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BIFURCATIONS OF NOVIKOV COMPLEXES FOR

CIRCLE-VALUED MORSE FUNCTIONS

TADAYUKI WATANABE

Abstract. We describe bifurcations of Novikov complexes for pairs of circle-valued Morse functions and metrics on a closed manifold. We also discuss the

rationality properties of the chain equivalences induced from generic homo-topies.

1. Introduction

The Novikov complex (CN∗, ∂) is defined for a generic pair (f, η) of S1-valued

Morse function f : M → S1 on a closed manifold and a Riemannian metric η on M ,

by counting gradient flow lines between critical points ([No]). The homology of the

Novikov complex is called the Novikov homology, that is a topological invariant of

pairs (M, [f ] ∈ [M,S1] = H1(M)). In this paper, we study algebraic properties of

generic homotopies between (M,f1) and (M,f2) with f1 ' f2 through the Novikov

complex functor for pairs (f, η).

In [Hu], M. Hutchings studied bifurcations in a generic 1-parameter family of

S1-valued smooth functions on M , generalizing the method of J. Cerf ([Ce]). Us-

ing his bifurcation analysis, Hutchings gave an a priori proof of invariance of the

Hutchings–Lee invariant, that is the product of the Reidemeister torsion of the

Novikov complex and the zeta function which counts closed orbits of the gradient

flow. In this paper, we give an explicit formula for the changes in the Novikov

complexes for the possible bifurcations of S1-valued Morse pairs that are listed in

[Hu] but partially not explicit there.

There is one more content in this paper, the rationality of the chain equivalences

induced from generic homotopies in the space of smooth S1-valued functions and

metrics on a manifold. In [Pa2], A. Pajitnov proved that the incidence coefficients

of the boundary of the Novikov complex for ‘generic’ S1-valued Morse pairs, i.e.

Pajitnov’s ‘condition (C′)’, are rational functions on t (see also [Pa1, Pa3]). The

condition (C′) is a convenient handle structures on a level surface of an S1-valued

function (see §7). We apply his idea of the proof to 1-parameter family of S1-

valued smooth pairs and proved that two Novikov complexes obtained from S1-

valued Morse pairs satisfying Pajitnov’s condition (C′) are related to each other by

a finite sequence of rational morphisms and that there exists a homotopy which

Date: February 7, 2013.

2000 Mathematics Subject Classification. 57M27, 57R57, 58D29, 58E05.

1

2 TADAYUKI WATANABE

gives the rational morphisms. We give an example which demonstrate the proof of

the rationality of morphism.

2. Preliminaries on circle-valued Morse theory

We review some terminologies of circle-valued Morse theory.

2.1. Circle-valued Morse function. Let M be a d-dimensional closed manifold

and let f : M → S1 be a Morse map, i.e., a smooth map that has only nondegenerate

critical points. Let π : M → M be a connected infinite cyclic covering, which is

the pullback of the Z-covering R → R/Z = S1 by f . Namely, there exists a lift

F : M → R, which makes the following diagram commutative:

MF //

π

��

R

��M

f // S1

The group Z = 〈t〉 ∼= π1(S1) acts naturally on M by the covering transformation

that is the downward shift and Z also acts on R by tn · x 7→ x − n. Then F is

Z-equivariant with respect to these actions. Throughout this paper, we will write

an S1-valued map by a lower-case letter, like f , and its Z-equivariant lift by the

corresponding capital letter, like F .

2.2. Novikov complex. Let η be a Riemannian metric on M . If (f, η) is such that

all the descending manifolds and the ascending manifolds of the gradient gradηf

intersect transversally, then the space M ′(F ; p, q) of unparametrized flow lines be-

tween p and q has a canonical compactification to a compact smooth manifold with

corners, denoted M ′(F ; p, q). The boundary of M ′(F ; p, q) consists of broken flow

lines broken at other critical points. In particular, when ind(p) = ind(q) + 1, the

space M ′(F ; p, q) is a compact 0-dimensional manifold. So it can be counted with

signs if M ′(F ; p, q) is oriented suitably. For the topology of the space of flow lines

M ′(F ; p, q) etc. and their compactifications, see e.g. [BH].

Let L denote the formal Laurent series ring Z((t)) = lim←−nZ[t, t−1]/tnZ[t] and

the Novikov complex for a generic Morse pair (f, η) is defined by

CNk(f) = ZPk(F ) ⊗Z[t,t−1] L,

∂(p) =∑

q∈Pk−1(F )

#M ′(F ; p, q) · q if p ∈ Pk(F ),

where Pk(F ) is the set of critical points of F of index k. We will also write PN∗(f) =

P∗(F ). Then the Novikov homology H(CN∗(f), ∂) is defined.

3. Results

3.1. Bifurcations of Novikov complexes. Since the Novikov ring and Novikov

complexes are defined by inverse limits, they possess the inverse limit topology and

convergences of elements, morphisms etc. make sense. For a 1-parameter family

A(s) of objects parametrized by a real number s, we write A−(λ) = lims↑λ A(s)

BIFURCATIONS OF NOVIKOV COMPLEXES 3

and A+(λ) = lims↓λ A(s) if the limits exist. If moreover A−(λ) = A+(λ), we write

A(λ) = A−(λ) = A+(λ). A 1-parameter family of pairs is a family (fJ , ηJ) =

{(fs, ηs); s ∈ J = [s0, s1]}, such that the map fJ : J × M → S1 defined by

fJ (s, x) = fs(x) is smooth and ηJ forms a (smooth) metric of J ×M .

Theorem 3.1. Let {(fs, ηs); s ∈ [0, 1]} be a 1-parameter family of pairs of S1-valued

smooth functions and metrics with (f0, η0) and (f1, η1) satisfying the transversality

condition. Then after a perturbation fixing the endpoints, we may arrange that

there is a countable subset Λ ⊂ (0, 1) such that (fs, ηs) is transversal on [0, 1] \ Λand at each point λ ∈ Λ, one of the following holds:

(1) ∂+λ = (1+hpq)

−1∂−λ (1+hpq) or vice versa, where CN−

∗ (λ) = CN+∗ (λ) and

hpq : CN∗(λ)→ CN∗(λ) is a L-linear map defined for p, q ∈ PN∗(λ) by

hpq(x) =

{±q if x = p

0 otherwise

(2) CN+∗ (λ) = CN−

∗ (λ) ⊕ CN elem∗ , ∂+

λ = ∂−λ + ∂elem, or vice versa, where

(CN elem∗ , ∂elem) is the acyclic chain complex 0 → L

{x} → L{y} → 0,

∂elem(x) = y, for some i ∈ Z.

What is new in Theorem 3.1 is the formula for the change at the self i/i-

intersection (for q = tkp in (1), see Lemma 5.1). (An i/i-intersection for a Morse

pair (fs, ηs) is the intersection of Dp(fs) and Aq(fs) for a pair p, q with ind(p) =

ind(q) = i.) Other assertions have been proved by Hutchings in [Hu]. We remark

that Theorem 3.1 is still inefficient to classify Novikov complexes for a manifold

since the ‘composition’ of the chain equivalences of Theorem 3.1 may be countable

infinite products and it is not completely determined what kind of infinite products

can be the induced chain equivalences for homotopies. However it would be useful

in proving the well-definedness of various circle-valued Morse theoretic invariants

of manifolds via a topological field theory method, such as the Novikov homology

or as the invariants in [Hu], [Wa2]. Thus Hutchings’s proof of well-definedness of

the Hutchings–Lee invariant can be simplified by Theorem 3.1. We will give in §8a concrete example of Novikov complex and chain equivalence correponding to a

homotopy (fs, ηs).

Corollary 3.2. The Novikov complexes (CN∗(0), ∂0) and (CN∗(1), ∂1) for two

homotopic transversal S1-valued Morse pairs on a manifold are related by a finite

sequence of the following changes: (CN∗, ∂) 7→ (CN ′∗, ∂

′), where

(1) CN ′∗ = CN∗, ∂′ = Φ−1∂Φ, Φ : CN∗ → CN∗ is a L-linear isomorphism

defined by an admissible product (See §A.1).(2) CN ′

∗ = CN∗ ⊕ CN elem∗ , ∂′ = ∂ + ∂elem, or vice versa.

This corollary has been essentially proved in [Pa4, Hu].

3.2. Rationality of chain equivalence. Although the chain equivalence Φ in

Corollary 3.2 is given by a (possibly infinite) product, it turns out that it has only

finite information if the endpoints satisfy some finiteness condition. The Novikov

complex (CN∗, ∂) of a transversal circle-valued Morse pair is said to be rational

4 TADAYUKI WATANABE

if for some L-basis B of CN∗, that is a finite subset of PN∗, all the incidence

coefficients n(p, q) ∈ L in the expansion

∂(p) =∑

q∈B∩PNi−1(f)

n(p, q) · q (p ∈ B ∩ PNi(f))

are rational functions, namely of the form P (t)Q(t) for polynomials P (t), Q(t) ∈ Z[t]

with Q(0) = 1. S. Novikov’s exponential growth conjecture says that for any

transversal Morse pair (f, η), the power series n(p, q) grow at most exponential

for all p, q with ind(p) = ind(q) + 1. M. Farber and P. Vogel conjectured (unpub-

lished) stronger statement that the rationality of the Novikov complex holds for

every pair (f, η) satisfying the transversality ([Pa1, page 371]). Pajitnov proved

that most of transversal Morse pairs satisfy the Farber–Vogel conjecture:

Theorem 3.3 (Pajitnov [Pa1]). For any compact closed manifold M , there is a pair

of circle-valued Morse function and a metric on M such that its Novikov complex is

rational. Moreover, the rationality property is C0-generic, i.e. C0-open and dense

in the space of gradient-like vector fields.

In view of the possibilities of complicated boundary operators as in Theorem 3.1,

this result is striking. Pajitnov considered what he calls ‘the condition (C′)’, a

condition for the gradient(-like) vector field (see §7.1), and proved that if a S1-

valued Morse pair satisfies the condition (C′), then the associated Novikov complex

is rational. Moreover, he showed that the condition (C′) is C0-generic in the space

of gradient(-like) vector fields.

We consider that Pajitnov’s method would also be useful for a homotopical

version of the conjecture of Farber and Vogel. For a ‘generic’ homotopy h =

{(fs, ηs); s ∈ J} of S1-valued Morse pairs with the endpoints satisfying the condi-

tion (C′), we define the endomorphism Φh : CN∗ → CN∗ of homogeneous degree 0

as follows:

Φh(p) =∑q

#M ′(H; p, q) · q,

where p is a critical locus of H = {Fs} of index k and q ranges over all critical loci

of {Fs} of index k − 1, and M ′(H; p, q) is the space of flow lines going from ps to

qs along −gradηsfs for some s ∈ J . This definition can be generalized to any paths

h in the space of (not necessarily Morse) pairs with endpoints (f0, η0) and (f1, η1),

namely the morphism Φh : CN∗(f0, η0) → CN∗(f1, η1) is defined up to homotopy

fixing the endpoints, that corresponds to the addition of ∂′g + g∂ for some degree

1 map g : CN∗(f0, η0)→ CN∗+1(f1, η1).

The following theorem is an analogue of the theorem of Pajitnov to 1-paramter

families.

Theorem 3.4. If the pairs (f0, η0) and (f1, η1) of S1-valued Morse functions

and metrics both satisfy the condition (C′), then the chain equivalence of Corol-

lary 3.2(1) can be realized as the rational morphism. Moreover, there is a homotopy

h = {(fs, ηs); s ∈ [0, 1]} whose Φh agrees with the rational chain equivalence.

Let FM be the (simplicial) space of generalized S1-valued Morse pairs on M

that are homotopic to a base point pair (f0, η0), where a generalized Morse function


is a smooth function with only Morse or birth-death singularities (see [Ig] for the

homotopy type of the space of R-valued generalized Morse functions). Let FC′

M [1] ⊂FM be the 1-dimensional subcomplex whose set of 0-simplices consists of Morse

pairs satisfying the condition (C′) and whose set of 1-simplices consist of homotopies

satisfying the condition (C′) generalized to 1-parameter families (see Lemma 7.6

for the definition) as in Theorem 3.4. Pajitnov’s theorem above implies that the

inclusion i1 : FC′

M [1]→ FM is 0-connected, i.e.,

π0(FM ,FC′

M [1]) = 0.

Theorem 3.4 can be restated as follows.

Theorem 3.5. The inclusion i1 : FC′

M [1]→ FM is 1-connected, i.e.,

π1(FM ,FC′

M [1]) = 0.

We could extend FC′

M [1] to a simplicial set FC′

M with the set of k-simplices

consists of families of generalized Morse pairs over the k-simplex ∆k satisfying

k-parameter analogue of the condition (C′).

Conjecture 3.6. The inclusion i∞ : FC′

M → FM is a weak homotopy equivalence,

i.e., we have

πk(FM ,FC′

M ) = 0 for k ≥ 0.

One could ask the following stronger conjecture.

Conjecture 3.7. FM = FC′

M .

The conjecture of Farber and Vogel follows immediately from Conjecture 3.7.

In this respect, Conjecture 3.6 can be considered as a homotopical version of the

conjecture of Farber and Vogel.

4. Bifurcation analysis of Cerf and Hutchings

We shall briefly review Hutchings’s bifurcation analysis in [Hu] with sketch proofs

since similar inductive argument will be necessary in §5 and §6.

4.1. Bifurcations in 1-parameter family of real-valued smooth functions

on compact manifold.

Definition 4.1. We say that a pair (f, η) of a smooth function (valued in S1 or

R) and a metric is admissible if it is Morse and satisfies transversality condition. A

bifurcation of a family {(fs, ηs); s ∈ [0, 1]} of pairs is a time λ ∈ [0, 1] such that the

pair (fλ, ηλ) fails to be admissible.

The following proposition is a straightforward generalization of a fundamental

fact in the theory of [Ce], which determines possible bifurcations in a generic 1-

parameter family of smooth functions and which can be proved by a general position

argument in the jet bundle. See [Lau] for a proof.

6 TADAYUKI WATANABE

Proposition 4.2. Let {fs; s ∈ [0, 1]} be a 1-parameter family of R or S1-valued

smooth functions on a compact manifold, with f0 and f1 Morse. Then after a

perturbation of the family fixing the endpoints, we may arrange that fs is Morse for

s in the complement of a finite subset {s1, s2, . . . , sr} of (0, 1), and for each j, fsjhas only one birth-death point and some nondegenerate critical points.

We shall study what happens on the complement of {s1, . . . , sr} of Proposi-

tion 4.2 first and then study each si. By definition, a change of the Novikov

complex (CN∗(s), ∂s) is caused by a change of the topology of the moduli spaces

M ′(fs; p, q) of (unparametrized) flow lines. We study the topology of a cobordism

of the moduli spaces in a 1-paramter family. The following proposition gives a

compact cobordism in R-valued case.

Proposition 4.3. Suppose that a 1-parameter family (fJ , ηJ) = {(fs, ηs); s ∈ J =

[s0, s1]} of R-valued Morse pairs on a compact manifold satisfies the generalized

Morse–Smale condition, i.e., the 1-parameter families of the ascending manifolds

and the descending manifolds, that form disk bundles over J , intersect transversally

in J ×M . Then M ′(fJ ; p, q) has a natural compactification to a compact smooth

manifold with corners, whose boundary consists of the spaces of broken trajectories.

See e.g., [Wa, §3] or [BH] for 0-parameter case. The proof for 1-parameter family

is almost the same as the 0-parameter case. Note that it can be shown by using

the genericity of the tranversality that the generalized Morse–Smale condition is

generic. We denote the compactification by M ′(fJ , p, q).

We say that a bifurcation λ is principal if λ is a bifurcation on which the topology

of the moduli space M ′(fs; p, q) for a pair p, q with ind(p) = ind(q)+1 changes, i.e.,∪s M ′(fs; p, q) is not a (trivial) covering space over any neighborhood of λ. We

say that a flow line of (fs, ηs) between p and q is degenerate if Dp(fs) and Aq(fs)

fails to be transversal in M . The following lemma follows immediately from the

proposition.

Lemma 4.4. Let {(fs, ηs); s ∈ [s0, s1]} be a 1-parameter family of pairs of R-valuedMorse functions and metrics on a compact manifold, with (fs0 , ∂s0) and (fs1 , ∂s1)

admissible. Then after a perturbation of the family fixing the endpoints, we may

arrange that there are only finitely many principal bifurcations over [s0, s1] and that

each of them is one of the following types:

(1) A degenerate flow line from p ∈ Pi to q ∈ Pi−1,

(2) A gradient i/i-intersection between p ∈ Pi and q ∈ Pi, where p 6= q.

We may assume that the perturbation is C∞-small.

4.2. Bifurcations in 1-parameter family of circle-valued Morse pairs. In

a 1-parameter family {(fs, ηs); s ∈ J} of circle-valued Morse pairs, the number of

points in the critical set PN∗(s) does not change through J . We write the set of

critical loci as PNi(J) =∪

s∈J{s} × PNi(s) and CNi(J) = Zπ0PNi(J) in such a

case.

Lemma 4.5. Let (fs, ηs), s ∈ J = [s0, s1], be a 1-parameter family of S1-valued

Morse pairs with (fs0 , ηs0) and (fs1 , ηs1) admissible. Then after a perturbation


fixing the endpoints, we may arrange that there is a contable subset Λ of the interval

(s0, s1) such that (fs, ηs) is admissible on J \Λ and at each point λ ∈ Λ, one of the

following occurs:

(1) A degenerate flow line from p ∈ PNi(J) to q ∈ PNi−1(J),

(2) A gradient i/i-intersection between p ∈ PNi(J) and q ∈ PNi(J), where

π(p) 6= π(q),

(3) A gradient i/i-intersection between p ∈ PNi(J) and tkp ∈ PNi(J) for some

k.

Proof. We assume that s1−s0 is so small that there is a point a ∈ S1 such that a is

regular for fs for all s ∈ J . If s1−s0 is not so small, we may take a finite subdivision

of [s0, s1] with each interval sufficiently small. We put Mi = F−1s0 [a− i− 1, a− i].

By a similar reason as Lemma 4.4, we see that for each N ≥ 0, there is a small

(bouded by ε2N

) perturbation of the family (fs, ηs) fixing the endpoints such that

there is a finitely many principal bifurcations in M0N = M0 ∪M1 ∪M2 ∪ · · · ∪MN

over J . Moreover, this property can be extended to M0,N+1 by additional small

perturbation (bounded by ε2N+1 ) without changing the topologies and times of the

moduli spaces of flow lines in M0N . Then taking N → ∞ by such an inductive

extension, we obtain the desired result. �

Lemma 4.6. Let {(fs, ηs); s ∈ J} be a 1-parameter family of S1-valued Morse pairs

as obtained in Lemma 4.5. Then for each point λ ∈ J , the limits

(CN−∗ (λ), ∂−

λ ) = lims↑λ

(CN∗(s), ∂s), (CN+∗ (λ), ∂+

λ ) = lims↓λ

(CN∗(s), ∂s)

are well-defined, where s ranges over non-bifurcations.

Proof. It is easy to check that CN−∗ (λ) = CN∗(λ − ε) for a small number ε > 0.

For each λ ∈ (0, 1) and for each N ≥ 0, there exists εN > 0 such that on [λ−εN , λ)

there are no principal bifurcations of flow lines in M0N , where M0N is as in the

proof of Lemma 4.5. Let B = M0 ∩ PN−∗ (λ) and let CN ′−

∗ (λ) be the subgroup of

CN−∗ (λ) defined by

CN ′−∗ (λ) =

∑pj∈B

npjpj

∣∣∣∣∣npj ∈ Z[[t]]

.

This is a Z[[t]]-submodule of CNi(λ). Then the limit lims↑λ(CN∗(s), ∂s) is well-

defined in CN ′−∗ (λ)/tN+1CN ′−

∗ (λ) since the Z[[t]]-complex (CN ′−∗ (λ)/tN+1CN ′−

∗ (λ), ∂s)

is constant on [λ−εN , λ). Then one can take inverse limit lim←−Nto get a well-defined

limit in CN ′−∗ (λ). Now, extend CN ′−

∗ (λ) to CN−∗ (λ) by the L-action. The complex

(CN+∗ (λ), ∂+

λ ) is obtained in a similar way. �

4.3. Degenerate flow line. We write

Dp(FJ ) =∪s∈J

Dp(Fs), Ap(FJ) =∪s∈J

Ap(Fs),

both equipped with natural topologies induced from J ×M .

8 TADAYUKI WATANABE

Lemma 4.7. Let {(fs, ηs); s ∈ J} be a 1-parameter family of S1-valued Morse

pairs as obtained in Lemma 4.5 and let λ ∈ Λ be a principal bifurcation at which

a degenerate flow line between p ∈ PNi(λ) and q ∈ PNi−1(λ) occurs, as in (1) of

Lemma 4.5. Then we have

∂+λ = ∂−

λ .

Proof. Take a small number ε > 0 and put J = [λ − ε, λ + ε] so that the change

of the topology of M (Fs; p, q) occurs only at λ. We may assume for the pair

(p, q) that the restriction of the projection to the transversal intersection (Dp(FJ)∩Aq(FJ)) ∩ F−1

J (a), that is a compact oriented 1-manifold of J ×M , is Morse. So

the bifurcation corresponds to a Morse point of the projection of the 1-manifold

on J . When time passes through J , a pair of points with opposite signs appears

or disappears. In each case, the sum does not change, so does not the incidence

coefficient n(Fs; p, q). �

4.4. Gradient i/i-intersection between p and q with π(p) 6= π(q).

Lemma 4.8. Let {(fs, ηs); s ∈ J} be a 1-parameter family of S1-valued Morse

pairs as obtained in Lemma 4.5 and let λ ∈ Λ be a principal bifurcation at which

a gradient i/i-intersection between p ∈ PNi and q ∈ PNi with π(p) 6= π(q) occurs,

as in (2) of Lemma 4.5. Then we have

∂+λ = (1 + hpq)

−1∂−λ (1 + hpq).

Proof. For each sufficiently large N , there is a small number εN > 0 such that for

any 0 < θ ≤ εN the following identity holds in CN ′∗(λ)/t

N+1CN ′∗(λ):

∂λ+θ ∼ (1− hpq)∂λ−θ(1 + hpq),

where CN ′∗(λ) is as above. This can be verified by a similar argument as the R-

valued case for the compact manifold M0N (see [Hu, Lemma 3.4], [Wa, §5]). Note

that (1 + hpq)−1 = 1− hpq since h2

pq = 0. Then take N →∞. �

5. Self i/i-intersection

In this section, we shall prove the following lemma, which can be considered as

an analogue of Lemma 4.8.

Lemma 5.1 (Main Lemma). Let {(fs, ηs); s ∈ J} be a 1-parameter family of S1-

valued Morse pairs as obtained in Lemma 4.5 and suppose that s = λ is a principal

bifurcation at which a self i/i-intersection between critical points (loci) p and tkp

occurs, as in (3) of Lemma 4.5. Let h : CN∗(λ) → CN∗(λ) be the endomorphism

defined by

h = ±tk + t2k ± t3k + · · · ,where t : CN∗(λ)→ CN∗(λ) is the L-linear map defined for x ∈ PN∗(λ) by

t(x) =

{tp x = p

0 x 6= p

Then we have

∂+λ = (1 + h)−1∂−

λ (1 + h) or ∂+λ = (1 + h)∂−

λ (1 + h)−1.


Figure 1

To study the change of the incidence coefficients n(Fs; p, tr q), r ∈ Z, for q with

ind(q) = ind(p)− 1, we study how the numbers of intersections of Dp(Fs) ∩ L and

Atr q(Fs) ∩ L for some level surface L that lies just above tr q changes. To see this,

we need to study the change of Dp(Fs) ∩ L and its natural compactification. We

also study the change of n(Fs; t−r r, p) = n(Fs; r, t

rp), ind(r) = ind(p) + 1 in a

similar way.

5.1. Compactification of Dp(FJ ) in case that fs has only one critical point.

There is a small number ε > 0 such that there are no other principal bifurcation

objects in M0N over J = [λ−ε, λ+ε] than a self i/i-intersection. We put f = fλ−ε,

F = Fλ−ε. First, we consider the movement of the descending manifold at self i/i-

intersection for a Morse function f : M → S1 with only one critical point p, to

explain the idea of the proof. Note that such a Morse function may exist only if M

is not closed.

We denote one of lifts of p in M by p. We assume that the level surface Σ =

f−1(a) lies just above p so that f(p) = a − δ for a small number δ and put Σi =

F−1(a− i) (i ∈ Z), so that Σi+1 = tΣi. Let Mi = F−1[a− i− 1, a− i], which is the

part located between Σi and Σi+1. We put ∂ = ∂λ−ε and ∂′ = ∂λ+ε for simplicity.

We put Lj = F−1(a− j − 2δ).

5.1.1. Movement of Dp(Fs) in two successive compartments. Suppose that in the

1-parameter family {Fs}s∈J the descending manifold of p slides over that of tp at

s = λ. We choose a lift p of p so that F (p) = a − δ. For the 1-parameter family

Fs, s ∈ J , we consider the changes of Dp(Fs) and Atp(Fs) around the time s = λ.

By Morse’s lemma, there is a local coordinate on a fiberwise tubular neighborhood

around the critical point locus of p on which Fs agrees with the standard quadratic

form and we may assume after a suitable perturbation of metric ηs that the gradient

gradηsFs agrees with the gradient of the standard quadratic form with respect to

the local coordinate. So gradηsFs is stationary near the critical points and we may

assume that Atp(Fs)∩Σ1 and Dp(Fs)∩L0 are stationary in Σ1 and L0 respectively.

10 TADAYUKI WATANABE

Moreover, We may assume that the 1-parameter family {Dp(Fs) ∩ Σ1} intersects

Atp(F ) ∩ Σ1 transversally in Σ1.

We shall describe the movement of Dp(Fs) in L1. Let S1 = Dtp(Fs) ∩ L1 =

Dtp(F ) ∩ L1, which is a submanifold diffeomorphic to the (i − 1)-sphere. Let

S(1)0 (s) = Dp(Fs) ∩ L1, s ∈ J \ {λ}. We may assume that the support of the

change of the (negative) gradient vector field in F−1[a − 1 − 2δ, a − δ], which in-

duces the movement of Dp(Fs)∩F−1[a−2, a−δ], is included in F−1(a−1, a−1+δ).

Moreover, if ε is small enough, we may assume that the change of the gradient vec-

tor field is induced from the 1-parameter family of submanifolds Dp(Fs)∩Σ1 of Σ1,

that is diffeomorphic at each s to the (i − 1)-sphere and that the family is an ‘el-

ementary’ deformation (elementary path in [Ce], which can be described explicitly

by using a cloche function). Thus, by using the local chart around tp of Morse’s

lemma, the movement of Dp(Fs) in L1 can be described explicitly as follows. (See

[Wa3, §2.3.2.] for detail.)As s ↑ λ from λ − ε, a part of the sphere S

(1)0 (s) come close to S1 and at

the limit s = λ from below, an (i − 1)-dimensional disk in S(1)0 (s) converges to a

disk R in S1 with respect to the Hausdorff metric. As s ↓ λ from λ + ε, a part

of the sphere S(1)0 (s) come close to S1 and at the limit s = λ from above, an

(i − 1)-dimensional disk in S(1)0 (s) converges to the disk Q = S1 \R in S1, and

lims↓λ S(1)0 (s) \Q = lims↑λ S

(1)0 (s) \R. See Figure 2.

If there were q with ind(q) = ind(p) − 1, satisfying a − 2 < Fs(q) < a − 1 − 2δ

(against the assumption that there are no such things, so the following arguments

shows nothing, although it will help understand the general case), then the incidence

coefficient n(Fs; p, q) would change as follows. The locus of the intersection of

S(1)0 (s) = Dp(Fs) ∩ L1 and Aq(Fs) ∩ L1 in s < λ forms a submanifold of J ×

L1 that has a canonical compactification to a submanifold with boundary {λ} ×lims↑λ S

(1)0 (s) ∩ (Aq(Fλ) ∩ L1). Note that we may assume that the topology of

Aq(Fs) ∩ L1 does not change on (λ − ε, λ + ε). Similarly, the intersection locus

in s > λ has a canonical compactification to a submanifold with boundary {λ} ×lims↓λ S

(1)0 (s)∩ (Aq(Fλ)∩L1). These give the compactification of the moduli space

M ′(FJ ; p, q), namely, M ′(FJ ; p, q) is canonically diffeomorphic to{∪s<λ

(Dp(Fs) ∩ L1) ∪ lims↑λ

S(1)0 (s) ∪ lim

s↓λS(1)0 (s) ∪

∪s>λ

(Dp(Fs) ∩ L1)}∩Aq(Fs).

By counting the points in ∂M ′(FJ ; p, q) with orientation, one obtains the identity:

n(Fλ−ε; p, q)− n(Fλ+ε; p, q)

−#lims↑λ

S(1)0 (s) ∩ (Aq(Fλ) ∩ L1) + # lim

s↓λS(1)0 (s) ∩ (Aq(Fλ) ∩ L1) = 0,

hence

n(Fλ+ε; p, q)− n(Fλ−ε; p, q) = (# lims↓λ

S(1)0 (s)−#lim

s↑λS(1)0 (s)) ∩ (Aq(Fλ) ∩ L1)

= S1 ∩ (Aq(Fλ) ∩ L1) = ±n(Fλ−ε; tp, q).

See [Hu, Lemma 3.4] or [Wa, Lemma 5.5] for the sign in the formula.


Figure 2. Movement of S(1)0 (s) = Dp(Fs) ∩ L1 in s ∈ (λ− ε, λ+ ε)

Figure 3

5.1.2. Movement of Dp(Fs) in all compartments. Next we consider three successive

compartments M02 = M0∪M1∪M2. We assume that the change of Fs on M0∪M1

with respect to s ∈ J is as observed above. Then by Z-equivariance of Fs, the

change of Dp(Fs) on M0 ∪M1 induces a change of Dtp(Fs) on M02.

Let S2 = Dt2p(F ) ∩ L2, S(1)1 (s) = Dtp(Fs) ∩ L2 and S

(2)0 (s) = Dp(Fs) ∩ L2,

s ∈ J \{λ}. Suppose that Dp(Fs) changes in M0∪M1 as §5.1.1 under the change of

gradηsFs. Then by the Z-equivariance of Fs, Dtp(Fs) changes in M1 ∪M2 exactly

the same way as the change of Dp(Fs) ∩ (M0 ∪ M1). On the range s < λ (or


s > λ), the three spheres S2, S(1)1 (s) and S

(2)0 (s) are disjoint and they are deformed

by an isotopy, which means that the topologies of them are unchanged under the

variation of s within the range s < λ (resp. s > λ). Hence the only time on which

the topologies of the three spheres change is s = λ.

Let us describe the change of the three spheres that occurs at s = λ. Due to the

change of gradηsFs, the locus of the sphere At2p(Fs)∩L1 intersects S1 transversally

at s = λ at a single point, say x1, which is disjoint from R(1)0 = R. Hence, there is

an open neighborhood U1 of (S1 \ Bµ(x1)) ∪ lims↑λ S(1)0 (s) in L1, where Bµ(x1) is

an open ball in S1 of radius µ around x1, such that the negative gradient induces a

diffeomorphism φU1 from U1 to an open subset U ′1 of L2, which sends S1 \ Bµ(x1)

to the complement of an open ball in lims↑λ S(1)1 (s) and which sends lims↑λ S

(1)0 (s)

to lims↑λ S(2)0 (s), i.e.,

lims↑λ

S(2)0 (s) = φU1

(lims↑λ

S(1)0 (s)

).

The positions of S2 and S(1)1 (s) in L2 are exactly the same as the positions of S1

and S(1)0 (s) in L1 if projected by π : M →M . Thus we have

S2 = tS1, lims↑λ

S(1)1 (s) = t lim

s↑λS(1)0 (s), lim

s↓λS(1)1 (s) = t lim

s↓λS(1)0 (s).

It remains to describe lims↓λ S(2)0 (s). If we restrict the movement of S

(1)0 (s),

s > λ, to a neighborhood of Bµ(x1), the corresponding restriction of lims↓λ S(2)0 (s)

agrees with a part of lims↓λ S(1)1 (s) which is disjoint from R

(2)0 = φU1(R

(1)0 ). There-

fore,

lims↓λ

S(2)0 (s) =

(lims↓λ

S(1)1 (s) ∪ lim

s↑λS(2)0 (s)

)\ IntR(2)

0 .

See Figure 3. Now all the limits of the three spheres from above/below have been

described. If there were q with ind(q) = ind(p) − 1, satisfying a − 3 < Fs(q) <

a− 2− 2δ, then the intersection of S(2)0 (s) = Dp(Fs)∩L2 and Aq(Fs)∩L2 in s < λ

is a submanifold of L2 that has a canonical compactification to a submanifold

with boundary, that is given by lims↑λ S(2)0 (s) ∩ (Aq(Fλ) ∩ L2). Note that we

may assume that the topology of Aq(Fs) ∩ L2 does not change on (λ − ε, λ +

ε). Similarly, the intersection in s > λ has a canonical compactification to a

submanifold with boundary, that is given by lims↓λ S(2)0 (s)∩ (Aq(Fλ)∩L2). These

give the compactification of the moduli space M ′(FJ ; p, q). By a similar argument

as above, one obtains the identity:

n(Fλ+ε; p, q)− n(Fλ−ε; p, q) = (# lims↓λ

S(2)0 (s)−#lim

s↑λS(2)0 (s)) ∩ (Aq(Fλ) ∩ L2)

= t lims↓λ

S(1)0 (s) ∩ (Aq(Fλ) ∩ L2)

= ±n(Fλ−ε; tp, q) + n(Fλ−ε; t2p, q).

For the movement of Dp(Fs) in M0r = M0 ∪M1 ∪M2 ∪ · · · ∪Mr, we iterate a

similar argument as above. Namely, we have the recursive formula

lims↓λ

S(r)0 (s) =

(lims↓λ

S(r−1)1 (s) ∪ lim

s↑λS(r)0 (s)

)\ IntR(r)

0 ,


where S(j)k (s) = Dtkp(Fs) ∩ Lj+k and

R(j)k = lim

s↑λS(j−1)k+1 (s) ∩ lim

s↑λS(j)k (s).

Hence if there were q with ind(q) = ind(p)−1, satisfying a−r−1 < Fs(q) < a−r−2δ,then one obtains the identity:

n(Fλ+ε; p, q)− n(Fλ−ε; p, q)

= (# lims↓λ

S(r)0 (s)−# lim

s↑λS(r)0 (s)) ∩ (Aq(Fλ) ∩ Lr)

= t lims↓λ

S(r−1)0 (s) ∩ (Aq(Fλ) ∩ Lr)

= ±n(Fλ−ε; tp, q) + n(Fλ−ε; t2p, q)± · · ·+ (±1)rn(Fλ−ε; t

rp, q).

(5.1)

This gives the formula

∂′p− ∂p ∼ ∂hp = (±t+ t2 ± t3 + · · · )∂p

in CN ′i(λ)/t

r+1CN ′i(λ). By considering the same for Ap(Fs), one obtains the for-

mula

∂′r − ∂r ∼ n(Fλ−ε; r, p) · (∓t+ t2 ∓ t3 + · · · )p.

for ind(r) = ind(p) + 1.

5.2. Compactification of Dp(FJ ) for general case. In a general case, p slides

over tkp for some k ≥ 1 and there may be other critical points than those of the

forms tj p, j ∈ Z. The result will be essentially the same as k = 1 case by considering

successive k compartments as a big single compartment, so we may assume k = 1 as

above. Although S(j)k (s) may not be spheres in general, one can prove the formula

(5.1) by an exactly the same argument as above, since we may assume that there

are no other principal bifurcations of height ≤ N than λ in J = [λ− ε, λ+ ε] for ε

small.

As in the previous subsection, the union of∪

s<λ{s}×S(r)0 (s) and {λ}×lims↑λ S

(r)0 (s)

gives a partial compactification of∪

s<λ{s} × S(r)0 (s) into a smooth submanifold

of J × Lr with corners. Similarly, the union of∪

s>λ{s} × S(r)0 (s) and {λ} ×

lims↓λ S(r)0 (s) gives a partial compactification of

∪s>λ{s} × S

(r)0 (s) into a smooth

submanifold of J × Lr with corners. From this and a similar argument as §5.1, weobtain the following identities:

∂′p ∼ (1± tk + t2k ± t3k + · · · )∂p,

∂′r ∼ ∂r + ∂rp · (∓tk + t2k ∓ t3k + · · · )p

in CN ′i(λ)/t

N+1CN ′i(λ). If we put h = ±tk + t2k ± t3k + · · · , then the following

identity follows from the identities above, as for the compact manifold case in [Hu,

Lemma 3.4], [Wa, §5].∂′ − ∂ ∼ h∂ − ∂′h.

Taking the inverse limit, we obtain the identity of Lemma 5.1. This completes the

proof of Lemma 5.1.


6. Birth-death bifurcation

Let λ ∈ [0, 1] be a birth-death bifurcation in a 1-parameter family found in

Proposition 4.2. We consider a 1-parameter family {(fs, ηs); s ∈ J = [λ− ε, λ+ ε]}for a small number ε > 0 so that

• (fλ±ε, ηλ±ε) are admissible,

• there is only one birth-death bifurcation in [λ− ε, λ+ ε].

If ε is small enough, one can find a ∈ S1 such that a is regular for fs for all s ∈ J .

Then we put Mi = F−1J [a− i− 1, a− i]. Let

B− = M0∩PN∗(λ−ε) = M0∩PN−∗ (λ), B+ = M0∩PN∗(λ+ε) = M0∩PN+

∗ (λ).

Definition 6.1. A birth-death point v of a generalized Morse function fλ is inde-

pendent up to height ≤ N if for each critical point p of Fλ with |dB−(p, v)| ≤ N ,

there is no flow line between p and v. We define the height of a principal bifurcation

λ to be min(p,q)dB±(p, q) (see §A.1), where (p, q) ranges over pairs of critical points

of index i that are in positions of i/i-intersection at s = λ.

Lemma 6.2. After a perturbation of the family of metrics {ηs; s ∈ J} fixing the

endpoints, we may arrange that for each birth-death bifurcation λ ∈ J , there is a

small number εN > 0 such that on J = [λ− εN , λ+ εN ],

(1) There is no principal bifurcation of height ≤ N with respect to both B− and

B+.

(2) The birth-death bifurcation is independent up to height ≤ N .

Proof. There is an interval [a, b] such that all the flow lines between critical points of

height ≤ N are included in F−1J [a, b], that is compact. Then as in the case of com-

pact manifold ([HW, page 62]), there is a perturbation that realizes independence

within F−1J [a, b] without affecting the properties for ≤ N − 1. �

Corollary 6.3. After the perturbation as in Lemma 6.2, one of the following holds:

CN∗(λ+ εN ) = CN∗(λ− εN )⊕ CN elem∗ , ∂λ+εN ∼ ∂λ−εN + ∂elem

CN∗(λ− εN ) = CN∗(λ+ εN )⊕ CN elem∗ , ∂λ−εN ∼ ∂λ+εN + ∂elem,

for any N . Therefore, we have

CN+∗ (λ) = CN−

∗ (λ)⊕ CN elem∗ , ∂+

λ = ∂−λ + ∂elem, or

CN−∗ (λ) = CN+

∗ (λ)⊕ CN elem∗ , ∂−

λ = ∂+λ + ∂elem.

Now Theorem 3.1 follows from the lemmas in §4, Lemma 5.1 and Corollary 6.3.

7. Condition (C′) for homotopies

In this section, we prove Theorem 3.4 on the basis of . The proof mainly consists

of the construction of a homotopy satisfying the condition (C′).


7.1. Morse–Smale filtrations of ∂0W and ∂1W , fitting C′-structures. Let

f : M → S1 be a Morse function. Suppose that a pair (f, η) is Morse–Smale

and that 0 ∈ S1 is a regular value of f . We obtain a cobordism (W,∂0W,∂1W )

by cutting M along Σ = f−1(0), in other words, W = F−1[0, 1], ∂0W = F−1(0)

and ∂1W = F−1(1) for the lift F : M → R of f . Let f� : W → [0, 1] denote

the induced function on the cobordism W from f . Then the descending manifolds

Dp(f�) define a stratification {Dind≤k(f

�)} of (W,∂0W ), where Dind≤k(f�) is the

union of the descending manifolds coming from critical points of indices ≤ k. Since

Dp(f) is transversal to Σ, the intersection Dp(f�) ∩ ∂0W is a smooth submanifold

of ∂0W . Thus we obtain a stratified subspace {Dind≤k+1(f�)∩ ∂0W} of ∂0W . We

shall extend a refinement of this stratified subspace to a Morse–Smale stratification

of ∂0W .

For filtered spaces X =∪

k Fk(X) and Y =∪

k Fk(Y ), we will say that a map

f : X → Y is filtration preserving, or f preserves filtration, if for each k we have

f(Fk(X)) ⊂ IntFk(Y ).

If a Morse–Smale pair (φ0, ρ0) on ∂0W is given, then the thickenings of the descend-

ing manifolds define a filtration of ∂0W , with the k-th term given by the thickenings

of the descending manifolds of critical points of indices ≤ k (Morse–Smale filtra-

tion).

Lemma 7.1. There exists a Morse–Smale pair (φ0, ρ0) on ∂0W such that the in-

clusion of the stratified subspace {Dind≤k+1(f�) ∩ ∂0W} into ∂0W equipped with

the Morse–Smale filtration with respect to (φ0, ρ0) is filtration preserving. (We will

say that such a Morse–Smale filtration is adapted to {Dind≤k+1(f�) ∩ ∂0W}.)

Proof. Such a Morse–Smale pair (φ0, ρ0) can be constructed as follows. First choose

a Morse–Smale pair on a closed neighborhood of Dind≤m−1(f�)∩∂0W , m = dimW ,

with values in [0, ε] such that the inclusion of its Morse–Smale filtration is filtration

preserving with respect to the given stratification and such that φ0 takes the value

ε on the boundary, and then extending to the whole of M . Then perturb the metric

ρ0 suitably to extend the Morse–Smale property so that the extension ∂0W → [0, 1]

together with ρ0 is Morse–Smale. �

Choosing (φ0, ρ0) as in the lemma, we obtain a stratification D ′ind≤0(φ0) ⊂

D ′ind≤1(φ0) ⊂ · · · ⊂ D ′

ind≤m−1(φ0) of ∂0W . Correspondingly, we have a Morse–

Smale filtration {Fk(∂0W )} of ∂0W by δ-thickenings of the descending manifolds

of (φ0, ρ0) for a small δ. Let {F k(∂0W )} be the dual filtration to {Fk(∂0W )}, i.e.,the filtration defined by thickenings of the ascending manifolds of (φ0, ρ0).

Similarly, we choose a Morse–Smale pair (φ1, ρ1) on ∂1W so that the inclusion of

the stratified subspace {Aind∗≤k+1(f�)∩∂1W} into ∂1W , where ind∗ = m− ind, is

filtration preserving with respect to a filtration of ∂1W by thickenings of ascending

manifolds. Then as above we obtain a Morse–Smale filtration {F k(∂1W )} of ∂1Wby δ-thickenings of the descending manifolds of (φ1, ρ1). Let {Fk(∂1W )} be the

dual filtration to {F k(∂1W )}.After a perturbation of (f, η) and (φj , ρj) that modifies {Fk(∂iW )}, {F k(∂iW )},

{Dind≤k(f�) ∩ ∂0W} and {Aind∗≤k(f

�) ∩ ∂1W}, we may assume the following


condition (the condition (C′) of [Pa1]): For each k ≥ 0,

(1) (−gradηf�) (Fk(∂1W )) ⊂ IntFk(∂0W ) ⊃ Dind≤k+1(f�) ∩ ∂0W,

(2) (gradηf�) (F k(∂0W )) ⊂ IntF k(∂1W ) ⊃ Aind∗≤k+1(f

�) ∩ ∂1W,

(3) t−1Fk(∂0W ) ⊂ IntFk(∂1W ),

where (−gradηf�) is the (partially defined) map that assigns to each point x on

∂1W \∪

p Ap(f�) the point of ∂0W that is the intersection of the integral curve

of −gradηf� passing through x with ∂0W . The map (gradηf�) is similarly

defined by the intersection of the positive gradient flow line with ∂1W . (1) can be

realized by a perturbation of {Fk(∂1W )}, (2) can be realized by a perturbation of

{F k(∂0W )}, and (3) can be realized by a perturbation of the metric η. See [Pa1,

Ch. 8.4] for the detailed proof of the realizability of the condition (C′). Roughly, the

image of the gradient descent (−gradηf�) (Fk(∂1W )) is a thickening of the union

of at most k-dimensional objects, so after a small perturbation, it can be made

disjoint from F k(∂0W ), that is at most (m− k − 2)-dimensional object. Thus by

carrying the image by the negative gradient −gradη0(φ0) for a period of time, the

image eventually pushed into the interior of Fk(∂0W ).

We call a pair of filtrations {Fk(∂0W )}, {Fk(∂1W )} as obtained above a fitting

C′-structure with respect to (f, η). The conditions (1), (2), (3) given above imply

that (−gradηf�) induces a well-defined continuous map

(−gradηf�)� : Fk(∂1W )/Fk−1(∂1W )→ Fk(∂0W )/Fk−1(∂0W ).

Roughly, by the condition (2), the subset of Fk(∂0W ) where (−gradηf) is not

defined consists of the intersection with the ascending manifolds of critical points

of f� of indices ≤ k − 1, whose descending manifolds land on Fk−1(∂0W ) by the

condition (1). Moreover, by the condition (3), the return map

t−1 : Fk(∂0W )/Fk−1(∂0W )→ Fk(∂1W )/Fk−1(∂1W )

is well-defined too. Here Fk(∂iW )/Fk−1(∂iW ) is homotopy equivalent to the

wedge of k-spheres since the filtration {Fk(∂iW )} is cellular.Pajitnov’s formula for the Novikov boundary ∂ : CNk(f) → CNk−1(f) is as

follows ([Pa2]):

∂(pi) =r∑

j=1

(n0(pi, qj ; f) + t〈(1− tA)−1[Npi ], [Sqj ]〉

)· qj ,

where pj is a critical point of f of index k, qj (j = 1, . . . , r) are the critical points

of f of index k − 1, Npi = Dpi(f�) ∩ ∂0W , Sqj = t(Aqj (f

�) ∩ ∂1W ) and

A : Hk(Fk(∂1W ),Fk−1(∂1W ))→ Hk(Fk(∂1W ),Fk−1(∂1W ))

is the induced map from t−1 ◦ (−grad f�)�.

7.2. 1-parameter family near a transition point. From now on we shall con-

struct an analogue of the fitting C′-structure for a 1-parameter family of S1-valued

pairs. First we decompose a 1-parameter family into pieces. We will call the set of

graphs of critical values of fs the graphic, as in [Ce].


Lemma 7.2. Let {(fs, ηs); s ∈ J = [s0, s1]}, be a 1-parameter family of S1-valued

Morse pairs such that the critical values of fs are distinct at s = s0, s1. Then

there are finitely many times u1, u2, . . . , ur in J at which a ‘transition’ occurs, i.e.,

a critical value of fs and a fixed regular value a ∈ S1 of fs0 and fs1 coincide.

Moreover, after a perturbation fixing the endpoints and fixing the graphic, we may

arrange that on a neighborhood Ni of each ui, there are no principal bifurcations.

Proof. The first statement is easy. For the second statement, suppose that λ ∈ J

be a time at which a transition of a critical locus p over the level a occurs. We

may assume that (fλ, ηλ) is admissible and the values of fλ at critical points other

than p are not equal to a. It is possible since the set of times of admissible pairs

is dense. Fix a small δ � s1 − s0 and take a smooth function g : R → R with the

following properties:

(1) g(x) = λ for λ− δ ≤ x ≤ λ+ δ,

(2) g(x) = x for x ≤ λ− 2δ or x ≥ λ+ 2δ,

(3) ddxg(x) > 0 for λ− 2δ < x < λ− δ or λ+ δ < x < λ+ 2δ.

Also, for a small number ε > 0, take a smooth function $ : R → R (‘cloche

function’) with the following properties

(1) $(−x) = $(x) for all x,

(2) $(x) = 0 for |x| ≥ ε,

(3) $(x) = 1 for |x| ≤ ε2 ,

(4) ddx$(x) < 0 for ε

2 < x < ε.

Let Up ⊂ J ×M be a small tubular neighborhood of the critical locus p on which

the parametrized Morse lemma hold. Then we put

hs(x) =

{fg(s)(x) + (fs(p)− fg(s)(p))$(|x|) on Up

fg(s)(x) otherwise

vs = gradηg(s)fg(s).

Fix metric η′s so that it is adapted to the pair (hs, vs), i.e., vs = gradη′shs. Then

by definition, hs(p) = fs(p). The homotopies (1 − µ)g(x) + µx : id ' g and

(1 − µ)$(x) : $ ' 0 induce homotopies hs ' fs, vs ' gradηsfs, η

′s ' ηs. Since

there are no principal bifurcation of vs for λ − δ ≤ s ≤ λ + δ, this homotopy is a

desired perturbation. �

Let ui ∈ J be as in Lemma 7.2, at which a transition occurs and choose ε > 0 so

small that [ui − ε, ui + ε] ⊂ Ni. Suppose that a critical locus p intersects the level

surface f−1J (a) =

∪s∈J{s} × Σs ⊂ J ×M , Σs = f−1

s (a) at s = ui from below and

that fui−ε(p) = a−µ for a small µ > 0. We may also assume that a−2µ is a regular

value of fs for each s ∈ Ni. Let Σui−εµ = f−1

ui−ε(a− 2µ). Let (W s, ∂0Ws, ∂1W

s) be

the cobordism obtained by cuttingM along Σs. Similarly, let (W sµ, ∂0W

sµ, ∂1W

sµ) be

the cobordism obtained by cutting M along Σsµ for s close to ui− ε. The following

lemma says that one can deform the 1-parameter family of pairs on J so that the

condition (C′) is satisfied for both of the two cobordisms at the left side s = ui − ε

and the composition satisfies the condition (C′) too.


Lemma 7.3. Let (fs, ηs) be a 1-parameter family as in Lemma 7.2. Then after a

perturbation of (fs, ηs) in Ni without affecting the property of Lemma 7.2, we may

find Morse–Smale filtrations {Fk(∂iWui−ε)} and {Fk(∂iW

ui−εµ )}, i = 0, 1, and

their duals {F k(∂iWui−ε)} and {F k(∂iW

ui−εµ )}, such that

(i) (−grad fui−ε) [a,a−2µ](Fk(∂1W

ui−ε)) ⊂ IntFk(∂0Wui−εµ ) ⊃ Dind≤k+1(f

�ui−ε)∩

∂0Wui−εµ ,

(ii) (−grad fui−ε) [a−2µ,a−1](Fk(∂1W

ui−εµ )) ⊂ IntFk(∂0W

ui−ε) ⊃ Dind≤k+1(f�ui−ε)∩

∂0Wui−ε,

(iii) (grad fui−ε) [a−2µ,a](F

k(∂0Wui−εµ )) ⊂ IntF k(∂1W

ui−ε) ⊃ Aind∗≤k+1(f�ui−ε)∩

∂1Wui−ε,

(iv) (grad fui−ε) [a−1,a−2µ](F

k(∂0Wui−ε)) ⊂ IntF k(∂1W

ui−εµ ) ⊃ Aind∗≤k+1(f

�ui−ε)∩

∂1Wui−εµ ,

(v) t−1Fk(∂0Wui−ε) ⊂ IntFk(∂1W

ui−ε),

(vi) t−1Fk(∂0Wui−εµ ) ⊂ IntFk(∂1W

ui−εµ ).

Proof. Consider Wui−ε as the composition of two cobordisms F−1[a−1, a−2µ] andF−1[a− 2µ, a] that are separated by Σui−ε

µ . Then after perturbations of the Morse

pair on both cobordisms, we may assume that the conditions (1), (2) of (C′) are

satisfied on both cobordisms and the conditions (i)–(iv) follow. The perturbations

needed are exactly the same as the ones needed for the condition (C′) for a single

cobordism (as in [Pa1, Ch. 8-4.1]). The condition (3) of (C′) will be satisfied after

an extra perturbation of the metric near Σui−ε and Σui−εµ and (v), (vi) follow. �

Remark 7.4. One can check that under the conditions of Lemma 7.3, both cobor-

disms (Wui−ε, ∂0Wui−ε, ∂1W

ui−ε) and (Wui−εµ , ∂0W

ui−εµ , ∂1W

ui−εµ ) produce the

same Novikov complex.

Since a−2µ is a regular value of fs for s ∈ Ni, the family of level surfaces at the

level a− 2µ over [ui − ε, ui + ε] forms a product structure [ui − ε, ui + ε]× Σui−εµ .

We define the filtration Fk(∂iWµ; [ui − ε, ui + ε]) on [ui − ε, ui + ε]× ∂iWµ by the

product

Fk(∂iWµ; [ui − ε, ui + ε]) = [ui − ε, ui + ε]×Fk(∂iWµ).

Since there are no i/i-intersections for (fs, ηs) in Ni, we may assume that the

restriction of Fk(∂iWµ; [ui − ε, ui + ε]) on each fiber satisfies the condition (C′)

with respect to the vertical gradients of (fs, ηs). Then by definition, the chain

equivalence that is induced from the homotopy (fs, ηs) is the identity.

7.3. 1-parameter family without transitions. Let {(fs, ηs); s ∈ J = [s0, s1]}be a 1-parameter family of S1-valued Morse pairs without transitions, namely,

there is a value a ∈ S1 that is a regular value of fs for all s ∈ J . Suppose

that on the endpoints s = s0, s1, the pair (fs, ηs) possesses a fitting C′-structure.

(We have not assumed that (f0, η0) and (f1, η1) possess fitting C′-structure. The

adjustment near s = 0, 1 will be considered later.) We may suppose without loss of

generality that a = 0 ∈ S1. Put Σs = f−1s (0) and cutting M along Σs, we obtain

a cobordism (W s, ∂0Ws, ∂1W

s). We put W = W s0 . We shall consider the fiber

bundle W =∪

s∈J{s} ×W s over J , that is obtained from J ×M by cutting along


the subbundle Σ =∪

s∈J{s} × Σs. The projection W → J is a trivial W -bundle

so we identify W with J ×W via some trivialization. We remark that this direct

product structure may not be consistent with that of J ×M . In the rest of this

section, we mainly deal with the product structure W ∼= J ×W .

Suppose that the loci of the descending manifolds Dp(fJ) =∪

s∈J{s} × Dp(fs)

and of the ascending manifolds Ap(fJ ) =∪

s∈J{s} × Ap(fs) satisfy the transver-

sality condition in the 1-parameter family. In particular, for each pair (p, q), the

intersection of Dp(f�J ) and Aq(f

�J ) is transversal in J ×W . The proof is similar as

the proof of the genericity of the Morse–Smale transversality condition. If J ′ ⊂ J

is a sufficiently small interval, the loci of the descending and ascending manifolds

define filtrations

{Dind≤k(fJ′)} and {Aind∗≤k(fJ ′)} of J ′ ×W,

where Dind≤k(fJ ′) etc denote the union of the loci of the descending manifolds

of critical points of indices ≤ k etc., and ind∗(x) = m − ind(x), m = dimW .

Here, for example, Dind=k(fJ′) = Dind≤k(fJ′) \ Dind≤k−1(fJ′) is the union of some

(k + 1)-dimensional submanifolds of J ′ ×W .

7.3.1. Handle filtration near cj. From now on we will define some filtrations of J ×∂iW . Let {Fk(∂iW, s)}, {F k(∂iW, s)}, s = s0, s1, be the Morse–Smale filtrations

of the given fitting C′-structures at the endpoints {s0, s1} × ∂iW . We shall extend

this to a filtration of J×∂iW that is convenient for analyzing homological properties

of the vertical gradient descent map.

By Lemma 4.4, we may assume that there are finitely many points c1, c2, . . . , cr ∈(s0, s1) at each of which a gradient i/i-intersection for (f�s , η�s ) occurs. We put

c0 = s0 and cr+1 = s1. At each point cj , the sequence {Dind≤k(f�cj )∩∂0W

cj} (resp.{Aind∗≤k(f

�cj ) ∩ ∂1W

cj}) forms a stratified subspace of ∂0Wcj (resp. ∂1W

cj ). One

can find Morse–Smale filtrations {Fk(∂iW, cj)}, {F k(∂iW, cj)} of ∂iW cj that are

adapted to the stratified subspaces {Dind≤k(f�cj ) ∩ ∂0W

cj} and {Aind∗≤k(f�cj ) ∩

∂1Wcj}, as in §7.1. For a sufficiently small number ε > 0, we define Fk(∂iW ; [cj −

ε, cj + ε] ∩ J) and F k(∂iW ; [cj − ε, cj + ε] ∩ J) to be the product filtrations

Fk(∂iW ; [cj − ε, cj + ε] ∩ J) = ([cj − ε, cj + ε] ∩ J)×Fk(∂iW, cj),

F k(∂iW ; [cj − ε, cj + ε] ∩ J) = ([cj − ε, cj + ε] ∩ J)×F k(∂iW, cj).

These are just thickenings of {Fk(∂iW, cj)} and {F k(∂iW, cj)}. Here we as-

sume that ε is so small that the restrictions of Fk(∂iW ; [cj − ε, cj + ε] ∩ J) and

F k(∂iW ; [cj − ε, cj + ε] ∩ J) on each fiber satisfies the condition (C′) with respect

to the vertical gradient of (f�s , η�s ). We will call these kinds of filtrations border

type filtrations. We also have families (φi,s, ρi,s), s ∈ [cj − ε, cj + ε] ∩ J , of Morse

pairs on ∂iW , whose vertical gradients generate the filtrations above.

7.3.2. Handle filtration away from cj. Now we define handle filtrations on [cj +

3ε, cj+1−3ε]×∂iW for each j ≥ 0. We may assume that (f�s , η�s ) is Morse–Smale at

s = cj+3ε. So the sequence {Dind≤k(f�cj+3ε)∩∂0W cj+3ε} (resp. {Aind∗≤k(f

�cj+3ε)∩

∂1Wcj+3ε}) forms a stratified subspace of ∂0W

cj+3ε (resp. ∂1Wcj+3ε). Then let


{Fk(∂iW, cj+3ε)} and {F k(∂iW, cj+3ε)} be Morse–Smale filtrations of ∂iWcj+3ε

that are adapted to the stratified spaces {Dind≤k(f�cj+3ε)∩∂0W cj+3ε} and {Aind∗≤k(f

�cj+3ε)∩

∂1Wcj+3ε}. We define Fk(∂iW ; [cj+3ε, cj+1−3ε]) and F k(∂iW ; [cj+3ε, cj+1−3ε])

to be the product filtrations

Fk(∂iW ; [cj + 3ε, cj+1 − 3ε]) = [cj + 3ε, cj+1 − 3ε]×Fk−1(∂iW, cj + 3ε),

F k(∂iW ; [cj + 3ε, cj+1 − 3ε]) = [cj + 3ε, cj+1 − 3ε]×F k−1(∂iW, cj + 3ε).

We remark that [cj + 3ε, cj+1 − 3ε] is not short, so in the right hand side Fk−1 is

appropriate to get a thickening of a k-dimenaional object. We will call these kinds

of filtrations filling type filtrations. The distinction between the border type and

filling type will correspond to the types of the perturbations of the filtrations for

the condition (C′) given later. We have families (φi,s, ρi,s), s ∈ [cj + 3ε, cj+1 − 3ε],

of Morse pairs on ∂iW , whose vertical gradient generates the filtrations above.

7.3.3. Extending handle filtrations. We extend the handle filtrations on [cj−1 +

3ε, cj − 3ε] × ∂iW obtained above slightly to handle filtrations on [cj−1 + 3ε, cj −ε] × ∂iW . First, we extend the filtration of [cj−1 + 3ε, cj − 3ε] × ∂iW to that of

[cj−1 +3ε, cj − 2ε]× ∂iW , by attaching a border type filtration. Namely, we define

Fk(∂iW ; [cj − 3ε, cj − 2ε]) = [cj − 3ε, cj − 2ε]×Fk(∂iW, cj−1 + 3ε),

F k(∂iW ; [cj − 3ε, cj − 2ε]) = [cj − 3ε, cj − 2ε]×F k(∂iW, cj−1 + 3ε).

This can be defined by a trivial extension of the Morse pairs (φi,s, ρi,s), s ∈ [cj−1+

3ε, cj − 3ε] to [cj−1 + 3ε, cj − 2ε].

To complete the extension, we define a filling type filtration on [cj − 2ε, cj −ε]× ∂iW . Although the interval [cj − 2ε, cj − ε] is short, we formally consider this

part as a filling type. We construct a filtration preserving homotopy between the

Morse–Smale pairs (φi,cj−2ε, ρi,cj−2ε) and (φi,cj−ε, ρi,cj−ε) on ∂iW chosen above.

The following lemma holds.

Lemma 7.5. There is a self-diffeomorphism g of ∂0W (resp. h of ∂1W ) that is

isotopic to the identity such that for each k, the following conditions are satisfied:

(1) gFk(∂0W, cj − 2ε) ⊂ IntFk(∂0W, cj − ε),

(2) g−1F k(∂0W, cj − ε) ⊂ IntF k(∂0W, cj − 2ε),

(3) hF k(∂1W, cj − 2ε) ⊂ IntF k(∂1W, cj − ε),

(4) h−1Fk(∂1W, cj − ε) ⊂ IntFk(∂1W, cj − 2ε).

Proof. The self-diffeomorphism g can be constructed by a similar deformation as

that Pajitnov used to prove the genericity of the condition (C′) [Pa1, Ch. 8.4].

Namely, let ht : ∂0W → ∂0W be the 1-parameter group of diffeomorphisms gen-

erated by the negative gradient for the pair (φcj−ε, ρcj−ε). For a sufficiently large

t, we have ht(Fk(∂0W, cj − 2ε)) ⊂ IntFk(∂0, cj − ε) and we may take g = ht.

A similar argument is valid for F k(∂0W, cj − ε) and g−1. So it remains to show

that F k(∂0W, cj − ε), k < m − 1, can be made sufficiently thin, i.e. small thick-

enings of the ascending manifolds of (fcj−ε, ηcj−ε), by a diffeomorphism of ∂0W

without changing Fk(∂0W, cj − ε), k < m − 1. The claim follows from the facts


that {Fk(∂0W, cj − ε)} and {F k(∂0W, cj − ε)} are defined by thickenings of the

descending and ascending manifolds, that are transversal. �

Take a self-diffeomorphism g : ∂0W → ∂0W as in the lemma and take an isotopy

gθ : ∂0W → ∂0W , θ ∈ [cj − 2ε, cj − ε], with gθ = id∂0W near θ = cj − 2ε and gθ = g

near θ = cj−ε. We define for each k the handle filtrations on [cj−2ε, cj−ε]×∂0W

as follows.

Fk(∂0W ; [cj − 2ε, cj − ε]) =∪

θ∈[cj−2ε,cj−ε]

{θ} × gθFk−1(∂0W, cj − 2ε)

F k(∂0W ; [cj − 2ε, cj − ε]) =∪

θ∈[cj−2ε,cj−ε]

{θ} × gθg−1F k−1(∂0W, cj − ε).

This is of filling type. We may define Fk(∂1W ; [cj −3ε, cj − ε]) and F k(∂1W ; [cj −3ε, cj − ε]) in a similar way, using the self-diffeomorphism h of ∂1W of Lemma 7.5.

Now we define the handle filtrations on [cj + 3ε, cj − ε]× ∂iW by the unions

Fk(∂iW ; [cj + 3ε, cj − ε]) = Fk(∂iW ; [cj + 3ε, cj+1 − 3ε]) ∪ Fk(∂iW ; [cj − 3ε, cj − 2ε])

∪ Fk(∂iW ; [cj − 2ε, cj − ε]),

F k(∂iW ; [cj + 3ε, cj − ε]) = F k(∂iW ; [cj + 3ε, cj+1 − 3ε]) ∪ F k(∂iW ; [cj − 3ε, cj − 2ε])

∪ F k(∂iW ; [cj − 2ε, cj − ε])

for all k. By a similar method one can extend handle filtrations over the blank

[cj + ε, cj + 3ε] as well.

As a consequence of the above, we will obtain handle filtrations {Fk(∂iW )} and{F k(∂iW )} of J × ∂iW by the union of all the handle filtrations made above. We

write

Fk(∂iW, s) = Fk(∂iW ) ∩ ({s} × ∂iW ),

F k(∂iW, s) = F k(∂iW ) ∩ ({s} × ∂iW ).

We will also denote by Fk(∂iW, s) the image of Fk(∂iW, s) under the obvious

diffeomorphism {s} × ∂iW → ∂iW . We now check the condition (C′) for the

filtrations obtained.

Lemma 7.6. Let (fJ , ηJ) be a 1-parameter family having no transitions, as above.

After perturbations of ηJ and of the handle filtrations {Fk(∂iW )}, {F k(∂iW )}, wemay assume the following conditions

(1) (−gradvηJfJ ) (Fk(∂1W )) ⊂ Int Fk(∂0W ) ⊃ Dind≤k(fJ) ∩ (J × ∂0W ),

(2) (gradvηJfλ) (F k(∂0W )) ⊂ Int F k(∂1W ) ⊃ Aind∗≤k(fJ) ∩ (J × ∂1W ),

(3) t−1Fk(∂0W ) ⊂ Int Fk(∂1W ),

where gradvηJfJ denote the gradient vector field considered along the fibers. (This

is a 1-parameter analogue of the condition (C′) of [Pa1].)

We will say that a 1-parameter family (fJ , ηJ ) satisfies the condition (C′) if for

some a ∈ S1, a is regular for fJ and the conditions of Lemma 7.6 is satisfied for

some family of level surfaces.


Proof. We check the first one. The proof is an almost straightforward generalization

of the argument of [Pa1, Ch. 8.4]. Put u = gradvηJfJ for simplicity. Then the set

Σk = Dind≤k(u) ∩ (J × ∂0W )

is a union of smooth submanifolds of dimensions ≤ k. We show that the set

Zk = Σk ∪ (−u) (Fk(∂1W ))

can be collapsed into the interior of Fk(∂0W ) by a fiber-preserving perturbation of

the metric ηJ .

We consider a dual filtration to {Fk(∂0W )} of J×∂0W . The filtration {F k(∂0W )}is not appropriate as a dual filtration of J×∂0W to {Fk(∂0W )}. Instead, we define

Fk(∂0W )∗ =r+1∪j=0

[cj − ε′, cj + ε′]×Fm−k−2(∂0W, cj)

∪r∪

j=0

[cj +

5

2ε− ε′, cj +

5

2ε+ ε′

]×Fm−k−2

(∂0, cj +

5

2ε)

∪r+1∪j=1

[cj −

5

2ε− ε′, cj −

5

2ε+ ε′

]×Fm−k−2

(∂0, cj −

5

2ε)

∪r∪

j=0

[cj + cj+1

2− ε′,

cj + cj+1

2+ ε′

]×Fm−k−1

(∂0W,

cj + cj+1

2

)

∪r+1∪j=1

[cj −

3

2ε− ε′, cj −

3

2ε+ ε′

]×Fm−k−1

(∂0W, cj −

3

2ε)

∪r∪

j=0

[cj +

3

2ε− ε′, cj +

3

2ε+ ε′

]×Fm−k−1

(∂0W, cj +

3

2ε)

(7.1)

for a sufficiently small number ε′ � ε. The first three correspond to border type

filtrations and the last three correspond to filling type filtrations. This is adequate

as a dual to the filtration {Fk(∂0W )} in the sense that Fk(∂0W )∗ is a deformation

retract of the complement of Fk(∂0W ) in J × ∂0W .

We shall see that Zk can be deformed by an isotopy in J × ∂0W so that it

is disjoint from Fk(∂0W )∗. Since Zk is a union of tubular neighborhoods of

submanifolds of dimensions ≤ k, we may arrange that it does not intersect the

(m−k−1)-dimensional object Aind∗≤m−k−1(φ0,cj+cj+1

2

) in the m-dimensional man-

ifold J × ∂0W after a slight perturbation of Zk by an isotopy. Then for δ > 0

sufficiently small, Zk does not intersect the δ-thickening A δind∗≤m−k−1(φ0,

cj+cj+12

),

δ � ε, in J × ∂0W . We have similar results for A δind∗≤m−k−1(φ0,cj± 3

2 ε). Moreover,

we may assume after a perturbation that Zk∩(([cj−ε, cj+ε]∩J)×∂0W ) is disjoint

from the δ-thickening A δind∗≤m−k−2(φ0,cj ) in ([cj − ε, cj + ε] ∩ J)× ∂0W . Here we

consider the disjunction in an (m− 1)-dimensional manifold {cj} × ∂0W . We have

similar results in other border type filtrations. Hence Zk can be separated from


Fk(∂0W )∗ by an isotopy in J × ∂0W . Since Fk(∂0W )∗ is a deformation retract of

the complement of Fk(∂0W ), there is an ambient isotopy of J × ∂0W that carries

Zk inside Int Fk(∂0W ).

Moreover, we claim that the ambient isotopy for the disjunction can be taken to

be fiber-preserving. To prove this, we use Lemma 7.7 below. By Lemma 7.7, we

may first perturb Zk so that it falls into Fk(∂0W ) at least in the subset [cj−1 +

3ε, cj − 3ε]× ∂0W . This can be done by using the trivial product structure of the

filtration in [cj−1 + 3ε, cj − 3ε] × ∂0W . Moreover, the perturbation that carries

Zk ∩ ([cj−1 +3ε, cj − 3ε]× ∂0W ) into Fk(∂0W ) can be extended to a perturbation

that carries Zk ∩ ((cj−1 + 3ε − 110ε, cj − 3ε + 1

10ε) × ∂0W ) into Fk(∂0W ) since

the handle filtration is consistent on the fiber {cj − 3ε} × ∂0W , i.e., k-handles are

attached to handles of indices ≤ k−1. We may perturb Zk similarly so that it falls

into Int Fk(∂0W ) on [cj − 2ε− 110ε, cj − ε+ 1

10ε]× ∂0W and on [cj + ε− 110ε, cj +

2ε+ 110ε]× ∂0W .

We then perturb Zk by a fiber-preserving isotopy so that it falls into Int Fk(∂0W )

in the subset [cj − ε, cj + ε] × ∂0W , again by using the product structure of the

filtration Fk(∂0W ; [cj − ε, cj + ε]), and similar for other border type filtrations.

After a perturbation of {Fk(∂0W )} and ηJ as above, we may assume that Zk

lies in the interior of Fk(∂0W ). Now we have

(−u) (Fk(∂1W )) ⊂ Int Fk(∂0W ) ⊃ Dind≤k(u) ∩ (J × ∂0W ).

The condition (1) of the lemma is fulfilled. A similar argument can make the

condition (2) fulfilled after a perturbation of {F k(∂0W )}. For the condition (3),

since t−1Fk(∂0W ) is a k-dimensional object, we can perturb the metric ηJ so that

it is disjoint from the (m − k − 1)-skeleton of the stratification {Aind∗≤k(φ1,cj )}.Hence it is also disjoint from the δ-thickening, with δ small. �

Lemma 7.7. Suppose that X is an (m−1)-dimensional manifold, Y is an (m−k−2)-dimensional submanifold of X. Let f : A→ [0, 1]×X be a smooth map from a k-

dimensional manifold A. Let pr1 : [0, 1]×X → [0, 1] and pr2 : [0, 1]×X → X be the

projections. Then one can perturb f so that the image of A under pr2 ◦f is disjoint

from Y and that the perturbation can be chosen arbitrarily small with respect to the

C∞-topology. Moreover, we may assume that the perturbation is fiber-preserving,

i.e., pr1 ◦ f : A→ [0, 1] is invariant under the perturbation.

If f is an embedding, then the perturbation of the image of f can be realized by

a fiber-preserving ambient isotopy of [0, 1]×X.

Lemma 7.7 is an immediate consequence of the transversality theorem and the

isotopy extension theorem.

Lemma 7.8. Let (fJ , ηJ) be as above. After the perturbation of Lemma 7.6, the

graded return map

t−1 ◦ (−gradvfJ )� : Fk(∂1W )/Fk−1(∂1W )→ Fk(∂1W )/Fk−1(∂1W ),

is well-defined as a continuous map.


Proof. By construction, if F `(∂1W ) intersects Fk(∂1W ), then ` ≥ m − k. So

the condition (2) of Lemma 7.6 implies that if Ap(f�J ) intersects Fk(∂1W ), then

the index of p must be ≤ k − 1. In particular, Dp(f�J ) ∩ ∂0W is included in

Int Fk−1(∂0W ) by (1). Then by setting the image of the points at which the

gradient descent is not defined to be the base point ω0 ∈ F0(∂0W ), one obtains a

continuous map

(−gradvfJ)� : Fk(∂1W )/Fk−1(∂1W )→ Fk(∂0W )/Fk−1(∂0W ).

By condition (3), the map

t−1 : Fk(∂0W )/Fk−1(∂0W )→ Fk(∂1W )/Fk−1(∂1W )

is well-defined as well. �

Proof of Theorem 3.4. By Theorem 3.1, it suffices to prove the theorem for 1-

parameter families without birth-death points. By Lemma 7.2, we may assume

that the chain equivalence on a small interval including a transition point is the

identity. So we assume that fJ has no transition points, i.e., 0 ∈ S1 is a regular

value of fs for all s ∈ J .

We first assume that on the endpoints s ∈ ∂J , the pair (fs, ηs) possesses a fitting

C′-structure. Then as above we may find filtrations {Fk(∂iW )} and {F k(∂iW )}satisfying the condition (C′) by Lemma 7.6. Let

H k(−gradvfJ) : Hk(Fk(∂1W ), Fk−1(∂1W ))→ Hk(Fk(∂1W ), Fk−1(∂1W )).

be the induced map on homology from t−1◦(−gradvfJ )�. Here,Hk(Fk(∂1W ), Fk−1(∂1W ))

is free abelian since the filtration {Fk(∂1W )} is a thickening of a cellular stratifi-

cation of J × ∂1W . Then by the proof of Theorem 3.1, it follows that the chain

equivalence Φ : CNk → CNk is given explicitly by the formula

Φ(pi) =r∑

j=1

(n0(pi, pj ; fJ ) + t〈(1− tA)−1[Npi ], [Spj ]〉

)· pj ,

where p1, p2, . . . , pr be the critical loci of f�J of index k, Npi = Dpi(f�J ) ∩ ∂0W ,

Spj = t(Apj (f�J )∩∂1W ) and A = H k(−gradηJ

fJ). This completes the proof when

fitting C′-structures are given on the endpoints.

In the general case, one can still find filtrations {Fk(∂iW, s)} and {F k(∂iW, s)},at s ∈ ∂J , satisfying the condition (C′). Moreover, we may assume that the cut

surface for the cobordism W at s ∈ ∂J agrees with the level surface f−1s (0) and

that (fs, ηs) at s = s0 + 2ε, s1 − 2ε possess fitting C′-structures. Then we may

find filtrations {Fk(∂iW )} and {F k(∂iW )} over [s0 + 2ε, s1 − 2ε] satisfying the

condition (C′). Then as before, the filtrations can be extended to that over [s0, s1],

by using Lemma 7.5. The rest is the same as above. �


8. An example

We shall illustrate a bifurcation of Novikov complex by a simple example. The

reader may find more non-trivial, interesting examples of a Novikov complex, in

[Pa3]. We shall give here an example of Novikov complexes for R2 × S1 and its

change under a bifurcation. Let M = R2 × S1 and let f0 : M → S1 be the

projection. Let η0 be the standard metric on M induced from the standard one on

R2 × R2. Then a gradient vector field on M is induced from the standard vector

field ∂∂t on S1 = R/Z. Now for a small number ε > 0 let

B0 = {(x1, x2) ∈ R2;x21 + x2

2 ≤ ε2}.

We modify (f0, η0) in B0 × S1 ⊂ M as follows. For a smaller positive number

ε′ � ε, introduce a cancelling pair (p, q) of critical points of indices (1, 2) inside

the ball B(0,0,1/2)(ε′) ⊂ B0 × S1, by adding to f0 the standard model for the birth

bifurcation. Then by choosing a metric on M appropriately, we may assume that

the support of the change of the gradient vector field lies inside B0×S1. We denote

the resulting Morse pair by (f1, η1).

8.1. The Novikov complex. Let Σ = R2 × {0} ⊂ R2 × S1. Let W be the

cobordism obtained from M by cutting along Σ, with ∂W = ∂1W∐

∂0W = R2 ×{1}

∐R2×{0}. LetM ′ be the mapping cylinder of the diffeomorphism g : R2 → R2,

g(x1, x2) = (x1 + 1, x2): M′ = W/(∂0W ∼g ∂1W ). Then the set

∪k∈Z t

k(Dp(F1) ∪Dq(F1)) ∩ Σi, where Σi is a lift of Σ, is as follows.

We also denote by (f1, η1) the pair of a map M ′ → S1 and a metric on M ′ induced

from the pair (f1, η1). By construction, the Novikov complex for (f1, η1) is

0→ L{p} ∂1→ L

{q} → 0,

∂1(p) = q.

If (f1, η1) is deformed so that p slides over tp through s = 1 to s = 2, as in §5,the set

∪k∈Z t

k(Dp(F1) ∪Dq(F1)) ∩ Σ4 changes as follows.

⇓

This can be checked by using the explicit local coordinate description of the Morse

lemma around critical points. We denote the resulting Morse pair by (f2, η2). Since

there are no other bifurcations than the self 2/2-intersection during the sliding


which replaces (f1, η1) with (f2, η2), Lemma 5.1 implies that the Novikov complex

for (f2, η2) is as follows.

0→ L{p} ∂2→ L

{q} → 0,

∂2(p) = (1± t+ t2 ± t3 + · · · )∂1(p)

= (1± t+ t2 ± t3 + · · · )q = (1∓ t)−1q.

8.2. Handle decomposition for the homotopy. We shall demonstrate a proof

of Theorem 3.4 for the 1-parameter family connecting (f1, η1) and (f2, η2) consid-

ered above.

8.2.1. Fk(∂iW, s) and F k(∂iW, s). We define the Morse–Smale filtrations {Fk(∂iW, s)},{F k(∂iW, s)} for the pair (fs, ηs) at a generic time s ∈ [1, 2] as follows.

F 0(∂1W, s) = ,F 1(∂1W, s) = ,F 2(∂1W, s) =

F0(∂1W, s) = ,F1(∂1W, s) = ,F2(∂1W, s) =

F0(∂0W, s) = ,F1(∂0W, s) = ,F2(∂0W, s) =

F 0(∂0W, s) = ,F 1(∂0W, s) = ,F 2(∂0W0, s) =

We have

H1(F1(∂1W, s),F0(∂1W, s)) ≈ Z2, H1(F1(∂0W, s),F0(∂0W, s)) ≈ Z3.

After a horizontal perturbation of the metric and the filtrations near ∂0W∐

∂1W ,

we may assume that the condition (C′) is satisfied.


Indeed, the gradient descent (−gradηsfs) takes F0(∂1W, s) to a subset of ∂0W

as in the left hand side of the following picture.

and then perturb φ1 and {Fk(∂1W, s)} so that the condition (−gradηsfs) (F0(∂1W, s)) ⊂

IntF0(∂0W, s) is satisfied (the right hand side). We perturb φ0 and {F k(∂0W, s)}in a similar manner to obtain (gradηs

fs) (F 0(∂0W, s)) ⊂ IntF 0(∂1W, s). The

image of F0(∂0W, s) under t−1 is in the left hand side of the following picture.

and then perturb {Fk(∂0W, s)} so that the condition t−1F0(∂0W, s) ⊂ IntF0(∂1W, s)

is satisfied.

The gradient descent takes F1(∂1W, s) to a subset of ∂0W as in the left hand

side of the following picture.

and then perturb {Fk(∂1W, s)} without affecting the previously arranged proper-

ties, so that the condition (−gradηsfs) (F1(∂1W, s)) ⊂ IntF1(∂0W, s) is satisfied.

The image of F1(∂0W, s) under t−1 is as follows:

and then perturb the metric ηs so that the condition t−1F1(∂0W, s) ⊂ IntF1(∂1W, s)

is satisfied. And so on.

Then the gradient descent

(−gradηsfs)� : Fk(∂1W, s)/Fk−1(∂1W, s)→ Fk(∂0W, s)/Fk−1(∂0W, s)

is well-defined. By using Morse’s lemma, one can check that the homological gra-

dient descent

H 1(−gradηsfs) : H1(F1(∂1W, s),F0(∂1W, s))→ H1(F1(∂0W, s),F0(∂0W, s))

takes Z2 onto two summands in Z3.


8.2.2. t−1Fk(∂0W, 2).

t−1F∗(∂0W, 2) = ⊂

⊂

The homology class [t−1(Dp(f2) ∩ ∂0W )] ∈ H1(F1(∂1W, 2),F0(∂1W, 2)) generates

a Z summand of Z2. So the homological return map

t−1◦H 1(−gradη2f2) : H1(F1(∂1W, 2),F0(∂1W, 2))→ H1(F1(∂1W, 2),F0(∂1W, 2))

is given by the matrix

(±1 0

0 1

). This shows that

n(f2; p, tkq) = (±1)k

and that

∂(p) = (1± t+ t2 ± t3 + · · · )q.

This agrees with the computation in §8.1.

8.2.3. Fk(∂iW ) and F k(∂iW ). Now we define handle filtrations of J×∂iW . First,

we define the border type filtrations as follows.

Fk(∂0W ; [1, 1 + ε]) = [1, 1 + ε]×Fk(∂0W, 1),

Fk(∂0W ; [2− ε, 2]) = [2− ε, 2]×Fk(∂0W, 2).

Then we define the filling type filtration on [1 + ε, 2 − ε] × ∂0W . For k = 0, 1, we

define

F0(∂0W ; [1 + ε, 2− ε]) =∪

s∈[1+ε,2−ε]

F−1(∂0W, s) = ∅,

F1(∂0W ; [1 + ε, 2− ε]) =∪

s∈[1+ε,2−ε]

F0(∂0W, s) ∼= [1 + ε, 2− ε]×F0

(∂0W,

3

2

).


We define F2(∂0W ; [1 + ε, 2− ε]) as follows. The thickenings of Dind≤1(fJ) ∩ (J ×∂0W ) is as in the following picture.

If we defined F2(∂0; [1 + ε, 2− ε]) to be a thickening of Dp(fJ), then the filtration

would not satisfy the condition (C′). So we shall perturb Dp(fJ) so that its thicken-

ing gives a filtration satisfying the condition (C′). The dual filtration {Fk(∂0W )∗}of (7.1) is as follows.

Here only the 1-skeleton is shown. We may then perturb the family (fJ , ηJ) by

using Lemma 7.7 so that the thickenings of Dind≤1(fJ) ∩ (J × ∂0W ) becomes as

follows.

The core Dp(fJ )∩(J×∂0W ) may intersect tAp(fJ )∩(J×∂0W ) only in [1, 1+ε]×∂0W(the border on the left side). Then F2(∂0W ; [1 + ε, 2− ε]) is defined by

F2(∂0W ; [1 + ε, 2− ε]) = [1 + ε, 2− ε]×F1(∂0, 2− ε).

{F k(∂0W )} is defined by

F 3−k−1(∂0W ) = (J × ∂0W ) \ Fk(∂0W ).

By a similar argument for the ascending manifolds of (fJ , ηJ), we define a filtration

{F k(∂1W )}. Then we define {Fk(∂1W )} by

F3−k−1(∂1W ) = (J × ∂1W ) \ F k(∂1W ).


We have

H2(F2(∂0W ), F1(∂0W )) ≈ Z, H2(F2(∂1W ), F1(∂1W )) ≈ Z2.

Now the conditions (1) and (2) of Lemma 7.6 are satisfied for (φi,sj , ρi,sj ), j = 1, 2.

Moreover, after a horizontal perturbation of the metric fJ near J × (∂0W∐

∂1W ),

we may assume that (φi,sj , ρi,sj ), j = 1, 2, satisfy the condition (3) of Lemma 7.6.

Then the graded gradient descent map is well-defined. We putH2(F2(∂1W ), F1(∂1W )) =

Z{α1,α2}. By using the parametrized version of Morse’s lemma, one can check that

the image of αi under the gradient descent

H 2(−gradvηJfJ) : H2(F2(∂1W ), F1(∂1W ))→ H2(F2(∂0W ), F1(∂0W ))

is a generator. The homology class [t−1(Dp(fJ )∩(J×∂1W ))] ∈ H2(F2(∂1W ), F1(∂1W ))

generates one of the two Z’s, say the first one Z{α1}. Then the homological return

map

t−1 ◦H 2(−gradvηJfJ) : H2(F2(∂1W ), F1(∂1W ))→ H2(F2(∂1W ), F1(∂1W ))

is given by the matrix

(±1 0

0 1

). This shows that

n(fJ ; p, tkp) = (±1)k

and that

Φ(p) = (1± t+ t2 ± t3 + · · · )p = (1∓ t)−1p.

Appendix A. Inverse limit

A.1. Admissible product. The critical set PN∗(f) of a Novikov complex CN∗(f)

of an S1-valued Morse pair (f, η) is a poset with respect to the partial ordering

defined by negative gradient trajectories: p > q if there is a flow line of (F, η) from

p to q.

Choose a L-basis B = {p1, p2, . . . , pN} of CN∗(f) that is a subset of PN∗(f). For

a pair (x, y) of critical points of F , we define the distance dB(x, y) with respect to

B as follows: if x = tkpi and y = t`pj for some i, j, then we define dB(x, y) = `−k.

Then the following identity holds:

dB(x, y) + dB(y, z) = dB(x, z).

Indeed, if x = tapi, y = tbpj , z = tcpk, then

LHS = (b− a) + (c− b) = c− a = RHS.

For a pair (p, q) of critical points of F of the same index, we define a L-linear

map hpq : CN∗(f)→ CN∗(f) by

hpq(x) =

{±q if x = p

0 otherwise

Here it may happen that π(p) = π(q).

In the following, we consider a formal product

Φ =∏j∈Λ

(1 + ϕj)εj ,


where

• Λ is a countable, totally ordered set,

• ϕj is hpq for some (p, q),

• εj = ±1.Note that (1 + hpq)

−1 = 1− hpq if π(p) 6= π(q). We define the distance of ϕj with

respect to B by

dBϕj = dB(p, q) if ϕj = hpq.

We say that Φ is admissible with respect to B if for each integer N ≥ 0 the subset

of {ϕj ; j ∈ Λ} consisting of objects of dB ≤ N is finite.

Lemma A.1. Admissibility of Φ does not depend on the choice of B.

Proof. Let B and B′ are two choices for the L-basis. We assume that Φ is admissible

with respect to B and shall prove that Φ is also admissible with respect to B′.

Since B and B′ are lifts of the set of critical points of f , a bijection β : B → B′

is determined by the condition π(β(x)) = π(x). We put B = {p1, p2, . . . , pn} andB′ = {p′1, p′2, . . . , p′n} so that p′j = tajpj for some aj ∈ Z. We claim that for any

N ≥ 0, there exists N ′ ≥ 0 such that for any pairs (x, y) of critical points of F ,

dB′(x, y) ≤ N ⇒ dB(x, y) ≤ N ′.

Indeed,

dB(p′i, t

mp′j) = dB(taipi, t

m+ajpj) = m+ aj − ai = dB′(p′i, tmp′j) + aj − ai

≤ dB′(p′i, tmp′j) + max

1≤k,`≤n(ak − a`).

Then put N ′ = N + maxk,`(ak − a`). The admissibility with respect to B′ is

straightforward from the claim. �

Lemma A.2. If Φ is admissible with respect to some B = {p1, p2, . . . , pn}, thenΦ is well-defined as an element of End0(CN∗(f)), i.e., for each x ∈ PNi(f), the

element

Φ(x) =∏j∈Λ

(1 + ϕj)εj (x)

is well-defined as an element of CNi(f). Moreover, Φ ∈ End0(CN∗(f)) does not

depend on the choice of B.

Proof. Let Φ(N) be the partial product of all the terms (1 +ϕj)εj with dBϕj ≤ N ,

that is finite by the admissibility. Let CN ′i be the subgroup of CNi(f) defined by

CN ′i =

∑pj∈B∩PNi(f)

n(pj)pj

∣∣∣∣∣n(pj) ∈ Z[[t]]

.

Then CN ′i is a Z[[t]]-submodule of CNi(f). For each pj ∈ B ∩ PNi(f), we define

Φ(pj) to be the inverse limit of Φ(N)(pj) ∈ CN ′i/t

N+1CN ′i in the inverse system

qN+1 : CN ′i/t

N+1CN ′i → CN ′

i/tNCN ′

i , N ≥ 0.

For this to be well-defined, it suffices to check that

(A.1) qN+1Φ(N)(pj) = Φ(N−1)(pj).


If dBϕk = N , then (1 + ϕk)(tap`) ∼ tap` in CN ′

i/tNCN ′

i and similarly, (1 +

ϕk)−1(tap`) = (1− ϕk + ϕ2

k − ϕ3k + · · · )(tap`) ∼ tap` in CN ′

i/tNCN ′

i . This proves

(A.1).

We should check that the inverse limit Φ = lim←−NΦ(N) does not depend on the

choice of B. We will write Φ′(N) for the partial product Φ(N) of Φ taken with

respect to another choice B′ for the L-basis of CN∗(f), and write lim←−B

NΦ(N) and

lim←−B′

NΦ(N) for the inverse limits considered with respect to B and B′ respectively.

Then we should check the following identity:

(A.2) lim←−N

BΦ(N) = lim←−N

B′Φ′(N).

Let CN ′′i be the subgroup of CNi(f) defined similarly as CN ′

i for B′. For

f ∈ End0(CN ′i) (resp. f ∈ End0(CN ′′

i )), the norm |f |B (resp. |f |B′) is defined as

follows:

|f |B := 2−dB(f) (resp. |f |B′ := 2−dB′ (f)).

This satisfies the following properties:

|f |B = 0⇔ f = 0, | − f |B = |f |B, |f + g|B ≤ max(|f |B, |g|B).

By a similar argument as Lemma A.1, we see that

dB′ϕj ≤ N ⇒ dBϕj ≤ N +∆,

where ∆ = maxk,`(ak−a`), which appeared in the proof of Lemma A.1. This shows

that Φ′(N) =∏

dB′ϕj≤N (1+ϕj)εj is a partial product of Φ(N+∆) =

∏dBϕj≤N+∆(1+

ϕj)εj and that Φ(N+∆) is a partial product of Φ′(N+2∆).

Now we claim that

(A.3) lim←−N

B′Φ(N+∆) = lim←−

N

B′Φ′(N).

This is verified as follows:

|Φ′(N+2∆) − Φ(N+∆)|B′ ≤ max(|Φ′(N+2∆)|B′ , |Φ(N+∆)|B′)

= |Φ′(N+2∆)|B′ = 2−N−2∆,

and this converges to 0 as N →∞. Hence we have

lim←−N

B′Φ(N+∆) = lim←−

N

B′Φ′(N+2∆) = lim←−

N

B′Φ′(N).

This completes the proof of (A.3).

Since Φ(N) is a partial product of Φ′(N+∆), Φ(N) defines an endomorphism of

CN ′′i /t

N+∆+1CN ′′i , that makes the following diagram commutative

t−∆CN ′i

tN+1CN ′i

Φ(N)//

ιN

��

t−∆CN ′i

tN+1CN ′i

ιN

��CN ′′

i

tN+∆+1CN ′′i

Φ(N)// CN ′′

i

tN+∆+1CN ′′i


where ιN is the induced map by the inclusion ι : t−∆CN ′i ⊂ CN ′′

i . Note that ι

induces the inclusion tN+1CN ′i ⊂ tN+∆+1CN ′′

i , so ιN is well-defined. The sequence

of the square diagrams as above for N ≥ 0 forms an inverse system of square

diagrams and its inverse limit is the following commutative square.

t−∆CN ′i

ΦB //

ι

��

t−∆CN ′i

ι

��CN ′′

i

ΦB′ // CN ′′i

where ΦB = lim←−B

NΦ(N) and ΦB′ = lim←−

B′

NΦ(N). The commutativity of the diagram

shows

(A.4) lim←−N

BΦ(N) = lim←−N

B′Φ(N).

Now (A.2) follows from (A.3) and (A.4). �

Lemma A.3. If Φ is admissible with respect to some B = {p1, p2, . . . , pn}, then Φ

is invertible.

Proof. Let Φ(N) : CN ′i/t


i/tN+1CN ′

i be as in the proof of the previ-

ous lemma. Since Φ(N) is defined by a finite product of invertible terms (1+ϕj)εj ,

Φ(N) is invertible on CN ′i/t

N+1CN ′i . Let Ψ

(N) : CN ′i/t


i/tN+1CN ′

i

be the inverse of Φ(N) and let

Ψ = lim←−N

Ψ(N).

This is the inverse of Φ since ΨΦ = lim←−NΨ(N)Φ(N) = id and ΦΨ = lim←−N

Φ(N)Ψ(N) =

id. �

The following lemma can be easily proved.

Lemma A.4. Any L-module isomorphism CNi → CNi can be realized by an ad-

missible product. �

Acknowledgments. The author would like to thank Professor Andrei Pajitnov for

valuable comments and for terminological and historical remarks on circle-valued

Morse theory. The author would also like to thank Takahiro Kitayama for informa-

tion on circle-valued Morse theory. The author is supported by JSPS Grant-in-Aid

for Young Scientists (B) 70467447.

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Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sap-

poro, Hokkaido, 060-0810, Japan

E-mail address: [email protected]

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