + j), 0 ≤ j ≤ p − 1

∪ (0, i), (i, q − 1), 0 ≤ i ≤ p − 1

∪ (0, i), (p − 1, i − p), p ≤ i ≤ q − 1.

If q ≤ p

E(Sp,q) = E(H) ∪ (j, 0), (0, q − 1 − j), 0 ≤ j ≤ q − 1

∪ (p − 1 − i, q − 1), (p − 1, i), 0 ≤ i ≤ q − 1

∪ (i, q − 1), (i + q, 0), 0 ≤ i ≤ p − q − 1.

13

12

11

10

9

13

12

11 10 9

6 7 8

1 2 3 4 5 10

10

9 8 7 6

9 8 7 6

6

7

8

9

10

11

12

13

6 7 8 9 10 5 4 3 2 1

11

12

13

1

2

3

4

5

1

2

3

4

5

6 7 8

b)

a)

Figure 4: a) S5,8 b) S8,5

Theorem 1.1 [4] If G is a locally grid graph with N vertices, then ex-actly one of the following holds:

a) G ∼= T δp,q with pq = N , p ≥ 5, δ ≤ p/2 and δ + q ≥ 5 if q ≥ 4 or

δ + q ≥ 6 if q = 2, 3 or 4 ≤ δ < p/2, δ = p/3, p/4 if q = 1.


b) G ∼= Kip,q with pq = N , p ≥ 5, i ≡ p (mod 2) for i ∈ 0, 1, 2 and

q ≥ 4 + i/2 .

c) G ∼= Sp,q with pq = N , p ≥ 3 and q ≥ 6.

2 Tutte Uniqueness

Let G = (V,E) be a graph with vertex set V and edge set E. The rankof a subset A ⊆ E is defined by r(A) = |A| − k(A), where k(A) is thenumber of connected components of the spanning subgraph (V,A). Therank-size generating polynomial is defined as:

R(G;x, y) =A⊆E

xr(A)y|A|

The coefficient of xiyj in R(G;x, y) is the number of spanning subgraphsin G with rank i and j edges. This polynomial contains exactly the sameinformation about G as the Tutte polynomial, which is given by:

T (G;x, y) =A⊆E

(x − 1)r(E)−r(A)(y − 1)|A|−r(A)

hence, the Tutte polynomial tells us for every i and j the number of edge-sets in G with rank i and size j. This fact is going to be essential in orderto prove the Tutte uniqueness of locally grid graphs. Given a locallygrid graph G, we show that for every locally grid graph H differentfrom G and with |V (G)| = |V (H)| there is at least one coefficient of therank-size generating polynomial in which both graphs differ.

Let S be the surface in which a locally grid graph G is embedded,that is, S is a torus or a Klein bottle [4]. Given two cycles C and Cof G, we say that C is locally homotopic to C if there exists a cycle oflength four, say H, with C ∩H connected and C is obtained from C byreplacing C − (C ∩H) with H − (C ∩H). A homotopy is a sequence oflocal homotopies. A cycle of G is called essential if it is not homotopicto a cycle of length four.

Let lG be the minimum length of an essential cycle of G. Notethat lG is invariant under isomorphism. The number of essential cyclesof length lG contributes to the coefficient alG−1,lG of R(G;x, y), whichcounts the number of edges sets with rank lG − 1 and size lG.

In order to show the Tutte uniqueness of locally grid graphs we aregoing to use the following results proved in [4]:


Lemma 2.1 [4] Given two graphs G and G , if G is locally grid andT (G;x, y) = T (G ;x, y) then G is locally grid.

Lemma 2.2 [4] Let G, G be a pair of locally grid graphs with pq ver-tices then: a) lG = lG implies T (G;x, y) = T (G ;x, y).b) If lG = lG but G and G do not have the same number of shortestessential cycles, then T (G;x, y) = T (G ;x, y).

The process we are going to follow is to pairwise compare all thegraphs given in the classification theorem of locally grid graphs. In thosecases for which the minimum length of essential cycles or the numberof cycles of this minimum length are different we have that both graphsare not Tutte equivalent, thus the relevance of the following result.

Lemma 2.3 If G is a locally grid graph with pq vertices, then the lengthlG of the shortest essential cycles and the number of these cycles aregiven in the following table:

G lG number of essential cycles

Tp,q minp, qq

2p

p

if p < q

if p = q

if p > q

T δp,q minp, q + δ

q

q + pq + δ − 1

δ

pq + δ − 1

δ

if p < q + δ

if p = q + δ

if p > q + δ

K0p,q minp, q + 1

q

5q

4q

if p < q + 1

if p = q + 1

if p > q + 1

K1p,q minp, q

q

q + 1

1

if p < q

if p = q

if p > q

K2p,q minp, q

q

q + 2

2

if p < q

if p = q

if p > q

Sp,q min2p, q

2p p−1j=0

q − 1j

q(q − p)2p − 1

p

2q

if p ≤ q ≤ 2p

if 2p ≤ q

if q ≤ p


Proof: The cases Tp,q, T δp,q and Ki

p,q are proved in [4], where it is alsoshown that lSp,q = min(2p, q) and that if q ≤ p the number of shortestessential cycles is 2q. Hence, we are only left with two cases in whichwe are given lower bounds on the number of essential cycles of lengthlSp,q . We are interested in calculating the exact number.

Locally grid graphs with pq vertices are constructed by adding edgesto the p× q grid. These edges are called exterior edges. Essential cyclesof shortest length are obtained by joining the two ends of an exterioredge by a path contained in the grid p × q. In Sp,q we distinguish twocases.

Case 1 If 2p ≤ q, every exterior edge of the form (0, i), (p−1, i−p)

determines2p − 1

pessential cycles of length 2p. We have q−p edges

of this kind and each of them can use up to q different vertices, therefore

the number of essential cycles of length 2p is q(q − p)2p − 1

p.

Case 2 If p ≤ q ≤ 2p, (0, i), (i, q − 1) and (i, 0), (p− 1, q − p+ i)

with 0 ≤ i ≤ p − 1 generateqi

essential cycles of length q. These

edges can use up to p different vertices, hence the number of essential

cycles of length q is 2p p−1j=0

q − 1j

.

Theorem 2.4 Let p, q ≥ 6 verify the following conditions:

a) pq + δ − 1

δ= 2n for n ∈ N.

b) pq = p q for all p , q ≥ 6 with p = q + δ = q + δ < p and

q + pp − 1

δ= p

p − 1δ

.

Then T δp,q is Tutte unique for all δ ≤ p/2.

Proof: Let p, q ≥ 6 and G be a graph with T (G;x, y) = T (T δp,q, x, y). By

Lemma 2.1, G is a locally grid graph, hence G has to be isomorphic toexactly one of the following graphs: Tp ,q , T δ

p ,q , Kip ,q , Sp ,q . We prove

that G is isomorphic to T δp ,q with p = p , q = q and δ = δ assuming

that G is isomorphic to each one of the previous graphs and obtaininga contradiction in all the cases except in the aforementioned case. In[4] Tp,q was shown to be Tutte unique, thus we can consider δ > 0 andG not isomorphic to Tp ,q .


Case 1 Suppose G ∼= K0p ,q . By Lemma 2.2, lT δ

p,q= lK0

p ,qand the

number of shortest essential cycles has to be the same in both graphs.Case 1.1 lT δ

p,q= p, lK0

p ,q= p with p < q + δ and p < q + 1.

As a result of Lemma 2.2, p = p and q = q . Our aim is to provethat the number of edge sets with rank q and size q + 1 is different foreach graph. This would lead to a contradiction since this number is thecoefficient of xqyq+1 in the rank-size generating polynomial.

If T δp,q has k essential cycles of length q (δ > 1) or k + pq (δ = 1),

then K0p,q would have k + 4q such cycles. Therefore if we can show that

there exits a bijection between edge sets with rank q and size q + 1 thatare not essential cycles, we would have proved what we want.

For every r with 0 ≤ r ≤ q − 2 denote by Er the set ((i, r), (i, r +1)) ; 0 ≤ i ≤ p − 1. Let A be an edge set that is not an essential cyclewith rank q and size q + 1 in T δ

p,q. Define s(A) as minr ∈ [0, q − 2] ;

A∩Er = ∅. If A ⊂ E(T δp,q) the minimum always exits. For every r with

0 ≤ r ≤ q − 2 we define the bijection, ϕr between A ⊆ E(T δp,q)|r(A) =

q, |A| = q+1, s(A) = r and A ⊆ E(K0p,q)|r(A) = q, |A| = q+1, s(A) =

r as follows:If A ⊂ E(T δ

p,q), ϕr(A) = ∪ψ(((i, j), (i , j ))); ((i, j), (i , j )) ∈ Awhere

ψ((h, k)) =

⎧⎪⎪⎨

⎪⎪⎩

(h, k) if j = q − 1, j = 0

(h, k) if j, j ∈ [0, r]

(h, k) if r + 1 ≤ j, j ≤ q − 1

with h = (i, j), k = (i , j ), h = (p−1−i+δ, j) and k = (p−1−i +δ, j ).Case 1.2 lT δ

p,q= p, lK0

p ,q= p = q + 1 with p < q + δ or lT δ

p,q=

q + δ < p, lK0p ,q

= p with p < q + 1.

The contradiction in these two cases is produced due to the equal-ity of shortest essential cycles, number of these cycles and number ofvertices on each graph.

Case 1.3 lT δp,q

= p, lK0p ,q

= q + 1 with p < q + δ and q + 1 < p .

To obtain a contradiction, we are going to prove that there are moreedge-sets with rank q +2 and size q + 3 in T δ

p,q than in K0p,q. Basically,

we are going to follow the same procedure that was developed in [4].The previous sets can be classified into three groups:

1.- Normal edge-sets (they are edge-sets that do not contain anyessential cycle).


2.- Sets containing an essential cycle of length q + 1 and two otheredges (Figure 5a).

3.- Essential cycles of length q + 3 (Figures 5b and 6).

)b)a

A

A

A A

BB

Figure 5: a) A set of edges in K0p ,q containing an essential cycle of

length q + 1 and two other edges. b) Essential cycles of length q + 3 inT δ

p,q.

A

A

A

A

Figure 6: Essential cycles of length q + 3 in K0p ,q

(1) By Corollary 16 of [4] we know that T δp,q and K0

p ,q have thesame number of normal edge-sets with rank q + 2 and size q + 3 thatdo not contain a cycle of length four. We are going to prove that thenumber of normal edge-sets with rank q + 2 and size q + 3 containinga cycle of length four is greater in T δ

p,q than in K0p ,q .

Again by Corollary 16 of [4], the number of edge-sets with rankq + 1 and size q + 2 containing a cycle of length four is the same inboth graphs, call it sq +1. Add one edge to each of these sets in orderto obtain a set with rank q + 2 and size q + 3. This set can be one ofthe following types depending on which edge we are adding:

(a) A normal edge set with rank q + 2.(b) A normal edge set containing two non essential cycles and having

rank q + 1.(c) An edge set containing an essential cycle of length q + 1 and a

non essential cycle of length four.Let A(G), B(G) and C(G) (where G is either T δ

p,q or K0p ,q ), the

number of edge-sets in G that belong to the groups A, B and C respec-


tively. We recall the following equality from [4]:

sq +1(2pq− q −2) = A(G)(q −1)+B∈B(G)

(q +3− δ(B))+C(G)(q −1)

where δ(B) is the number of edges of B which do not belong to anycycle of length four in B. Since C(T δ

p,q) = 0 and C(K0p ,q ) = 0 we have:

A(T δp,q)(q − 1) + B∈B(T δ

p,q)(q + 3 − δ(B)) =

A(K0p ,q )(q − 1) + B∈B(K0

p ,q)(q + 3 − δ(B)) + C(K0

p ,q )(q − 1).

Applying Corollary 16 several times we get that:

B∈B(T δp,q)

(q + 3 − δ(B)) =B∈B(K0

p ,q)

(q + 3 − δ(B))

hence

A(T δp,q)(q − 1) = A(K0

p ,q )(q − 1) + C(K0p ,q )(q − 1).

(2) In T δp,q, every essential cycle of length q + 1 plus two edges has

rank q + 2, but in K0p ,q there are essential cycles for which if we add

two edges we obtain sets with rank q + 1. By hypothesis, both graphshave the same number of shortest essential cycles therefore the numberof edge-sets in this case is greater in T δ

p,q than in K0p ,q .

(3) For every essential cycle of length p = q + 1 in T δp,q we have

2p2

ways of adding two edges in order to obtain a new essential

cycle, hence in T δp,q there are 2q

p2

essential cycles of length q + 3.

In [4] it is proved that in K0p ,q there are 4q

q2

+ 4q + 2

3essential cycles of length q + 3. Since p = q + 1 and q = 4q , thenumber of essential cycles of length q + 3 is greater in T δ

p,q than inK0

p ,q .

Case 1.4 lT δp,q

= p = q + δ, lK0p ,q

= p with q + 1 > p .

Suppose p = q + δ = p then q = q , hence δ < 1. We get acontradiction because δ ≥ 1.

Case 1.5 lT δp,q

= p = q + δ, lK0p ,q

= p = q + 1.


If the length of the shortest essential cycles and the number of thesecycles coincide in both graphs, we would have p = p , q = q , δ = 1 andq + pq = 5q therefore p = 4.

Case 1.6 lT δp,q

= p = q + δ, lK0p ,q

= q + 1 with p > q + 1.

q + 1 = p = q + δ ⇒ 4q = q + pqδ

,

p > 4, q =q

q − 1<

qδ

⇒ 4q < pqδ

.

Case 1.7 lT δp,q

= q + δ < p, lK0p ,q

= p = q + 1.

q + 1 = p = q + δ and 5q = pq + δ − 1

δ.

If δ = 1 then q = q , p = p = q + δ < p hence δ > 1.

p > 5 and

q + δ − 1

δ=

q

δ> q ⇒ 5q < p

q + δ − 1

δ.

Case 1.8 lT δp,q

= q + δ < p, lK0p ,q

= q + 1 < p .

Now, q + 1 = q + δ, so we can assume that δ > 1 because if δ = 1,then q = q , p = p and the number of shortest essential cycles wouldnot be the same in both graphs. The contradiction in this case is similar

to the one obtained in the previous case because 4q = pqδ

.

After these eight cases we can conclude that T δp,q is not isomorphic

to K0p,q.

Case 2 Suppose G isomorphic to T δp ,q .

Case 2.1 lT δp,q

= p, lT δp ,q

= p with p < q + δ and p < q + δ .

As a result of Lemma 2.2, p = p and q = q . Suppose δ < δ, as incase 1.1 our purpose is to prove that the number of edge sets with rankq + δ −1 and size q + δ is different in each graph. If T δ

p,q has x essential

cycles of length q+δ , T δp ,q has x+p

q + δ − 1δ

, therefore if we show

that there exits a bijection between the edge sets of non essential cycleswith rank q + δ − 1 and size q + δ , we would have proved what we


want. For every r with 0 ≤ r ≤ q − 2 we define the following bijection,ϕr between A ⊆ E(T δ

p,q)|r(A) = q + δ − 1, |A| = q + δ , s(A) = r and

A ⊆ E(T δp,q)|r(A) = q + δ − 1, |A| = q + δ , s(A) = r.

If A ⊂ E(T δp,q), ϕr(A) = ∪ψ(((i, j), (i , j ))); ((i, j), (i , j )) ∈ A

where

ψ((h, k)) =

⎧⎪⎪⎨

⎪⎪⎩

(h, k) if j = q − 1, j = 0

(h, k) if j, j ∈ [0, r]

(h, k) if r + 1 ≤ j, j ≤ q − 1

with h = (i, j), k = (i , j ), h = (i + δ − δ , j) and k = i + δ − δ , j ).

Case 2.2 lT δp,q

= p, lT δp ,q

= p = q + δ with p < q + δ.

Suppose T (T δp ,q ;x, y) = T (T δ

p,q;x, y) then p = p and q = q +

pq + δ − 1

δ. Since pq = p q we obtain q = q , a contradiction.

Case 2.3 lT δp,q

= p, lT δp ,q

= q + δ with p < q + δ and q + δ < p .

q + δ = p ⇒ q = pp − 1

δ

pq = p q ⇒ pq + δ − 1

δ= q = p − δ

δ < p− 1 then pq + δ − 1

δ> p. This contradiction was obtained

by having assumed that both graphs have the same Tutte polynomial.Because of hypothesis 2, we have that the case lT δ

p,q= p = q + δ,

lT δp ,q

= q + δ with q + δ < p cannot occur.

With an analogous process to the one followed in case 1.3 we provethat the number of edge-sets with rank q + δ +1 and size q + δ +2 aredifferent in T δ

p,q and T δp ,q . Therefore, lT δ

p,q= q + δ, lT δ

p ,q= q + δ < p

is not possible.The rest of the cases are analogous to the previous ones, hence just

one case can occur, namely, lT δp,q

= p = q + δ, lT δp ,q

= p = q + δ , which

implies p = p , q = q and δ = δ .

Case 3 Suppose G K1p ,q , then T (T δ

p,q;x, y) = T (K1p ,q ;x, y).

Because of Lemma 2.3, we cannot have p > q .


Case 3.1 lT δp,q

= p < q + δ, lK1p ,q

= p < q .

As in case 1.1 we have to obtain a bijection to prove that the numberof edge-sets with rank q − 1 and size q are different in each graph.

If A ⊂ A(T δp,q), ϕr(A) = ∪ψ(((i, j), (i , j ))); ((i, j), (i , j )) ∈ A

where

ψ((h, k)) =

⎧⎪⎪⎨

⎪⎪⎩

(h, k) if j = q − 1, j = 0

(h, k) if j, j ∈ [0, r]

(h, k) if r + 1 ≤ j, j ≤ q − 1

with h = (i, j), k = (i , j ), h = (p−1−i+δ, j) and k = (p−1−i +δ, j ).The rest of the cases cannot occur because the length of shortest essen-tial cycles, the number of these cycles and the number of vertices donot coincide. We omit the proof for the sake of brevity.

The case G K2p ,q is similar to the previous ones, hence we just

specify the bijection in the case p = p < q and q = q :If A ⊂ A(T δ

p,q), ϕr(A) = ∪ψ(((i, j), (i , j ))); ((i, j), (i , j )) ∈ Awhere

ψ((h, k)) =

⎧⎪⎪⎨

⎪⎪⎩

(h, k) if j = q − 1, j = 0

(h, k) if j, j ∈ [0, r]

(h, k) if r + 1 ≤ j, j ≤ q − 1

with h = (i, j), k = (i , j ), h = (p − i + δ, j) and k = (p − i + δ, j ).Case 4 Finally, we are going to assume that G Sp ,q . For the

cases for which lT δp,q

= p < q+δ and lSp ,q= 2p ≤ q or lT δ

p,q= p = q+δ

and lSp ,q= q with q ≤ p we have that the length of shortest essential

cycles, the number of these cycles and the number of vertices in bothgraphs, cannot coincide. Therefore we obtain a contradiction, since Gand Sp ,q do not have the same Tutte polynomial.

By hypothesis 1 we cannot have lT δp,q

= q + δ < p, lSp ,q= q with

q ≤ p .Case 4.1 lT δ

p,q= p < q + δ, lSp ,q

= q with p ≤ q ≤ 2p .

Given that p = q , q = p and the equality of the number of shortestessential cycles q ≥ 2q −1 we arrive to a contradiction, because: p ≥2q −1 ≥ 2p −1 ⇒ 2p ≥ 2p .

Case 4.2 lT δp,q

= p < q + δ, lSp ,q= q with q ≤ p

Using the same ideas as in case 1.3 we prove that the number ofedge-sets with rank q + 1 and size q + 2 is greater in T δ

p,q than in Sp ,q .For the sake of brevity we only give a sketch of the proof. These sets


are classified into three groups: normal edge-sets, sets containing anessential cycle of length q and two other edges and essential cycles oflength q + 2. We prove that T δ

p,q has more edge sets of each type thanSp ,q . The ideas are similar to case 1.3, so we just mention the last

type. In T δp,q we have 2

p2

ways of adding two edges in order to get

a new essential cycle, hence there are 2qp2

essential cycles of length

q + 2. In Sp ,q (Figure 7) there are exterior edges to which we can add

two edges inq2

+q − 1

2different ways. Since p = q we have

more essential cycles of length q + 2 in T δp,q than in Sp ,q .

A

A

B

B A

A B

B

b) a)

Figure 7: a) Edge sets in Sp ,q with p ≥ q containing an essential cycleof length q and two other edges. b) Essential cycles of length q + 2 inSp ,q .

Case 4.3 lT δp,q

= p = q + δ, lSp ,q= 2p with q ≥ 2p .

Given that 2p = p = q + δ, we have q = 2q. We will obtain acontradiction by assuming we have equality for the number of shortestessential cycles in both graphs. In this case and in the next ones we

are going to use the following property:2p − 1

n<

2p − 1m

if

n < m ≤ [(2p − 1)/2] = p − 1.

If q + pq + δ − 1

δ= q (q − p )

2p − 1p

then

q (q −p )2p − 1

p= q+p

2p − 1

δ< q+p

2p − 1

p − 1= q+p

2p − 1

p


< q + q2p − 1

p< q 1 +

2p − 1

p< q (q − p )

2p − 1

p.

Case 4.4 lT δp,q

= p = q + δ, lSp ,q= q with p ≤ q ≤ 2p .

q = p = q + δ then p = q. Suppose

q + pq + δ − 1

δ= 2p

p −1

j=0

q − 1

j.

δ ≤ p/2 = q /2 ≤ p = q. If δ < q ≤ p − 1 then:

q + pq + δ − 1

δ= q + p

q − 1δ

= p + qq − 1

δ

< q 1 +q − 1

δ≤ 2p 1 +

q − 1δ

< 2pp −1

j=0

q − 1j

.

If δ = q,q − 1

δ≤ q − 1

q − 1because [(q − 1)/2] = q − 1. The

difference between this case and the previous one is that the last boundis obtained as follows:

2p 1 +q − 1

δ≤ 2p 1 +

q − 1q − 1

< 2pp −1

j=0

q − 1j

.

Case 4.5 lT δp,q

= q + δ < p, lSp ,q= 2p with 2p ≤ q .

2p = q + δ and q (q − p )2p − 1

p= p

2p − 1δ

.

Since [(2p − 1)/2] = p − 1,2p − 1

δ≤ 2p − 1

p,

q (q − p )2p − 1

p≥ (2p q − p q )

2p − 1

p

≥ pq2p − 1

δ> p

2p − 1

δ.

Case 4.6 lT δp,q

= q + δ < p, lSp ,q= q with p ≤ q ≤ 2p .

q = q + δ and pq + δ − 1

δ= 2p p −1

j=0q − 1

j.


We will get a contradiction if we prove that

q − 1

δ< 2

p −1

j=0

q − 1

j.

q ≤ 2p ⇒ [(q − 1)/2] ≤ p − 1

then

∃j0 ∈ [0, p − 1],q − 1

[(q − 1)/2]

q − 1

j0

q − 1δ

≤ q − 1[(q − 1)/2]

< 2p −1

j=0

q − 1j

.

Theorem 2.5 K0p,q is Tutte unique for all p, q ≥ 6.

Proof: Let p, q ≥ 6 and G a graph with T (G;x, y) = T (K0p,q;x, y). Due

to Lemma 2.1 and Theorem 2.4, G has to be isomorphic to exactly oneof the following graphs: Ki

p ,q , Sp ,q . We are going to prove that G is

isomorphic to K0p ,q with p = p , q = q .

Suppose G isomorphic to K0p ,q then lT δ

p,q= lK0

p ,qand the number

of shortest essential cycles has to be the same in both graphs. We justhave to study the case in which lK0

p,q= p < q + 1, lK0

p ,q= q + 1 with

p > q + 1. This is so because, if lK0p,q

= q + 1 < p and lK0p ,q

= p

with p < q + 1 the reasoning would be analogous and in these cases itis easy to verify that the number of vertices and the length of shortestessential cycles can not coincide in both graphs.

If lK0p,q

= p < q +1, lK0p ,q

= q +1 with p > q +1 we can show that

the number of edge-sets with rank q + 2 and size q + 3 is different inK0

p,q and K0p ,q . We omit the proof because it uses the same arguments

as those in case 1.3.Suppose G ∼= K1

p ,q . Since p is even and p odd, all the cases in which

the length of shortest essential cycles in K0p,q is p and in K1

p ,q is p areproved. By Lemma 2.3 we know that the number of shortest essentialcycles in K0

p,q is always bigger than one, hence we obtain a contradictionin all those cases in which the number of shortest essential cycles in K1

p ,qis one. Therefore we just have to study two cases: lK0

p,q= q + 1 < p,

lK1p ,q

= p < q and lK0p,q

= q + 1 < p, lK1p ,q

= p = q . In the first


case we obtain a contradiction by proving that the number of edge-setswith rank p + 1 and size p + 2 is different in each graph (followingthe same reasoning as in case 1.3 of Theorem 2.4). In the second casewe show that if p = q = q + 1, pq = p q , p are even and p is oddit must then be the case that q is even. By Lemmas 2.2 and 2.3,T (K0

p,q;x, y) = T (K2p ,q ;x, y) if p > q , therefore G is not isomorphic

to K2p ,q . Following the same reasoning as in case 1.1 of Theorem 2.4

we show that it cannot be that lK0p,q

= p < q + 1 and lK2p ,q

= p < q .

We just specify the bijection between A ⊆ E(K0p,q)|r(A) = q, |A| =

q + 1, s(A) = r and A ⊆ E(K2p ,q )|r(A) = q, |A| = q + 1, s(A) = r.

If A ⊂ A(K0p,q), ϕr(A) = ∪ψ(((i, j), (i , j ))); ((i, j), (i , j )) ∈ A

where

ψ((h, k)) =

⎧⎪⎪⎨

⎪⎪⎩

(h, k) if j = q − 1, j = 0

(h, k) if j, j ∈ [0, r]

(h, k) if r + 1 ≤ j, j ≤ q − 1

with h = (i, j), k = (i , j ), h = (i + 1, j) and k = (i + 1, j ).

On the other hand, we prove (as in case 1.3 of Theorem 2.4) that iflK0

p,q= q + 1 < p and lK2

p ,q= p < q the number of edge-sets of rank

p + 1 and size p + 2 is different for each graph.

The other four cases obtained by considering the possible combina-tions of the lengths of shortest essential cycles in K0

p,q and K2p ,q , are

not possible since the length of shortest essential cycles, the number ofthese cycles and the number of vertices cannot coincide in both graphs.

Finally, suppose G Sp ,q .

Case 1 If lK0p,q

= p < q + 1, lSp ,q= q with p ≤ q ≤ 2p we obtain

a contradiction as follows:

p < q + 1 ⇒ p ≤ q < p + 1 ⇒ p = q ⇒ p = q = p = q .

q ≥ 2q −1 = 2q−1.

Case 2 lK0p,q

= p < q + 1 and lSp ,q= q ≤ p .

As we did in case 1.3 of Theorem 2.4 we show that there are differentnumber of edge-sets with rank q + 1 and size q + 2 in K0

p,q and Sp ,q ,

hence these graphs do not have the same Tutte polynomial.

Case 3 lK0p,q

= p = q + 1 and lSp ,q= 2p ≤ q .


2p = p = q + 1 then q = 2q. Since 5q = q (q − p )2p − 1

pwe

will obtain a contradiction if we prove that 5q < q (q − p ).

q (q − p ) = 2q(2q− (p/2)) = 4q2 − pq = 4q2 − q(q +1) = q(3q− 1) > q5

Case 4 lK0p,q

= q + 1 < p and lSp ,q= 2p ≤ q .

2p = q + 1. 4q = q (p − q )2p − 1

p≥ 2p 2 2p − 1

p>

8p > 8p − 4 = 4(2p − 1) = 4q.Similarly as in the comparisons between K0

p,q and K2p ,q , the rest of

the cases cannot occur because the length of shortest essential cycles,the number of these cycles and the number of vertices do not coincidein both graphs given that q ≥ 6.

Theorem 2.6 The graph K1p,q is Tutte unique for all p, q ≥ 6.

Proof: The argument of this proof is basically the same as those followedin Theorems 2.4 and 2.5. Because of the Tutte uniqueness of T δ

p,q and

K0p,q we only have to prove that T (G;x, y) = T (K1

p,q ;x, y) with G ∈K1

p ,q (except if p = p and q = q ), K2p ,q , Sp ,q . In every case we are

going to suppose that T (G;x, y) = T (K1p,q ;x, y) and we will obtain a

contradiction.Case 1 If G K1

p ,q it is easy to prove that the length of shortestessential cycles, the number of these cycles and the number of verticesonly coincide if p = p and q = q .

Case 2 If G K2p ,q , by Lemmas 2.2 and 2.3 we get a contradiction

in all those cases for which the number of shortest essential cycles inK1

p,q is one or the number of shortest essential cycles in K2p ,q is two.

In the other cases a contradiction is reached because p is odd and p iseven.

Case 3 If G Sp ,q we can consider p ≤ q because if p > q thenumber of shortest essential cycles in K1

p,q is one and p, q ≥ 6.

If lK1p,q

= p < q and lSp ,q= 2p ≤ q or lSp ,q

= q with p ≤ q ≤ 2p ,

by Lemmas 2.2 and 2.3 it is easy to obtain a contradiction. The sameis true if lK1

p,q= p = q and lSp ,q

= q with q ≤ p or lSp ,q= q with

p ≤ q ≤ 2p .

If lK1p,q

= p < q and lSp ,q= q ≤ p we prove (as in previous

cases) that there are different number of edge-sets with rank q + 1 and


size q + 2 in both graphs, hence they can not have the same Tuttepolynomial.

If lK1p,q

= p = q and lSp ,q= 2p ≤ q , 2p = p = q then q = 2q

therefore

q + 1 = q (q − p )2p − 1

p= 2q(q − p )

2p − 1p

> q + 1.

Theorem 2.7 The graph K2p,q is Tutte unique for p, q ≥ 6.

Proof: Due to Theorems 2.4, 2.5 and 2.6 we have to prove that T (G;x, y)= T (K2

p,q;x, y) with G ∈ K2p ,q (except if p = p and q = q ), Sp ,q

and pq = p q .By Lemmas 2.2 and 2.3, T (K2

p ,q ;x, y) = T (K2p,q;x, y) if p = p and

q = q because the length of shortest essential cycles, the number ofthese cycles and the number of vertices only coincide if p = p andq = q .

If G Sp ,q we can assume that p ≤ q otherwise if q < p, K2p,q has

two shortest essential cycles and by Lemma 2.2 we obtain a contradic-tion.

Case 1 If lK2p,q

= p < q and lSp ,q= q ≤ p we prove as in previous

cases that the number of edge-sets with rank q + 1 and size q + 2 isdifferent in both graphs. Hence these two graphs cannot have the sameTutte polynomial, therefore we get a contradiction and G can not beisomorphic to Sp ,q .

Case 2 If lK2p,q

= p = q and lSp ,q= 2p ≤ q then 2p = p = q and

q = 2q hence

q + 2 = q (q − p )2p − 1

p= 2q(q − p )

2p − 1p

> q + 2.

In the other four cases we obtain a contradiction because the lengthof shortest essential cycles, the number of these cycles and the numberof vertices cannot coincide in both graphs.

Theorem 2.8 The graph Sp,q is Tutte unique for p, q ≥ 6 and 2q =

pq + δ − 1

δfor all p , q with p q = pq and δ > 0.

Proof: Suppose that Sp,q is not Tutte unique. Then, by Theorems 2.4,2.5, 2.6, 2.7 and Lemma 2.2 Sp,q is isomorphic to Sp ,q with p = p,q = q and pq = p q .


Case 1 If lSp,q = 2p ≤ q and lSp ,q= q with p ≤ q ≤ 2p , by

Lemma 2.2 q = 2p and q = 2p then:

q(q − p)2p − 1

p= 2p (2p − (q /2))

q − 1

q /2

> (2p − (q /2))q − 1

q /2≥ (2p − p )

q − 1

q /2

= pq − 1

q /2>

p −1

j=0

q − 1

j.

Hence, the number of shortest essential cycles is different in eachgraph and by Lemma 2.2 we have that Sp,q is not isomorphic to Sp ,q .

Case 2 If lSp,q = 2p ≤ q and lSp ,q= q ≤ p then 2p = q and

q = 2p .

2q = 22p = 22p−1

j=0

2p − 1

j< 2 · 2p 2p − 1

p

≤ q2p − 1

p< q(q − p)

2p − 1

p.

We obtain a contradiction to the assumption that the number ofshortest essential cycles is equal in both graphs.

The other three cases are analogous to the previous ones.

3 Concluding Remarks

We have shown that locally grid graphs are Tutte unique for p, q ≥ 6,but our techniques do not apply to p = 3, 4, 5. An interesting open

problem is to prove that the number pq + δ − 1

δis not a power of

two. This would give a more general result about the Tutte uniquenessof T δ

p,q and Sp,q.

AcknowledgementWe are very grateful to I. Gitler of CINVESTAV-I.P.N. for carefully

reviewing the manuscript and for making useful suggestions.


D. GarijoDepartment of Applied Math.,University of Sevilla,Avenida Reina Mercedes S/N,Sevilla C. P. 41012, [email protected]

A. MarquezDepartment of Applied Math.,University of Sevilla,Avenida Reina Mercedes S/N,Sevilla C. P. 41012, [email protected]

M. P. RevueltaDepartment of Applied Math.,University of Sevilla,Avenida Reina Mercedes S/N,Sevilla C. P. 41012, [email protected]

References

[1] Brylawsky T.; Oxley J., The Tutte polynomial and its applica-tions, Matroid Applications, Cambridge University Press, Cam-bridge, 1992.

[2] Hall J.I., Locally Petersen graphs, J. Graph Theory, 4 (1980), 173-187.

[3] Marquez de Mier A.; Noy M., On graphs determined by their Tuttepolynomials, Graphs Combin., 20 (2004), 105-119.

[4] Marquez de Mier A.; Noy M.; Revuelta M.P., Locally grid graphs:Classification and Tutte uniqueness, Discr. Math., 266 (2003),327-352.

[5] Thomassen C., Tilings of the Torus and the Klein Bottle andvertex-transitive graphs on a fixed surface, Trans. Amer. Math.Soc., 323 (1991), 605-635.



Enrique Torres Giese 1

Resumen

Con herramientas basicas como la sucesion espectral de Serre ylos cuadrados de Steenrod se obtienen resultados de no-inmersionde espacios lente de dimension 2n+1 y torsion 2m. En la situacionα(n) = 1, donde α(n) es el numero de 1’s en la expansion binariade n, el resultado es optimo.

2000 Mathematics Subject Clasification: 57R42.Keywords and phrases: Inmersion de variedades, sucesion espectral deSerre, cuadrados de Steenrod.

1 Introduccion

Un problema clasico de la Topologıa Diferencial es el de conocer cuandouna variedad M admite una inmersion de manera optima en un espacioeuclideano, es decir, conocer el mınimo entero k para el cual M admiteuna inmersion en Rk. Este es un problema aun abierto, pues el decidircuando una variedad M admite una inmersion en una variedad N esen extremo complicado. Una primer contribucion en la solucion de esteproblema ocurrio en 1944, cuando Whitney demostro que toda variedadcompacta de dimension n admite una inmersion en un espacio euclideanode dimension 2n − 1. En otras palabras, Whitney acoto superiormentela dimension del espacio euclideano donde la variedad pudiera tener unainmersion de manera optima. En este trabajo analizaremos el problemade inmersion de espacios lente, de hecho concluiremos un resultado deno inmersion de tales espacios.

1Becario Conacyt 165576. El contenido de este artıculo esta basado en latesis de Maestrıa presentada por el autor en el Departamento de Matematicas delCINVESTAV-IPN.

57

58 Enrique Torres Giese

Si f : Xn → Rn+k es una inmersion de la variedad X de dimensionn, entonces existe un haz νf de dimension k tal que τX ⊕ νf es trivial.El haz νf se nombra el haz normal asociado a la imersion f . CuandoX es compacta Hirsch en [7] demostro el recıproco: si existe un haz νde dimension k tal que τX ⊕ ν es trivial, entonces existe una inmersionXn → Rn+k cuyo haz normal asociado es ν. Observe que en el casode tener una inmersion Xn → Rn+k, el haz tangente τX es el inversoestable del haz normal νX .

El trabajo de Hirsch tambien afirma que entre cualesquiera dos in-mersiones de Xn en R2n+1 existe una homotopıa X × I → R2n+1 talque cada X × t → R2n+1 es inmersion. Por lo que si f : Xn → Rn+k

y g : Xn → Rn+l son inmersiones, entonces Xn f→ Rn+k → R2n+1

y Xn g→ Rn+k → R2n+1 son homotopicas, ası νf ⊕ (n − k + 1) =νg ⊕ (n − l + 1). Es decir, cualesquiera dos haces normales asociados adistintas inmersiones determinan la misma clase estable, llamada el haznormal estable de X y denotado por νX .

En estos terminos, el decidir si una variedad admite una inmersionen codimension k es equivalente a conocer si la dimension geometrica deνX , denotada por gd(νX), es menor o igual que k, mientras que conocerla dimension optima de inmersion es equivalente a conocer exactamentegd(νX ). A su vez, el problema de encontrar la dimension geometrica deun haz α es equivalente al problema de levantamiento

BO(k)

Xα

BO

La idea clave en la demostracion del resultado principal, que a con-tinuacion enunciamos, se basa en este ultimo hecho. En nuestro casosupondremos una inmersion, lo cual producira un levantamiento de lafuncion clasificante del haz normal y algebraicamente se probara suinexistencia.

Escribimos X ⊆ Y si la variedad X admite una inmersion en lavariedad Y . Para n y m enteros positivos denotamos por L2n+1(2m) alespacio de orbitas asociado a la accion usual de Z/2m sobre los vectoresde norma uno en Cn+1.

Teorema Sea m ≥ 2 y l(n) = max1 ≤ i ≤ n − 1 : n+i+1n ≡ 0 (4).

No-inmersion de espacios lente 59

a) Si n = 2s + 1 y n ≥ 2, entonces L2n+1(2m) ⊆ R2n+1+2l(n).

b) Si n = 2s + 1, con s ≥ 1, entonces L2n+1(2m) ⊆ R2n+2l(n) =R4n−4.

Corolario Sea m ≥ 2

a) Si n = 2s con s ≥ 1, entonces L2n+1(2m) ⊆ R4n−1.

b) Si n = 2s + 2t con s > t ≥ 1, entonces L2n+1(2m) ⊆ R4n−3.

En la Subseccion 3.2 comparamos estos resultados con situacionesconocidas de inmersion de espacios complejos proyectivos.

2 Preliminares

La sucesion de Gysin es una sucesion en cohomologıa asociada a un

haz esferico Sk i→ Ep→ B orientable. Esta sucesion se obtiene a partir

de la sucesion exacta larga de la pareja (D,E) (con E → D y D → Bel haz de discos asociado) y el isomorfismo de Thom produciendo

· · · → Hr−1(E)φ→ Hr−k−1(B)

·e→ Hr(B)p∗→ Hr(E) → · · ·

donde e ∈ Hk+1(B) es la clase de Euler del haz esferico. La condicionde que el haz sea orientable puede cubrirse si suponemos que su basesea simplemente conexa.

Lema 2.1 Si en el haz esferico Sk i→ Ep→ B, B es 1-conexo y su clase

de Euler es cero, entonces

H∗(E; Z) ∼= H∗(B; Z) ⊕ a · H∗(B; Z)

como un H∗(B; Z)-modulo, donde a ∈ Hk(E; Z) es tal que φ(a) es ungenerador de H0(B; Z).

Demostracion: Es importante realizar dos comentarios. Primero, quela forma en que H∗(E) es visto como H∗(B)-modulo es a traves de p.Segundo, que el morfismo φ es la composicion del morfismo de conexiony el isomorfimo de Thom, de modo que φ es un morfismo de H∗(B)-modulos. Observe que la sucesion de Gysin toma la forma

0 → Hq(B)p∗→ Hq(E)

φ→ Hq−k(B) → 0.


Sea a ∈ Hk(E) tal que φ(a) = 1 ∈ H0(B) ∼= Z, entonces el morfismoH∗(B) → H∗(E) dado por b → b · a escinde esta ultima sucesion,obteniendo ası el resultado buscado.

Por otra parte, en el estudio de haces vectoriales, las clases de Stiefel-Whitney y de Chern juegan un papel esencial. En nuestro estudiomanejaremos otros elementos que surgen de la complejificacion de hacesreales. Estos elementos en cohomologıa son llamadas las clases de Pon-trjagin. Estas se definen para haces reales como pi(ξ) = (−1)ic2i(ξC) ∈H4i(X) con n = dimR(ξ), i ≤ [n2 ] y cj(ξC) la j-esima clase de Chernde ξC. En particular, las clases de Pontrjagin del haz universal λn →BO(n) (o de su version universal λ+

n → BSO(n)) se llaman clases uni-versales de Pontrjagin y se denotan por pk. En la Observacion 2.2estaremos de hecho interesados en las clases de λ+

2n cuya clase de Eulerdenotaremos por en (ver Observacion 2.3 (a)).

Observacion 2.2 A continuacion mencionamos algunas propiedades yrelaciones de las clases de Pontrjagin y la clase de Euler.

a) La funcionU(n) → SO(2n)

A + iB → A −BB A

induce una funcion rn : BU(n) → BSO(2n) la cual satisface quer∗n(λ+

2n) = r(γn). Este hecho junto con los isomorfismos canonicos

ξC = ξC, r(ξC) = ξ ⊕ ξ y r(η)C = η ⊕ η nos permiten obtener lasrelaciones

r∗n(pk) = c2k − 2ck−1 · ck+1 + . . . + (−1)k+12c2k−1 · c1 + (−1)k2c2k

r∗n(en) = cn

b) Si ξ es un haz orientable con dim(ξ) = n, entonces pn(ξ) = e(ξ)2.

Observacion 2.3 Algunas observaciones utiles en las siguientes sec-ciones:

a) Tanto S2n−1 → BU(n − 1) → BU(n) y Sn−1 → BSO(n −1) → BSO(n) son respectivamente los haces esfericos de los hacescanonicos γn → BU(n) y λn → BSO(n) [15], lo cual nos permiteconsiderar sus respectivas clases de Chern, Pontrjagin y Euler.


b) Puesto que la funcion BU(n)i→ BU(m) induce al haz universal

γm → BU(m) en γn ⊕ (m − n), entonces i∗ci(γm) = ci(γn ⊕(m − n)) = ci(γn) por lo que la funcion i∗ : H∗(BU(m)) →H∗(BU(n)) es un isomorfismo hasta dimension 2n. Similarmente,

la funcion BSO(n)i→ BSO(m) induce un isomorfismo en coho-

mologıa modulo 2 hasta dimension n.

Lema 2.4 Si n − k es par, entonces Hn−k(Vn,k) ∼= Z.

Demostracion: Considere la fibracion

Sn−k i→ Vn,k → Vn,k−1.

Puesto que Vn,k y Vn,k−1 son (n − k − 1) y (n − k) conexos respectiva-mente, podemos aplicar la sucesion exacta de Serre en cohomologıa. Enparticular observe que tenemos la siguiente sucesion exacta

Hn−k(Vn,k−1) → Hn−k(Vn,k)i∗→ Hn−k(Sn−k) ∼= Z

donde Hn−k(Vn,k−1) = 0. Veamos que en el caso n − k par el morfismoi∗ es un isomorfismo. Para ello veamos que el morfismo δ de conexionsiguiente es trivial. El morfismo δ se encuentra inducido por la dife-rencial dn−k+1 de la sucesion espectral de Serre y esta definido comomultiplicacion por la clase de Euler de la fibracion en cuestion. Ahora,considere el siguiente diagrama de fibraciones

Sn−k Vn−k+2 Sn−k+1

Sn−k i Vn,k Vn,k−1

∗ Vn,k−2 Vn,k−2

La sucesion espectral de Serre del segundo renglon se mapea en la delprimer renglon, de hecho utilizando la sucesion exacta de Serre se verificaque el morfismo Hn−k+1(Vn,k−1) → Hn−k+1(Sn−k+1) es un isomorfismolo cual, en vista de la naturalidad de la clase de Euler, implica que laclase de Euler buscada es trivial. Ası, Hn−k(Vn,k) ∼= Z.

Lema 2.5 Sean Vn,kα→ BSO(n − k) la funcion clasificante del haz

SO(n − k) → SO(n) → Vn,k y Sn−k = Vn−k+1,1i→ Vn,k la inclusion

natural. Si n−k es par, entonces α∗(e) = ±2u donde u es un generadorde Hn−k(Vn,k) y e ∈ BSO(n − k) es la clase de Euler.


Demostracion: Considere el siguiente diagrama

Sn−k−1 Vn−k+1,2 Sn−k

i

Sn−k−1 Vn,k+1 Vn,k

α

Sn−k−1 BSO(n − k − 1) BSO(n − k)

BSO(n) BSO(n)

donde el segundo renglon es el haz esferico canonico sobre BSO(n− k)inducido por α y el primer renglon el haz inducido por i. Observeque el haz Sn−k−1 → Vn−k+1,2 → Sn−k es el haz esferico del haz tan-gente de Sn−k el cual, cuando n − k es par, tiene por clase de Eulera 2 ∈ Hn−k(Sn−k). Escribimos a α∗(e) como ku, donde k ∈ Z y u esun generador de Hn−k(Vn,k). Como i∗ es un isomorfismo en este caso(Lema 2.4) y la clase de Euler es natural, entonces k = ±2.

Finalmente, recordamos que la cohomologıa de los espacios lente detorsion 2m esta dada por H∗(L2n+1(2m); Z/2) ∼= E(x) ⊗ Pn+1(z), con|x| = 1 y |z| = 2.

3 El espacio clasificante de haces reales estable-mente complejos

El material de esta seccion esta basado en el trabajo de B. Junod [9].

3.1 La cohomologıa de B(n, k)

En esta primer subseccion estudiaremos las propiedades cohomologicasde un espacio particular que nos permitira desarrollar nuestro trabajo.

Definicion 3.1 Considerese el siguiente diagrama

BSO(k)

BU(n) BSO(2n)


Decimos que el pull-back de este sistema es el espacio clasificante dehaces reales de dimension k establemente complejos sobre complejoscelulares de dimension ≤ 2n + 1 y lo denotamos por B(n, k). Observeque inductivamente podemos definir el espacio B(n, k) como el pull-backdel diagrama

BSO(k)

B(n, k + 1) BSO(k + 1)

Observacion 3.2 En la defincion anterior se debe cumplir que 1 ≤ k ≤2n, para el caso en que k = 2n se verifica que B(n, 2n) = BU(n). Porotra parte, si k = 2n − 1 y P es el pullback del sistema

SO(2n − 1)

U(n) SO(2n)

donde U(n) → SO(2n) esta dada por

A + iB → A −BB A

entonces existe una funcion U(n − 1) → P inducida por las funcionesU(n−1) → U(n) y U(n−1) → SO(2n−2) → SO(2n−1). De hecho, talsituacion forza a que U(n−1) → P sea un isomorfismo. Este argumentomuestra que B(n, 2n − 1) = BU(n − 1).

Trabajaremos frecuentemente con el siguiente diagrama

V2n,2n−2j

i1

V2n,2n−2j

i2

B(n, 2j)

p

f2jBSO(2j)

BU(n)rn BSO(2n)


Observe que existe una funcion h : BU(j) → B(n, 2j) determinada porla propiedad universal de B(n, 2j)

BU(j)rjh

iB(n, 2j)

p

f2jBSO(2j)

BU(n)rn BSO(2n)

Lema 3.3 Para cada n ≥ 1 y 1 ≤ j ≤ n − 1, existe un elementoaj ∈ H2j(B(n, 2j); Z) tal que

f∗2j(ej) = p∗(cj) − 2aj , i∗1(aj) = uj , h∗(aj) = 0

donde ej ∈ H2j(BSO(2j); Z) es la clase universal de Euler y uj ∈H2j(V2n,2n−2j ; Z) es un generador.

Demostracion: De acuerdo al Lema 2.5 podemos elegir

uj ∈ H2j(V2n,2n−2j ; Z) ∼= Z

generador tal que i2(ej) = −2uj. Puesto que V2n,2n−2j es (2j − 1)-conexo, BU(n) es 1-conexo y la cohomologıa de BU(n) esta concentradaen dimensiones pares, aplicando la sucesion exacta larga de Serre a lafibracion V2n,2n−2j → B(n, 2j) → BU(n) se tiene la sucesion exactacorta

0 → H2j(BU(n)) → H2j(B(n, 2j)) → H2j(V2n,2n−2j) → 0

Sea x ∈ H2j(B(n, 2j)) tal que i∗1(x) = uj . Puesto que i∗ es un iso-morfismo hasta dimension 2j podemos reemplazar a x por aj = x −p∗(i∗)−1h∗(x), ası h∗(aj) = 0 y i∗1(aj) = uj . Por otra parte, comoH2j(V2n,2n−2j ; Z) ∼= Z, entonces la sucesion exacta anterior se escindeproduciendo el isomorfismo

H2j(B(n, 2j)) ∼= im(p∗) ⊕ Zaj∼= H2j(BU(n)) ⊕ Zaj

Como h∗f∗2j(ej) = r∗j (ej) = cj , entonces f∗

2j(ej) = p∗(cj)+maj. Ademas,i∗1f

∗2j(ej) = i∗2(ej) = −2uj , ası m = −2 y f∗

2j(ej) = p∗(cj) − 2aj .


Una situacion particular de nuestro diagrama de trabajo es la si-guiente

S2j−1 S2j−1

B(n, 2j − 1)f2j−1

p2j−1

BSO(2j − 1)

B(n, 2j)f2j

BSO(2j)

Considere la sucesion de Gysin de la fibracion

S2j−1 → B(n, 2j − 1)p2j−1→ B(n, 2j)

la cual tiene clase de Euler (de acuerdo al Lema anterior) p∗(cj) − 2aj

→ Hq(B(n, 2j))·p∗(cj)−2aj−→ Hq+2j(B(n, 2j)) → Hq+2j(B(n, 2j − 1)) →

y por exactitud se satisface que p∗2j−1p∗(cj) = 2p∗2j−1(aj).

Denotaremos por bj a p∗2j−1(aj), mas generalmente bj denotara ap∗kp

∗k+1 · · · p∗2j−1(aj), mientras que cj al elemento (p∗kp

∗k+1 · · · p∗2j−1)p

∗(cj).Con esta notacion se satisface que 2bj = cj para 1 ≤ j ≤ n − 1. Lasiguiente figura muestra tal situacion.

B(n, 2j − 1) bj, bj+1, . . . , bn−1, c1, . . . , cj−1 2bj = cj

↓B(n, 2j) aj , bj+1, . . . , bn−1, c1, . . . , cj

↓B(n, 2n − 5) bn−2, bn−1, c1, . . . , cn−3 2bn−2 = cn−2

↓B(n, 2n − 4) an−2, bn−1, c1, . . . , cn−2

↓B(n, 2n − 3) bn−1, c1, . . . , cn−2 2bn−1 = cn−1

↓B(n, 2n − 2) an−1, c1, . . . , cn−1

↓B(n, 2n − 1) c1, . . . , cn−1

↓BU(n) c1, . . . , cn


Observacion 3.4 Hacemos un par de observaciones utiles en la de-mostracion del siguiente resultado.

a) De acuerdo al Lema anterior f∗2k(ek) = ck − 2ak en H2k(B(k +

1, 2k)), y en vista de la Observacion 2.2 (ck − 2ak)2 = f∗2k(e

2k) =

f∗2k(pk) = p∗r∗k+1(pk) = p∗(c2

k − 2ck−1ck+1) = c2k. Por lo tanto

4ckak = 4a2k en H∗(B(k + 1, 2k).

B(k + 1, 2k) ak, c1, . . . , ck

BU(k) = B(k + 1, 2k + 1) c1, . . . , ck

BU(k + 1) c1, . . . , ck+1

b) Utilizando la fibracion V2n,2n−k → B(n, k) → BU(n) y en vistade que V2n,2n−k y BU(n) son 1-conexo, se tiene que B(n, k) es1-conexo.

c) Sea h : B(n− 1, 2j) → B(n, 2j) la funcion que define la propiedaduniversal de B(n, 2j) y que es compatible con la misma funcionh del diagrama previo al Lema 3.3. En esta situacion se tiene eldiagrama

B(n − 1, 2j) h B(n, 2j) BSO(2j)

B(n − 1, 2n − 3) B(n, 2n − 3) BSO(2n − 3)

BU(n − 1) B(n, 2n − 2) BSO(2n − 2)

BU(n − 1) B(n, 2n − 1) BSO(2n − 1)

BU(n − 1) BU(n) BSO(2n)

el cual nos muestra que < bn−1 >⊆ Ker(h∗) ya que h∗(an−1) = 0(Lema 3.3).


d) En el Lema 3.3 construimos elementos aj ∈ H2j(B(n, 2j)), noteseque estos elementos en principio dependen de n, para ser precisosaj deberıa denotarse como an

j . Veamos que h preserva tales ele-

mentos, es decir h(anj ) = an−1

j . Para esto considere el siguientediagrama conmutativo

V2n,2n−2j−2

i

V2n,2n−2j−2

B(n − 1, 2j)

h

BSO(2j)

B(n, 2j) BSO(2j)

BU(n − 1) BSO(2n − 2)

BU(n) BSO(2n)

donde la funcion i es la inclusion canonica que es parte de la fi-

bracion V2n,2n−2j−2i→ V2n,2n−2j → V2n,2. Utilizando la sucesion

exacta de Serre se tiene que i∗ es un isomorfismo en dimension2j, lo cual implica que h∗(aj) al restringirlo en su correspondientevariedad de Stiefel es un generador, en particular h∗(aj) y aj sonlibres de torsion. La observacion es consecuencia de la conmuta-tividad del diagrama, del hecho de que h∗(aj) y aj son libres detorsion y de la relaciones del Lema 3.3 que definen dichos elemen-tos. En adelante usaremos indistintamente la notacion aj parareferirnos a los elementos del Lema 3.3 sin importar n.

Teorema 3.5 H∗(B(n, k); Z) es un Z-modulo libre determinado por elisomorfismo

H∗(B(n, k); Z) ∼= Z[c1, . . . , ct] ⊗ ∆(at, bt+1, . . . , bn−1) k = 2tZ[c1, . . . , ct] ⊗ ∆(bt+1, . . . , bn−1) k = 2t + 1


donde ∆(x1, . . . , xm) es el grupo abeliano libre generado por los elemen-tos

xi1xi2 · · · xis , 1 ≤ i1 < i2 < · · · < is ≤ m.

Demostracion: Procederemos por induccion decreciente sobre k. Loscasos en que k es 2n o 2n−1 son inmediatos pues H∗(B(n, k)) es en estoscasos, de acuerdo a la Observacion 3.2, H∗(BU(n)) y H∗(BU(n − 1))respectivamente. Analicemos ahora el caso k = 2n − 2 y para ello lasucesion de Gysin de la fibracion

S2n−2 → B(n, 2n − 2)p2n−2→ B(n, 2n − 1)

que es

0→H2q(BU(n−1))p∗2n−2→ H2q(B(n, 2n−2))

φ→H2q−2n+2(BU(n−1))→ 0

pues su clase de Euler es trivial al ser de grado impar en BU(n − 1).Ası, de acuerdo al Lema 2.1, tenemos el isomorfismo

H∗(B(n, 2n − 2)) ∼= H∗(BU(n − 1)) ⊕ a · H∗(BU(n − 1))

para a ∈ H2n−2(B(n, 2n− 2)) tal que φ(a) es generador de H0(BU(n−1)). Por otra parte, aplicando la sucesion exacta de Serre a la fibracion

V2n,2 → B(n, 2n − 2)p→ BU(n) se tiene la sucesion exacta corta

0 → H2n−2(BU(n))p∗→ H2n−2(B(n, 2n − 2)) → H2n−2(V2n,2) → 0

la cual produce el isomorfismo H2n−2(B(n, 2n − 2)) ∼= Im(p∗) ⊕ Zan−1

ya que H2n−2(V2n,2) ∼= Z (Lema 2.4). Como Im(p∗) = Im(p∗2n−2) =ker(φ), entonces φ(an−1) es generador de H0(BU(n − 1)), lo cual des-cribe el isomorfismo deseado. Observe que en este caso, de acuerdo a laObservacion 3.4, a2

n−1 = cn−1an−1.

Supongamos ahora el resultado cierto para k con r ≤ k ≤ 2n − 1 yprobemoslo para r − 1.

Si r es impar, digamos r = 2j + 1, en este caso procedemos comoarriba. Aplicamos la sucesion de Gysin a la fibracion S2j → B(n, 2j) →B(n, 2j+1), la cual tiene clase Euler trivial, para concluir el isomorfismo

H∗(B(n, 2j)) ∼= H∗(B(n, 2j + 1)) ⊕ ajH∗(B(n, 2j + 1))∼= Z[c1, . . . , cj ] ⊗ ∆(aj , bj+1, . . . , bn−1)


Para efectos de la demostracion del siguiente caso, en que r es par,demostraremos que el morfismo de grupos

Z[c1, . . . , cj−1, cj − 2aj ] ⊗ ∆(aj, bj+1, . . . , bn−1)ψn→ H∗(B(n, 2j))

x ⊗ y −→ xy

es un isomorfismo. Procedemos por induccion sobre n y para ello comen-zamos en n = j + 1 que es el primer valor que puede tomar n. Elmorfismo en consideracion toma la forma

Z[c1, . . . , cj−1, cj − 2aj ] ⊗ ∆(aj)ψj+1→ Z[c1, . . . , cj ] ⊗ ∆(aj)

y en vista de que a2j = ajcj las relaciones (cj − 2aj) ⊗ 1 + 2(1 ⊗

aj) → cj y (cj − 2aj)2 ⊗ 1 → c2j muestran que ψj+1 es suprayec-

tiva y en consecuencia un isomorfimo entre Z-modulos libres. Supon-gamos que la afirmacion es cierta para valores menores que n y seanA = Z[c1, . . . , cj ]⊗∆(aj , bj+1, . . . , bn−2), B = Z[c1, . . . , cj−1, cj − 2aj ]⊗∆(aj , bj+1, . . . , bn−2), xi base de A (por lo que xbn−1 es base deAbn−1) y h : B(n − 1, 2j) → B(n, 2j) la funcion de la Observacion 3.4.Observe por una parte que Ker(h∗) = Abn−1 y que H∗(B(n, 2j)) ∼=A ⊕ Abn−1. Ası, la hipotesis de induccion y la conmutatividad del dia-grama

B

ψn−1ψn|B

A ⊕ Abn−1h∗

A

muestran que ψn|B es monomorfismo, y de hecho podemos elegir yibase de B de tal suerte que ψn(yi) = xi + zibn−1 para zi ∈ A. Puestoque a2

n−1 = cn−1an−1 y 2bn−1 = cn−1, se cumple que b2n−1 = 0 en

H∗(B(n, 2j)). Por lo tanto ψn|Bbn−1 es biyectiva y su imagen esta enAbn−1. Ası, ψn es isomorfismo.

Si r es par, digamos r = 2j, entonces de acuerdo al Lema 3.3 la clasede Euler de la fibracion S2j−1 → B(n, 2j−1) → B(n, 2j) es cj−2aj y envista de que ψn es inyectiva, se cumple que el morfismo de multiplicacionpor la clase de Euler es inyectivo, luego el morfismo φ de la sucesion deGysin de la fibracion S2j−1 → B(n, 2j − 1) → B(n, 2j) es trivial lo cualproduce los isomorfismos de grupos

H∗(B(n, 2j − 1)) ∼= H∗(B(n, 2j))/ < cj − 2aj >∼= Z[c1, . . . , cj−1] ⊗ ∆(bj , bj+1, . . . , bn−1).


Lema 3.6 Para cada n ≥ 1 y 1 ≤ j ≤ n − 1, el elemento aj ∈H2j(B(n, 2j); Z) satisface

a2j = ajcj + (−1)j

min(2j,n−1)

r=j+1

(−1)rbrc2j−r.

Demostracion: Recordemos que la clase de Euler ej ∈ H2j(BSO(2j))satisface la relacion

e2j = pj

y que

r∗n(pj) = c2j + (−1)j

min(2j,n)

r=j+1

(−1)r2crc2j−r

en H4j(BU(n)) (Observacion 2.2). Ası,

f∗2j(e

2j ) = c2

j − 4ajcj + 4a2j

y

r∗n(pj) = c2j + (−1)j

min(2j,n−1)

r=j+1

(−1)r2crc2j−r

= c2j + (−1)j

min(2j,n−1)

r=j+1

(−1)r4brc2j−r.

El resultado es ya inmediato pues H∗(B(n, 2j)) es libre de torsion.

Observacion 3.7 De nuevo hacemos tres observaciones utiles para lademostracion del siguiente resultado.

a) La funcion h∗ de la Observacion 3.4 define un isomorfismo hastadimension 2n − 3. (o < 2(n − 1)).

b) Considere el siguiente diagrama

B(n, 2j − 2) × CP∞ f2j−2×1BSO(2j − 2)×CP∞

B(n + 1, 2j)f2j

BSO(2j)

BU(n) × CP∞ BU(n + 1) BSO(2n + 2)


donde el resto de las funciones son canonicas. En esta situacionel cuadro externo es homotopicamente conmutativo y en vista deque BSO(2j) → BSO(2n + 2) es una fibracion podemos sustituirla funcion B(n, 2j−2)×CP∞ → BSO(2j−2)×CP∞ → BSO(2j)por una que haga tal cuadro extrictamente conmutativo. De estamanera existe una funcion f de tal suerte que los cuadros

B(n, 2j − 2) × CP∞

f

BSO(2j − 2) × CP∞

B(n + 1, 2j) BSO(2j)

B(n, 2j − 2) × CP∞ fB(n + 1, 2j)

BU(n) × CP∞ BU(n + 1)

sean (homotopicamente) conmutativos.

Analizaremos cual es el efecto de f∗ sobre H∗(B(n + 1, 2j)). Ob-serve que la funcion BU(n) × CP∞ → BU(n + 1) muestra que

c(γn × γ1) = c(π∗n(γn) ⊕ π∗

1(γ1))

= c(π∗n(γn))c(π∗

1(γ1))

= (c(γn) ⊗ 1) · (1 ⊗ c(γ1))

= c(γn) ⊗ c(γ1)

donde πn : BU(n) × CP∞ → BU(n) y π1 : BU(n) × CP∞ →CP∞ son las proyecciones canonicas, ası f∗(ci) = ci + ci−1z enH∗(B(n, 2j − 2)× CP∞) ∼= H∗(B(n, 2j − 2)) ⊗H∗(CP∞), dondez es el generador de H∗(CP∞) y 1 ≤ i ≤ j.Recordemos que cj = 2bj en B(n, 2j − 2). Ahora, puesto quef∗

ej(aj) = cj − 2aj y la funcion BSO(2j − 2) × CP∞ → BSO(2j)envıa e(λ2j) en (e(λ2j−2)⊗ 1) · (1⊗ z), se tiene que 2f∗(aj) = cj +2aj−1 y en consecuencia f∗(aj) = bj + aj−1z (pues H∗(B(n, 2j −2) × CP∞) es libre de torsion).Por otra parte, la funcion B(n, 2j − 2) × CP∞ → B(n + 1, 2j) →B(n + 1, 2i) se factoriza B(n, 2j − 2) × CP∞ → B(n, 2i − 2) ×CP∞ → B(n+1, 2i) donde la ultima funcion de esta composiciones una version de f para j = i. Utilizando este hecho tenemos que


f∗(bi) = bi + bi−1z para j + 1 ≤ i ≤ n − 1 y que f∗(bn) = bn−1zpues bn = 0 en H∗(B(n, 2n− 2)) de acuerdo a la Observacion 3.4.

c) Sea G = Z/2[c1, . . . , cj−1] ⊗ (Z/2 < aj > ⊕Z/2 < bj+1 > · · · ⊕Z/2 < bn >) → H∗(B(n+1), 2j); Z/2) y observe que la restriccionde f∗ a G es inyectiva.

Teorema 3.8 Para cada n ≥ 1, 1 ≤ j ≤ n − 1 y cada 0 ≤ k ≤ j setiene la siguiente relacion en H∗(B(n, 2j); Z/2)

Sq2k(aj) =k−1

r=max(0,k+j+1−n)

j − r

k − rbk+j−rcr + ajck.

Demostracion: Procederemos por induccion sobre n. El caso en quen = 1 se verifica trivialmente, mientras que para n = 2 se tiene quej = 1 y k = 0, 1. Pero Sq2(a1) = a2

1 = a1c1 lo cual coincide con nuestrarelacion.

Suponga cierto el resultado para valores menores que n + 1. Paraaj ∈ H2j(B(n+1, 2j)) se cumple que Sq2k(aj) ∈ H2(k+j)(B(n+1, 2j)),ası de acuerdo al isomorfismo de la Observacion 3.7 nuestra relacion esvalida para k + j ≤ n − 1. Para k + j ≥ n consideraremos la situacionde la Observacion 3.7. Primero analicemos el caso en que j > 1, k < jy k + j ≥ n, de acuerdo a nuestra hipotesis de induccion se tiene que

h∗Sq2k(aj) = Sq2k(h∗(aj))

= Sq2k(aj)

=k−1

r=k+j+1−n

j − r

k − rbj+k−rcr + ajck

y puesto que ker(h∗) = bnH∗(B(n, 2j)) tenemos

Sq2k(aj) =k−1

r=k+j+1−n

j − r

k − rbj+k−rcr + ajck + bnp(c1, . . . , cj+k−n)

donde p ∈ Z/2[c1, . . . , cj−1]. Luego, Sq2k(aj) ∈ G y

f∗Sq2k(aj) = Sq2k(f∗(aj))

= Sq2k(bj + aj−1z)

= Sq2k(bj) + Sq2k(aj−1)z + Sq2k−2(aj−1)z2.


Aplicando de nuevo la hipotesis de induccion, vemos que f∗Sq2k(aj)esta dado por

k

r=k+j+1−n

j − r

k − rbj+k−rcr +

k−1

r=k+j−n

j − r − 1

k − rbj+k−r−1crz

+k−2

r=max(0,k+j−n−1)

j − r − 1

k − r − 1bj+k−r−2crz

2 + (aj−1ckz + aj−1ck−1z2)

y como j−r−1k−r ≡ j−r−1

k−r−1 + j−rk−r (2), f∗Sq2k(aj) ahora es

k

r=k+j+1−n

j − r

k − rbj+k−r(cr + cr−1z) +

k−1

r=k+j−n

j − r

k − rbj+k−r−1crz

+k−1

r=max1,k+j−n

j − r

k − rbj+k−r−1cr−1z

2 + aj−1z(ck + ck−1z)

Si k + j > n, se tiene

f∗Sq2k(aj) =k

r=k+j+1−n

j − r

k − rbj+k−r(cr + cr−1z)

+k−1

r=k+j−n

j − r

k − rbj+k−r−1z(cr + cr−1z) + aj−1z(ck + ck−1z)

=k−1

r=k+j+1−n

j − r

k − r(bj+k−r + bk+j−r−1z)(cr + cr−1z)

+(bj + aj−1z)(ck + ck−1z) +n − k

n − jbn−1z(cj+k−n + cj+k−n−1z)

= f∗

⎛

⎝k−1

r=max(0,k+j−n)

j − r

k − rbk+j−rcr + ajck

⎞

⎠

que es nuestro resultado ya que f∗|G es inyectiva. Si k + j = n, pro-cedemos de manera analoga como arriba. Restan dos casos, el primeropara j = 1 y el segundo para j = k. En el primer caso la relaciondeseada se verifica facilmente, mientras que el segundo es consecuenciadel Lema 3.6 y que Sq2j(aj) = a2

j .


3.2 El problema de inmersion

A continuacion mencionamos como un sistema de Moore-Postnikovesta relacionado con el problema de levantamiento.

Proposicion 3.9 Sea g : W → Y con W un CW -complejo finito. Si

Zn,αn,βn es un sistema de Moore-Postnikov de F → Xf→ Y , en-

tonces una condicion suficiente para que g levante a traves de f es queH∗(W,π∗−1(F )) = 0.

Una aplicacion de este resultado la podemos hacer al siguiente pro-blema. Cada haz vectorial complejo estable se clasifica a traves deuna funcion X → BU . Si X es un CW complejo finito, digamos dedimension N , la pregunta es si podemos levantar la funcion de clasifi-cacion a algun BU(n) y si es el caso como depende n de N . Notese queun tal levantamiento siempre es posible para n suficientemente grandepues X es compacto y BU esta dotado con la topologıa union de losBU(n).

Puesto que la fibra W de BU(n) → BU es ∪k≥0Wn+k,k y cadaWn+k,k es 2n-conexo, se tiene que W es 2n-conexo. Ası, para q ≤2n+1 tenemos Hq(X,πq−1(W )) = 0, y si suponemos que n ≥ 1

2(N − 1)entonces Hq(X,πq−1(W )) = 0 para q ≥ 2n + 2, por lo que tendremosun levantamiento a BU(n).

Un razonamiento analogo al anterior muestra que si X es un CW -complejo de dimension N y existe un haz complejo ξ sobre X con funcionclasificante X → BU(M) y M > N , entonces la funcion clasificante deξ levanta a BU(n), para 2n + 1 ≥ N , produciendo un isomorfismoξ = η ⊕ (M − n) para η un haz complejo sobre X.

En particular, cualquier haz complejo estable sobre L2n+1(2m) seclasifica por una funcion L2n+1(2m) → BU(n). El trabajo de estaseccion puede ser interpretado como un metodo para detectar obstruc-ciones no triviales para levantar levantar haces L2n+1(2m) → BU(n) aalgun BU(m) con m < n.

Enunciamos de nuevo el resultado principal de este trabajo.

Teorema 3.10 Sea m ≥ 2 y l(n) = max1 ≤ i ≤ n − 1 : n+i+1n ≡

0 (4).

a) Si n = 2s + 1 y n ≥ 2, entonces L2n+1(2m) ⊆ R2n+1+2l(n).


b) Si n = 2s + 1, con s ≥ 1, entonces L2n+1(2m) ⊆ R2n+2l(n) =R4n−4.

Corolario 3.11 Sea m ≥ 2

a) Si n = 2s con s ≥ 1, entonces L2n+1(2m) ⊆ R4n−1.

b) Si n = 2s + 2t con s > t ≥ 1, entonces L2n+1(2m) ⊆ R4n−3.

En la demostracion de este teorema utilizaremos los isomorfismoscanonicos τCP n ⊕ 1C = (n + 1)γn y τL2n+1(2m) ⊕ 1R = (n + 1)σ, dondeσ = q∗(r(γn)) y q : L2n+1(2m) → CPn la proyeccion canonica. Ob-serve que estos isomorfimos muestran que si gd(νCP n) = M , entoncesgd(νL2n+1(2m)) ≤ M . Esto ultimo quiere decir que si CPn ⊆ R2n+k,

entonces L2n+1(2m) ⊆ R2n+1+k. Por otra parte, Milgram probo en[11] que CPn ⊆ R4n−α(n)+1 y que CPn ⊆ R4n−α(n) si n es impar o siα(n) = 1. Ası, cuando α(n) = 1 tenemos que L2n+1(2m) ⊆ R4n lo cualimplica que en este caso nuestro resultado es optimo.

Por otra parte, en [13] Sanderson y Schwarzenberger demostraronque CPn ⊆ R4n−2α(n)−1 y que CPn ⊆ R4n−2α(n)+ si α(n) = 1 o sin es par con α(n) ≡ 0 (4), donde = 0 si α(n) ≡ 1 (4), y = 1 siα(n) ≡ 2, 3 (4).

Algunos casos del trabajo de Sanderson-Schwarzenberger puedenobtenerse (o compararse a partir de cierto punto) con el Teorema 3.10,por ejemplo:

a) Si α(n) = 1, de acuerdo a nuestro resultado CPn ⊆ R4n−2 el cuales optimo en vista de la inmersion de Milgram y coincide con lano-inmersion de Sanderson-Schwarzenberger.

b) Para n = 2s + 1 nuestro resultado afirma que L2n+1(2m) ⊆ R4n−4

y en consecuencia CPn ⊆ R4n−5 que coincide tambien con el re-sultado de Sanderson-Schwarzenberger.

c) Para n = 2r +2s nuestro resultado afirma que L2n+1(2m) ⊆ R4n−3

y en consecuencia CPn ⊆ R4n−4 mientras que el resultado deSanderson-Schwarzenberger afirma que CPn ⊆ R4n−3.

d) Para n = 2r + 2s + 1 nuestro resultado afirma que L2n+1(2m) ⊆R4n−7 y en consecuencia CPn ⊆ R4n−8 mientras que el resultadode Sanderson-Schwarzenberger afirma que CPn ⊆ R4n−7.


Finalmente, Davis y Mahowald [1] demostraron que para α(n) = 2y n impar CPn ⊆ R4n−3 y si n es par CPn ⊆ R4n−2. Por lo que en lasituacion α(n) = 2 nuestro resultado de no-inmersion dista a lo mas enuna unidad de la situacion optima.

Observacion 3.12 Haremos a continuacion algunas observaciones so-bre l(n). Denotamos por ν(n) al exponente de 2 en n. Usaremos lasidentidades ν a

b = α(b) + α(a− b)−α(a) y α(a− 1) = α(a)− 1 + ν(a).

a) Como ν 2nn = α(n), entonces l(n) = n − 1 para α(n) = 1.

b) Como ν 2n−1n = α(n) − 1, entonces l(n) = n − 2 para α(n) = 2.

c) Como ν 2n−2n = α(n − 2) + 1 − ν(n), entonces l(n) < n − 3 para

n impar y α(n) ≥ 3.

Lema 3.13 Si n = 2s1 + · · · + 2sk con s1 > · · · > sk ≥ 0 y k ≥ 3,entonces

l(n) = 2s1 + 2s2 − 2 − 2s3 − · · ·− 2sk .

Demostracion: Usaremos el hecho bien conocido de que ν ab es el

numero de acarreos que ocurren al realizar la resta de las expresionesbinarias de a y b. Sea r = 2s1 + 2s2 − 2 − 2s3 − · · · − 2sk . Entoncesn + r + 1 = 2s1+1 + 2s2+1 − 1, por lo que el maximo numero de acarreosen la resta de n+ r+1 y n es precisamente uno, en la posicion s1, comose muestra en la figura (ya sea que s1 = s2 + 1 o s1 > s2 + 1)

s1 + 1 s1 · · · s2 + 1 s2 · · · sk

1 0 0 1 1 n + r + 11 0 1 1 n

Por lo que r ≤ l(n). Veamos que r es de hecho l(n). Analicemos, paraejemplificar, la diferencia (n+ r +2)−n. En este caso (n+ r +2)−n =2s1+1 + 2s2+1 + 1 y en tal situacion la figura

s1 + 1 s1 · · · s2 + 1 s2 · · · sk

1 0 1 0 0 n + r + 21 0 1 1 n

muestra que hay un acarreo en la posicion sk y otro, en el peor de loscasos, en s2. Consideremos en adelante la diferencia (n+r+k+1)−n =


r + k + 1 para k ≥ 1 la cual, de acuerdo a nuestra definicion de l(n),debe ser a lo mas n.

Supongamos que s1 = s2+1 y no existen acarreos en (n+r+k+1)−n,entonces (n + r + k + 1) − n ≥ 2s1+1 > n que es una contradiccion. Deexistir solo un acarreo tal situacion forza a que n + r + k + 1 tenga 1 enla posicion s2 (de lo contrario habrıa dos o mas acarreos)

s1 + 1 s1 s2 · · · sk

1 1 1 n + r + k + 10 1 1 1 n

ası (n + r + k + 1) − n ≥ 2s1+1 > n.

Supongamos que s1 > s2 + 1, entonces existe un acarreo en (n + r +k + 1) − n en la posicion s1

s1 + 1 s1 · · · s2 + 1 s2 · · · sk

1 0 1 n + r + k + 11 0 1 1 n

y de no existir mas acarreos (n + r + k + 1) − n ≥ 2s1 + 2s2+1 > n. Porlo tanto l(n) = r.

Esta descripcion de l(n) para α(n) ≥ 3 nos dice que

l(n) ≡

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

2 (4) si n ≡ 0 (4)1 (4) si n ≡ 1 (4)0 (4) si n ≡ 2 (4)3 (4) si n ≡ 3 (4)

Demostracion del Teorema 3.10

Puesto que τL2n+1(2m) ⊕ 1 = (n + 1)σ, L2n+1(2m) ⊆ R2n+1+k si ysolo si, se tiene un levantamiento

BSO(k)

−(n + 1)σ) : L2n+1(2m) BSO


Al inicio de esta seccion se discutio la existencia de un levantamiento gde −(n+1)(γ) a BU(n+1), de modo que se tiene el siguiente diagrama

B(n + 1, k)

p

BSO(k)

L2n+1(2m)

fk

g BU(n + 1) BSO(2n + 2)

Observe que g∗(ci) = ci(−(n + 1)γ) = −n−1i zi = (−1)i n+i

n zi pues

c(−(n + 1)γ) = c(γ)−n−1 = (1 + z)−n−1 = i≥0−n−1

i zi.

Para i ≥ [k2 ] + 1 se tiene que p∗(ci) = 2bi ∈ H∗(B(n + 1, k)), luego

2f∗k (bi) = g∗(ci) = (−1)i

n + i

nzi (2m).

Esta ultima identidad implica que n+in es par y que

2f∗k (bi) = 2(−1)i

1

2

n + i

nzi (2m).

Ası

f∗k (bi) =

1

2

n + i

nzi (2m−1)

y dado que m ≥ 2 se obtiene en particular

f∗k (bi) =

1

2

n + i

nzi (2).(1)

Ahora, si k = 2i y ai ∈ H2i(B(n + 1, 2i)) con i ≤ n, entoncesf∗

k (ai) = λizi ∈ H2i(L2n+1; Z) con λi ∈ Z/2m.

Como Sq2(ai) = ibi+1 + aic1 y Sq2(λizi) = iλizi+1, al aplicar f∗2i a

Sq2(ai) se tiene la siguiente identidad

iλi ≡ (n + 1)λi + i1

2

n + i + 1

n(2)(2)

que es valida para i + 1 ≤ n.


Similarmente, como Sq4(ai) = i2 bi+2 + (i − 1)bi+1c1 + aic2 y

Sq4(λizi) = i2 λizi+2, al aplicar f∗

2i a Sq4(ai) se tiene la siguiente iden-tidad modulo 2

i

2λi≡

n+2

2λi+(i − 1)(n + 1)

1

2

n+i+1

n+

i

2

1

2

n+i+2

n(3)

que es valida para i + 2 ≤ n.

Tomemos en adelante i = l(n) y m ≥ 2, por lo que (1) para i + 1,y en consecuencia (2) y (3), es valida, en caso de existir f2i. De hecho,de acuerdo a la definicion de l(n), el tercer sumando de la ecuacion (3)es nulo, por lo que esta se transforma en

i

2λi ≡

n + 2

2λi + (i − 1)(n + 1)

1

2

n + i + 1

n(2).(4)

Si n = 2s con s ≥ 1, en este caso tenemos que i = l(n) = n − 1,entonces la ecuacion (2) nos dice que

1

2

n + i + 1

n≡ 0 (2)

lo cual contradice la definicion de l(n).

Si n es par y α(n) ≥ 2, entonces (ver los comentarios siguientes alLema 3.13 y (b) de la Observacion 3.12) i = l(n) es par y ≤ n − 2.Entonces, de acuerdo a la ecuacion (2), tenemos que λi ≡ 0 (2), y enconsecuencia la ecuacion (4) nos dice

0 ≡ 1

2

n + i + 1

n(2)

lo cual contradice la definicion de l(n).

Si n es impar y α(n) ≥ 3, tenemos que (ver (c) de la Observacion 3.12y los comentarios siguientes al Lema 3.13) i = l(n) < n − 3 es impar,que

λi ≡1

2

n + i + 1

n(2)(5)

y quei

2λi ≡

n + 2

2λi (2).(6)


Por otro lado n ≡ 1, 3 (4). Si n ≡ 1 (4), entonces l(n) ≡ 1 (4), ν i2 ≥ 1

y ν n+22 = 0. Utilizando la ecuacion (6) tenemos que λi ≡ 0 (2) y de

acuerdo a (5)

0 ≡ 1

2

n + i + 1

n(2)

lo cual contradice la definicion de l(n). Si n ≡ 3 (4) , entonces l(n) ≡3 (4), ν i

2 = 0 y ν n+22 ≥ 1. Utilizando (6) tenemos que λi ≡ 0 (2) lo

cual, de acuerdo a (5) contradice la definicion de l(n). Esto concluye lademostracion del primer inciso.

Si n = 2s + 1 con s ≥ 1 y L2n+1(2m) ⊆ R2n+1+2(n−2)−1 se tiene elsiguiente diagrama

B(n + 1, 2(n − 2) − 1)

p2n−5

BSO(2(n − 2) − 1)

B(n + 1, 2(n − 2))

L2n+1(2m)f2(n−2)

f2(n−2)−1

g BU(n + 1) BSO(2n + 2)

La ecuacion (2) toma ahora la forma (ver inciso (b) de la Obser-vacion 3.12)

λn−2 ≡ 1

2

n + i + 1

n≡ 1

2

2n − 1

n(2)

donde ν 2n−1n = 1, por lo que λn−2 ≡ 1 (2), y ademas

λn−2zn−2 = f∗

2(n−2)(an−2)

= f∗2(n−2)−1(p

∗2n−5(an−2))

= f∗2(n−2)−1(bn−2)

=1

2

2n − 2

nzn−2

donde ν 2n−2n = 2 + α(n − 2) − 1 ≥ 2, por lo que zn−2 = 0, que es una

contradiccion. Con esto concluye la demostracion del Teorema 3.10.


Observacion 3.14 En la demostracion del Teorema 3.10 solo se uti-lizaron relaciones obtenidas a partir de Sq2 y Sq4 mientras que en elTeorema 3.8 se obtuvo Sq2k(aj) para k ≥ 1. Surge la pregunta natural:que ocurre con las demas relaciones que proporcionan Sq2k para k ≥ 3.

Utilizando la misma notacion que en el Teorema anterior y de acuerdoal Teorema 3.8 se tiene que para k + i ≤ n

i

kλi ≡

i

k

1

2

n + i + k

n+

i − 1

k − 1

1

2

n + i + k − 1

n

n + 1

n+ . . .

+i − k + 2

2

1

2

n + i + 2

n

n + k − 2

n

+(i − k + 1)1

2

n + i + 1

n

n + k − 1

n+ λi

n + k

n(2).

Lo cual, de acuerdo a la definicion de l(n), se convierte en la relacion

i

kλi ≡ (i − k + 1)

1

2

n + i + 1

n

n + k − 1

n+ λi

n + k

n(2).(7)

Cuando n + 1 ≤ k + i la relacion que obtenemos es

0 ≡ (i − k + 1)1

2

n + i + 1

n

n + k − 1

n+ λi

n + k

n(2)(8)

pues Sq2k(ai) = Sq2k(zi) = 0. Cuando n es par y α(n) ≥ 2 vimos en

la demostracion del Teorema que i = l(n) es par ≤ n − 2 y que λi ≡ 0,luego las ecuaciones (7) y (8) se transforman en

0 ≡ (k + 1)1

2

n + i + 1

n

n + k − 1

n(9)

la cual solo tiene interes de ser estudiada para k par, ademas de quek esta restringido por la relacion k ≤ i = l(n). Recuerdese que 2i esla codimension de la inmersion supuesta. De modo que la ecuacion(9) no conduce a mejora alguna de nuestro resultado. Similarmente,para n impar y α(n) ≥ 3 los terminos que involucran a i = l(n) en lasecuaciones (7) y (8) no conducen a mejora alguna.

Cuando n = 2s + 1 se tiene que i = l(n) = n− 2 y la ecuacion (7) esvalida para k ≤ 2. El caso k = 1 fue estudiado en la prueba, mientras


que en el caso k = 2 la ecuacion (7) es la ecuacion (2.4) que se reduce ala relacion i

2 ≡ n+22 que no tiene mayor importacia. Por otro lado, la

ecuacion (8) toma la forma 0 ≡ k n+k−1n pues 1

22n−1

n ≡ 1, y esta solo

tendrıa interes para k impar, pero en tal caso ν n+k−1n ≥ 1.

Con este argumento queda eliminada la posibilidad de mejorar nue-stro resultado con las relaciones que producen Sq2k(ai) con k ≥ 3.

AgradecimientosEl autor agradece al Dr. Jesus Gonzalez su apoyo y valiosa direccion

durante la elaboracion de este trabajo.

Enrique Torres GieseMathematics Department,University of Wisconsin-Madison,480 Lincoln Dr,Madison, WI 53706,[email protected]

Referencias

[1] Davis D. M.; Mahowald M. E., Immersions of complex projec-tive spaces and the generalized vector field problem. Proc. LondonMath. Soc., 35 (1977), 333–344.

[2] Dold A., Lectures on Algebraic Topology. Springer-Verlag, Berlin-New York, 1980.

[3] Gonzalez J., Connective K-theoretic Euler classes and nonimmer-sions of 2k lens spaces. J. London Math. Soc., 63 (2001), 247-256.

[4] Gonzalez J., Inmersion de variedades: Una excursion homotopica,Aportaciones Matematicas. Comunicaciones 30. Memorias delXXXIV Congreso Nacional de la Sociedad Matematica Mexicana,(2002), 131-164.

[5] Hatcher A., Algebraic Topology. Disponible a traves de la red en:http://www.math.cornell.edu/˜hatcher.

[6] Hatcher A., Vector Bundles and K-theory. Disponible a traves dela red en: http://www.math.cornell.edu/˜hatcher.

[7] Hirsch M. W., Immersions of manifolds. Trans. Amer. Math.Soc., 93 (1959), 242–276.


[8] Husemoller D., Fibre Bundles. Graduate Texts in Mathematics,Springer–Verlag, 1993.

[9] Junod B., A non-immersion result for Lens Spaces Ln(2m). Math.J. Okayama Univ., 37 (1995), 137–151.

[10] McCleary J., A User’s Guide to Spectral Sequences. CambridgeStudies in Advanced Mathematics. Cambridge University Press,2000.

[11] Milgram R. J., Immersing projective spaces. Ann. of Math, 85(1967), 473-482.

[12] Mosher R. E.; Tangora M. C., Cohomology Operations and Appli-cations in Homotopy Theory. Harper & Row, New York-London,1968.

[13] Sanderson B. J.; Schwarzenberger R. L. E., Non-immersion the-orems for differentiable manifolds. Proc. Cambridge Philos. Soc.,59 (1963), 222–319.

[14] Spanier E., Algebraic Topology. Mc Graw–Hill. New York, 1966.

[15] Switzer R., Algebraic Topology–Homotopy and Homology.Springer-Verlag, New York, 1975.

Morfismos, Comunicaciones Estudiantiles del Departamento de Matematicas delCINVESTAV, se termino de imprimir en el mes de noviembre de 2004 en el tallerde reproduccion del mismo departamento localizado en Av. IPN 2508, Col. SanPedro Zacatenco, Mexico, D.F. 07300. El tiraje en papel opalina importada de 36kilogramos de 34 × 25.5 cm consta de 500 ejemplares en pasta tintoreto color verde.

Apoyo tecnico: Omar Hernandez Orozco.

Contenido

Multiobjective Markov Control Processes: a Linear Programming Approach

Onesimo Hernandez-Lerma and Rosario Romera . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Tutte uniqueness of locally grid graphs

D. Garijo, A. Marquez and M.P. Revuelta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35


Enrique Torres Giese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

morfismos, vol 8, no 1, 2004

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