multiplicative structures subm 130626

8/13/2019 Multiplicative Structures Subm 130626

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MULTIPLICATIVE STRUCTURES ON THE TWISTED

K-THEORY OVER FINITE GROUPS

CESAR GALINDO, ISMAEL GUTIERREZ, AND BERNARDO URIBE

Abstract. Let Kbe a finite group and let G be a finite group acting over Kby automorphisms. In this paper we study two different but intimate relatedsubjects: on the one side we classify all possible multiplicative and associative

structures which one can endow the twisted G-equivariant K-theory over K,and on the other, we classify all possible monodical structures which one en-

dow the category of twisted and G-equivariant bundles over K. We achievethis classification by encoding the relevant information in the cochains of asub double complex of the double bar resolution associated to the semi-direct

product K G; we use known calculations of the cohomology of K, G andK G to produce concrete examples of our classification.

In the case on which K= G and G acts by conjugation, the multiplicationmap G G G is a homomorphism of groups and we define a shuffle ho-

momorphism which realizes this map at the homological level. We show thatthe categorical information that defines the Twisted Drinfeld Double can berealized as the dual of the shuffle homomorphism applied to any 3-cocycle of

G. We use the pullback of the multiplication map in cohomology to classifythe possible ring structures that Grothendieck ring of representations of the

Twisted Drinfeld Double may have, and we include concrete examples of thisprocedure.

IntroductionThe purpose of this work is to investigate a relationship existing among certain

tensor categories attached to a semi-direct product of groups (they include theTwisted Drinfeld Double of a discrete group), their fusion algebras and the coho-mology groups of the semi-direct product. These tensor categories, as well as someof our results, are closely related to abelian extensions of Hopf algebras and thecohomological description of Opext(CG, CF)), given by Kac in [16], cf. [19].

The abelian extension theory of Hopf algebras was generalized to coquasi-Hopfalgebras by Masuoka in [20]. Some of our results and constructions can be framed inthe abelian extension theory of coquasi-Hopf algebras in the particular case wherethe matched pair of groups is a semi-direct product. However, our approach to thesetensor categories does not follow Masuokas point of view, instead we use the con-cept of pseudomonoids in a suitable 2-monoidal 2-category associated to a group.

2010 Mathematics Subject Classification. (primary) 19L50, 18D10, (secondary) 16L35, 20J06.Key words and phrases. Twisted Equivariant K-theory, Twisted Drinfeld Double, Fusion Cat-

egory, Multiplicative Structure, Fusion Algebra.C.G would like to thank the hospitality of the Mathematics Department at MIT where part

of this work was carried out. C.G. was partially supported by Vicerrectora de Investigaciones dela Universidad de los Andes. I.G acknowledges and thanks the financial support of the Deutscher

Akademischer Austausch Dienst. B.U. acknowledges and thanks the financial support of theAlexander Von Humboldt Foundation.

1


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2 CESAR GALINDO, ISMAEL GUTIERREZ, AND BERNARDO URIBE

There are some reasons why we prefer to use the pseudomonoid approach: First, itexhibits more clearly the relationship between the cohomology of semi-direct prod-ucts and the multiplicative structures of the G-equivariant twistedK-theory over afinite groupK. Second, some terms of a spectral sequence associated to the doublecomplex have a direct interpretation in terms of obstructions and classification ofthe possible pseudomonoid structures. Third, some of our constructions and re-sults make sense in categories different from the category of sets; in particular, ifwe change to the Cartesian categories of locally compact topological spaces andchange discrete group cohomology by the Borel-Moore cohomology [21], we have amuch general theory where the categories of coquasi-Hopf algebras may not captureall the desired information.

There are two main reasons for our interest in the fusion algebra of the tensorcategories defined in this paper. On the one side, semisimple tensor categories canbe encoded in a combinatorial structure divided in two parts: the fusion algebra(or the Gorthendick ring of the tensor category) and a non-abelian cohomological

information provided by the F-matrices of the 6j-symbols, [24].And on the other, these fusion algebras are generalizations of K([G/G]), thew-twisted stringyK-theory of the groupoid [G/G] (see [25]).

In the case that the semi-direct product is finite, the associated tensor categoriesare fusion categories, and they belong to a bigger family of fusion categories denotedgroup-theoretical fusion categories for which many interesting results have beenestablished cf. [12, 22]. Since an explicit description of the fusion rules of thetensor categories studied in this paper, via induction and restriction of projectiverepresentations of certain subgroups ofG already appear in [26, Theorem 4.8], ourapproach focuses in determining the number of fusion category and fusion algebrastructures associated to a fix semi-direct product. We accomplish this task in severalsteps. First, we show that the information encoding a pseudomonoid with strict unitin the 2-category ofG-sets with twists over the groupKis equivalent to a 3-cocycle

in Z3(Tot

(A,(K G, T))), where the double complexA,(K G, T) is the subdouble complex without the 0-th row of the double bar resolution C,(K G, T)and whose total cohomology calculates the cohomology ofK G. Second, we showthat the information encoded in a pseudomonoid with strict unit in the 2-categoryof G-sets with twists K, is precisely the precise information required to endowthe category BunG(K) of projective G-equivariant complex vector bundles with amonodical structure; hence the isomorphism classes of bundles Groth(BunG(K))becomes a fusion algebra, and this fusion algebra structure could be alternativelyunderstood as a twisted G-equivariant K-theory ring over K. Third, we definethe twisted G-equivariant K-theory over the group Kand we show the conditionsunder which this group could be endowed with a multiplicative structure making ita ring; we define the group of multiplicative structures by MSG(K) and we showthat this group could be calculated by the use of a spectral sequence associated

to the complex Tot(A,(K G, T)). We study the canonical homomorphism

H3(Tot(A,(K G, T))) MSG(K) and we give an explicit description of its

kernel and its cokernel; a multiplicative structure on M SG(K) not appearing in theimage of endows the twisted G-equivariant K-theory ofKwith a ring structure,which is not the fusion algebra of the tensor categories BunG(K), or in other wordsan algebra structure, which is not possible to categorify.


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MULTIPLICATIVE STRUCTURES ON TWISTED EQUIVARIANT K-THEORY 3

Of particular interest is the case on which G = K and G acts on itself byconjugation. In this case, the cohomological information which was used in [9] todefine the Twisted Drinfeld Double Dw(G) forw Z3(G, T) defines an element inZ3(Tot(A,(G G, T))). The formul defining this 3-cocycle were reminiscentof the formul appearing in [11, Theorem 5.2] on the proof of the Eilenberg-Zilbertheorem, and we conjectured that there had to exist a way to define for any cocyclein Zn(G, T) a n-cocycle in Zn(Tot(A,(G G, T))) having similar properties asthe ones defined for n = 3. We show in this paper that indeed this is the case andits proof is based on two facts: first that the multiplication map : G G G,(k, g) = kg is a homomorphism of groups, and second, on a construction of anexplicit Shuffle homomorphismat the chain level, whose dual : C(G, T) Tot(C,(G G, T)) applied tow recovers the cocycle defined in [9], and moreoverthat in cohomology equals the pullback of, e.i. = : H(G, T) H(G G, T). Since the group G G is isomorphic to the product G G we get that the

mapH3(Tot(A,(K G, T))) M SG(K) is surjective, and since we know that

the cohomology class of the 3-cocycle that is defined in [9] could be recovered fromthe cohomology class ofw, we give a simple procedure to determine the fusionalgebras of Rep(Dw(G)) which are isomorphic to the G-equivariant K-theory ringKUG(K); at the end of this work this procedure exemplified in some interestingcases.

This paper is organized as follows. In Section 1 we provide background materialon the semi-direct products and the double bar complex associated to its cohomol-ogy. In Section 2 the Shuffle homomorphism of a trivializable semi-direct productis defined and some of its properties are shown. In Section 3 the 2-monoidal 2-category of twistedG-sets with strict unit and the 2-category of pseudomonoids inthis 2-category are defined and described using the complex Tot(A,(K G, T)).In Section 4 the tensor category of equivariant vector bundles over a group, ten-sor functors and monoidal natural isomorphism associated to the 2-category of

pseudomonoids in the 2-category of twisted G-sets with strict unit are defined.In Section 5 the obstruction to the existence of multiplicative structures over thetwistedG-equivariant K-theory over a group Kis described using the spectral se-quence associated to the filtration Fr := A,>r of the double complex A,. InSection 6 several concrete examples are completely calculated. We finish with anappendix in Section 7 on which we give the explicit relation of our tensor categoriesand coquasi-bialgebras.

1. Preliminaries

1.1. Semi-direct products. Let Kbe a discrete group endowed with an actionof the discrete group G defined through a homomorphism : G Aut(K); forsimplicity, for g G and k K denote the action by g(k) := (g)(k). Denote

by K G the group defined by the semi-direct product of G with K; as a setK G:= K Gand the product structure is defined by (a, g)(b, h) := (a g(b), gh).

The group K Gfits in the short exact sequence

1 K K G 2G 1

and we say that K G is isomorphic to another split extension

1 K E pG 1


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whenever there is an isomorphism :K G =E such that 2 =p . In what

follows we will outline the conditions under which the semi-direct product K G

is isomorphic to the direct product K Gas split extensions.For this recall that Inn(K) is the group of inner automorphisms of the group K,i.e. the automorphisms ofKinduced by conjugation, and that it fits in the shortexact sequences

1 Z(K) K Inn(K) 1

1 Inn(K) Aut(K) Out(K) 1

whereZ(K) denotes the center ofKand Out(K) denotes the group of outer auto-morphisms ofK.

Proposition 1.1. The semi-direct product K G is isomorphic to the productK G if and only if there is a homomorphism : G Ksuch that the followingdiagram commutes

G

K

Inn(K)

where the action ofG onKis given by the inner automorphisms defined by.

Proof. Suppose that there is a homomorphism : G Kwith the desired prop-erties. Define the isomorphism of sets

: K G K G, (a, g)(a(g), g);

note that on the one hand

((a, g) (b, h)) = (a(g)(b), gh)

=(a(g)b(g)1

, gh)= (a(g)b(h), gh)

and on the other hand

(a, g)(b, h) = (a(g), g)(b(h), h)

= (a(g)b(h), gh);

therefore the map is the desired isomorphism of groups.For the converse let us suppose that : K G K G is an isomorphism of

split extensions. In particular we have that

2((a, g)) = g

since the projection on the second coordinate on both groups produce the anchor

map toG. Define the map

: G K, g1((1, g))

and hence we have that (1, g) = ((g), g). Since is an isomorphism we havethat

((gh), gh) = (1, gh) = (1, g)(1, h) = ((g), g)((h), h) = ((g)(h), gh)

and therefore we obtain that the map is moreover a homomorphism of groups.


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Denote by : K Kthe isomorphism of groups induced by whenis definedby the equation

(a, 1) = ((a), 1).Applying to both sides of the equation

(1, g) (a, 1) = (g(a), g)

we obtain on the one side

(1, g)(a, 1) = ((g), g)((a), 1) = ((g)(a), g)

and on the other

(g(a), g) = (g(a), 1) (1, g) = ((g(a)), 1)((g), g) = ((g(a))(g), g),

implying that

(g)(a)(g)1 =(g(a))

and applying 1 we get

1((g))a1((g))1 =g(a)

which implies the desired equation

(1((g))) =(g).

Hence the homomorphism

:= 1 : G K

fits into the diagram

G

K

Inn(K).

Corollary 1.2. The semi-direct productK Inn(K) is isomorphic as split exten-sions to the direct productKInn(K)if and only if the natural short exact sequence1 Z(K) K Inn(K)1 splits.

Corollary 1.3. If the semi-direct productK G is isomorphic to K G then theaction ofG onKis given by inner automorphisms ofK, and there is a homomor-phism: G K that realizes such inner automorphisms.

Example 1.4. LetK=G and consider the conjugation action ofK on itself. Inthis case = and we can take = IdK. Therefore the map

: K K K, (a, g)ag

is a homomorphism of groups and

2 : K K K K, (a, g)(ag,g)

is an isomorphism.


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Remark 1.5. Note that whenever we have a homomorphism : G Ksuch that= then the map

: K G K, (k, g)k(g)becomes a group homomorphism. Moreover, the map

K G K K, (k, g)(k, (g))

is a homomorphism of groups

1.2. Bar resolution. Let us find an explicit model for the homology of the groupK G. For this, let us first setup the notation for the explicit model for the barresolution that we will use.

Take Ha discrete group and define the complex C(EH, Z) with

Cn(EH, Z) := ZHn+1

and with differential H : Cn(EH, Z) Cn1(EH, Z) defined by the equation ongenerators

H(h1, h2,...,hn+1) = (h2, h3,...,hn+1) +n

i=1

(1)i(h1,...,hihi+1,...,hn+1).

The complex (C(EH, Z), H) becomes a complex in the category of ZH-modulesif we endow each Cn(EH, Z) with the left ZH-module structure defined by theequation

h (h1,...,hn+1) := (h1,...,hn+1h1).

The augmentation map

: C0(EH, Z) Z, (h) = 1

is a map ofZH-modules and the complex C(EH, Z) becomes a ZH-free resolutionof the trivial ZH-module Z,

: C(EH, Z) Z.

The elements(h1, h2,...,hn, 1)

generate the ZH-module Cn(EH, Z) and therefore we could write them using thebar notation

[h1|h2|...|hn] := (h1, h2,...,hn, 1);

the differential Hin this base becomes

H[h1|h2...|hn] = [h2|h3|...|hn] +n

i=1

(1)i[h1|...|hihi+1|...|hn]

+ (1)n+1h1n [h1|...|hn1].

For a left Z

H-module W, the homology groups ofHwith coefficients in W aredefined asH(H, W) := H(C(EH, Z) ZHW)

and the cohomology groups ofHwith coefficients in W as

H(H, W) := H(HomZH(C(EH, Z), W)).

Since we have a canonical isomorphism of Z-modules

HomZH(Cn(EH, Z), W)=Maps(Hn, W),


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the cohomological differential Hin terms of the bar notation becomes

(Hf)[h1|h2...|hn] =f[h2|h3|...|hn] +

n

i=1(1)

i

f[h1|...|hihi+1|...|hn]

+ (1)n+1h1n f[h1|...|hn1].

1.3. Cohomology ofK G. Consider the double complex

C(EG, Z) ZC(EK, Z)

with differentialsG 1 and 1K. Denote by (g1,...,gp+1||k1,...,kq+1) a generatorinCp(EG, Z) Z Cq(EK, Z) and define the action of (k, g) K Gby the equation

(k, g) (g1,...,gp+1||k1,...,kq+1) := (g1,...,gp+1g1||g(k1),...,g(kq+1)k

1).

A straightforward computation shows that indeed it is an action and thereforeCp(EG, Z) ZCq(EK, Z) becomes a free Z(K G)-module. Since the differentials

G 1 and 1 Kare also maps of Z(K G)-modules, we could take the totalcomplex

Tot(C(EG, Z) ZC(EK, Z))

whose degree n-component is

Totn(C(EG, Z) ZC(EK, Z)) :=

p+q=n

Cp(EG, Z) ZCq(EK, Z)

and whose differential is

G 1 (1)p1 K

thus obtaining a free Z(K G) complex. Since the homology of the total complexof this double complex is just Z in degree 0,

H(Tot(C(EG,Z

) Z

C(EK,Z

)), G 1 (1)

p

1 K) =Z

,we have that

Tot(C(EG, Z) ZC(EK, Z)) Z

is a free Z(K G) resolution of Z.Making use of the bar notation we take the elements

[g1|...|gp||k1|...|kq] := (g1,...,gp, 1||k1,...,kq, 1)

as a set of generators ofCp(EG, Z) ZCq(EK, Z) as a Z(K G)-module; in thisbase we have the equality

(g1,...,gp, g||k1,...,kq, k) = (k1, g1) [g1|...|gp||g(k1)|...|g(kq)].

This choice of base provides us with an isomorphism of Z-modules

HomZ(KG)(Cp(EG, Z) ZCq(EK, Z), Z)=Maps(Gp Kq, Z)

that allows us to transport the dual of the differentials G 1 and 1 K toMaps(GpKq, Z); the induced differentials will be denoteb by Gand K. Thereforewe obtain:

Definition 1.6. LetCp,q

(K G, Z), p, q 0, be the double complex

Cp,q

(K G, Z) := Maps(Gp Kq, Z)


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with differentials G : Cp,q

Cp+1,q

and K : Cp,q

Cp,q+1

defined by theequations

(GF)[g1|...|gp+1||k1|...|kq] =F[g2|...|gp+1||k1|...|kq]

+

pi=1

(1)iF[g1|...|gigi+1|..|gp+1||k1|...|kq]

+ (1)p+1F[g1|...|gp||gp+1(k1)|...|gp+1(kq)]

(KF)[g1|...|gp||k1|...|kq+1] =F[g1|...|gp||k2|...|kq+1]

+

qj=1

(1)j F[g1|...|gp||k1|...|kj kj+1|...|kq+1]

+ (1)q+1F[g1|...|gp||k1|...|kq],

where by convention G0 K0 is the set with one point.

Hence we haveLemma 1.7. The cohomology of the total complex of the double complexC

,(K

G, Z), G, K is the cohomology of the group K G, i.e.

H

Tot(C,

(K G, Z)), G (1)pK

=H(K G, Z).

We can take a smaller double complex, more suited for our work, which is calledthe normalized double complex. Let us define it

Definition 1.8. The normalized double complex ofC,

(K G, Z), G, K is thedouble complex

Cp,q(K G, Z) Cp,q

(K G, Z)

consisting of maps F : Gp Kq Z such that F[g1|...|gp||k1|..|kq] = 1 whenever

gi= 1 or kj = 1. The differentials onC

p,q

are alsoG andK. We setupC

0,0

=Z

.It is known in homological algebra that the normalized complex of the bar res-

olution is quasi-isomorphic to the bar resolution (see page 215 in [13]). Thereforewe have

Lemma 1.9. The induced map on total complexes

Tot(C,(K G, Z)) Tot(C,

(K G, Z))

is a quasi-isomorphisms. Then

H (Tot(C,(K G, Z)), G (1)pK)=H

(K G, Z).

We can generalize our definition of the double complex to other coefficients.Denote by T the groupS1 and consider the exact sequence of coefficients

0Z

R

T

0.We will denote the complex Cp,q(K G, T) as the complex generated by maps

F : Gp Kq T on which F is 1 if one of the entries is the identity, and thedifferentialsG andKare the same ones as Definition 1.6 but changing the sumsby multiplications. Note also that we set up C0,0(K G, T) := T.

Then

H

Tot(C,(K G, T)), G (1)p

K

=H(K G, T).


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1.3.1. Decomposition of the cohomology ofK G. Note that the 0-th row of thedouble complex is isomorphic to the normalized bar cochain complex ofG

(C,0

(K G, Z), G)=(C

(G, Z), G),and the action of the differential Kon this row is trivial. Therefore, if we define

Definition 1.10. Let

A,(K G, Z) := C,>0(K G, Z)

be the sub double complex ofC,(K G, Z) with trivial 0-th row, and differentialsG and K.

Then we obtain

Lemma 1.11. There is a canonical isomorphism of double complexes

(C,(K G, Z), G (1)pK)=(A

,(K G, Z), G (1)pK) (C

(G, Z), G),

which induces a canonical isomorphism in cohomologyH(K G, Z)=H(Tot(A,(K G, Z))) H(G, Z),

and the respective one with coefficients in T,

H(K G, T)=H(Tot(A,(K G, T))) H(G, T)

whereA,(K G, T) = C,>0(K G, T)

2. The case of the trivializable semi-direct product K G

Whenever the semi-direct product K G is isomorphic to K G we know byProposition 1.1 and Remark 1.5 that there exists a homomorphism : G KsuchthatK G K K, (k, g)(k, (g)) is a homomorphism of groups. In this casewe have the homomorphisms

K G K K K, (k, g)(k, (g))k(g)

which induce the following homomorphism at the level of their homologies

H(K G, Z) H(K K, Z) H(K, Z).

Since the interesting information lies on the homomorphism we will investigateits properties.

2.1. The Shuffle homomorphism. In what follows we will describe how to obtainthe map at the chain level using the explicit models for H(K K, Z) andH(K, Z) defined previously. Its definition needs some preparation.

Take a base element [g1|...|gp||k1|...|kq] inCp(EK, Z) Z Cq(EK, Z) and think ofit as a way to represent p q as the product of the simplices [g1|...|gp][k1|...|kq]in B K. For Shuff(p, q) a (p, q)-shuffle, i.e. an element in the symmetric groupSp+q such that (i)< (j) whenever 1 i < j p or p + 1 i < j p +q, wecan define an element

[g1|...|gp|k1|...|kq] Cp+q(EK, Z)

such that[g1|...|gp|k1|...|kq] := [s1|...|sp+q] with

s(i)=

gi if i p

gp+(i)i+1 . . . gp1gpkip(gp+(i)i+1 . . . gp1gp)1 if i > p.


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From the Eilenberg-Zilber theorem it follows that the set

{[g1|...|gp|k1|...|kq] : Shuff(p, q)}

is a subdivision in simplices of dimension p+ q of the product of the simplices[g1|...|gp] [k1|...|kq].

An equivalent way to see the elements [g1|...|gp|k1|...|kq] is the following: a(p, q)-shuffle can also be understood as a way to parameterize a lattice path ofminimum distance from the point (0, 0) to the point (p, q); one moves one unit tothe right in the steps (1),...,(p) and one unit up in the steps (p + 1),...,(p + q).We label the horizontal and vertical edges in the rectangular lattice defined by thepoints (0, 0), (p, 0), (p, q), (0, q) by the following rule:

The horizontal path between (i 1, j) and (i, j) is labeled with gi indepen-dent ofj .

The vertical path between (p, j 1) and (p, j) is labeled with kj and allthe other vertical paths are labeled in such a way that the squares become

commutative squares (when thinking of the labels as maps). This impliesthat the vertical edge from (i, j 1) to (i, j) is labeled with

gi+1 . . . gp1gpkj (gi+1 . . . gp1gp)1.

Since parameterizes a path in the lattice, the element[g1|...|gp|k1|...|kq] encodesthe labels that the path follow in order to go from (0, 0) to (p, q).

For example, consider the (3, 2)-shuffle

:=

1 2 3 4 51 3 5 2 4

then the element [g1|g2|g3|k1|k2] can be seen as the path

(2, 2) g3 (3, 2)

(1, 1) g2 (2, 1)

g3k2g1

3

g3 (3, 1)

k2

(0, 0) g1 (1, 0)

g2g3k1(g2g3)1

g2 (2, 0) g3 (3, 0)

k1

and therefore

[g1|g2|g3|k1|k2] = [g1|g2g3k1g13 g

12 |g2|g3k2g

13 |g3].

We can now define the map of complexes which in homology realizes :

Definition 2.1. Let

:Cp(EK, Z) ZCq(EK, Z) Cp+q(EK, Z)

be the graded homomorphism defined on generators by the equation

(g1,...gp, g||k1,...,kq, k) :=

Shuff(p,q)

(gk)1 (1)||[g1|...|gp|gk1g1|...|gkqg

1]

where|| denotes the sign of the permutation . This is usually called the Shufflehomomorphism.


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We claim that the Shuffle homomorphismis an admissible productin the sensedescribed in [7, IV-55, p. 118]

Theorem 2.2. The graded homomorphism: Tot(C(EK, Z) ZC(EK, Z)) C(EK, Z)

is a chain map that satisfies the equations

(a1, h1) (g1,...gp, g||k1,...,kq, k)

= (ha)1 (g1,...gp, g||k1,...,kq, k)

( )(k, g) = (kg).

Hence it induces a graded homomorphism in homology

: H(K K, Z) H(K, Z)

which is equal to the one induced by the pushforward.

Proof. Let us start by proving the equations. For the first one, we have on the oneside

((a1, h1) (g1,...gp,g||k1,...,kq, k))

=

g1,...gp, gh||h1k1h,...,h

1kqh, h1kha

=

Shuff(p,q)

(gkha)1 (1)||[g1|...|gp|gk1g1|...|gkqg

1]

and on the other

(ha)1 (g1,...gp,g||k1,...,kq, k)

= (ha)1

Shuff(p,q)

(gk)1 (1)||[g1|...|gp|gk1g1|...|gkqg

1]

= Shuff(p,q)

(gkha)1 (1)||[g1|...|gp|gk1g1|...|gkqg

1];

this in particular implies thatis a homomorphism which preserves the respectivemodule structures, namely that

((a, h)(g1,...gp, g||k1,...,kq, k)) = (a, h)(g1,...gp, g||k1,...,kq, k).

For the second, simply note that

( )(k, g) = 1 1 = 1 =(kg)

which implies that (k, g)) = ( )(k, g).The proof of the fact that is a chain map, namely that

K = (K 1 + (1)p1 K),

is essentially included in the proof of [11, Theorem 5.2]; the decomposition in sim-

plices of dimensionp + qof the product of the simplices [g1|...|gp] [k1|...|kq] inBKis done by choosing appropriately p+qedges with the use of the (p, q)-shuffles asfollows

[g1|...|gp||k1|...|kq] =

Shuff(p,q)

(1)[g1|...|gp|k1|...|kq].

With this decomposition of [g1|...|gp] [k1|...|kq], its boundary can be calculatedas (K1+(1)

p1K)[g1|...|gp][k1|...|kq] or alternatively as K([g1|...|gp||k1|...|kq]);then it follows that is a chain map.


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Now, since is a chain map which preserves the module structures, then itinduces a chain map at the level of the coinvariants

Tot(C(EK, Z) ZC(EK, Z)) Z(KK) Z C(EK, Z) ZK Z

which defines a homomorphism

: H(K K, Z) H(K, Z).

This homomorphism must be equal to the pushforward sincepreserves themodule structures defined by .

If we have a homomorphism of groups : G Kfor which the map : KG =

K G, (k, g)(k, (g)) induces an isomorphism of semi-direct products, then wehave that the map

( 1) : Cp(EG, Z) ZCq(EK, Z) Cp+q(EK, Z)

[g1|...|gp||k1|...|kq][(g1)|...|(gp)||k1|...|kq]

induces a chain map

( 1) : Tot(C(EG, Z) ZC(EK, Z))C(EK, Z)

preserving their respective module structures, and hence inducing a homomorphism

( ( 1)): H(K G, Z) H(K, Z)

which is equal to the pushforward map of the composition ; i.e.

( ) = ( ( 1)) : H(K G, Z) H(K, Z).

2.2. The dual of the Shuffle homomorphism. Dualizing the map, we obtaina homomorphism

:Cn

(K, Z))

p+q=n

Cp,q

(K K, Z)

(F)[s1|...|sn]

p+q=n

F([s1|...|sp||sp+1|...|sp+q])

which induces a cochain map

: (C

(K, Z), K) (Tot(C

,(K K, Z)), G (1)

pK);

here we have kept the notation of the differentials as G and Kin order to avoidconfussions. A straightforward calculation shows that the cochain map preservesnormalized cochains

: (C(K, Z), K) (Tot(C,(K K, Z)), G (1)

pK)

and therefore it induces a homomorphism at the level of cohomologies

:H(K; Z) H(K K, Z)

which by Theorem 2.2 is equal to the pullback of the homomorphism of groups .If we bundle up our previous discussion we have


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Theorem 2.3. The homomorphism in cohomology

:H(K, Z) H(K K, Z)

defined by the cochain map

:Cn(K, Z))

p+q=n

Cp,q(K K, Z)

(F)[s1|...|sn]

p+q=n

F([s1|...|sp||sp+1|...|sp+q])

is equal to

:H(K, Z) H(K K, Z)

which is the pullback of the group homomorphism: K K K, (k, g)kg .

2.3. Further properties of the Shuffle homomorphism. Consider the homo-morphisms

K :K K K, G: KK K

x(x, 1K) g (1K, g).

and note that they fit into the commutative diagram

K K

=

K K

KG

=K

which induces the commutative diagram

H(K, Z)=

=

H(K, Z) H(K K, Z)

K

G

H(K, Z).

(2.1)

From Lemma 1.11 we know that there is a canonical isomorphism

Hp(K K, Z)=Hp(Tot(A,(K K, Z))) Hp(K, Z)

and the homomorphism G is precisely the projection on the second component ofthe direct sum Hp(Tot(A,(K K, Z))) Hp(K, Z).

Moreover, by defining the chain map

1 :Cn(K, Z))

p+q=n, q>0

Ap,q(K K, Z)(2.2)

(1 F)[s1|...|sn]

p+q=n, q>0

F([s1|...|sp||sp+1|...|sp+q])

we have that the chain map

:C(K, Z)) Tot(A,(K K, Z)) C,0(K K, Z)

is isomorphic to the chain map

1 1 : C(K, Z)) Tot(A,(K K, Z)) C,0(K K, Z)


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and therefore we have that at the cohomological level we obtain the commutativediagram

Hp(K, Z)=

1

Hp(K, Z) Hp(Tot(A,(K K, Z)))

K |A

(2.3)

for all p >0, where at the cochain level K|A is simply the projection map on the0-th column

K|A: Tot(A,(K K, Z)) A,0(K K, Z)=C>0(K, Z).

Therefore, if

Definition 2.4. Let

B,(K G, Z) :=C>0,>0(K G, Z)

be sub double complex of C,(K G, Z) with trivial 0-th row and trivial 0-thcolum, and differentials G andK.

We obtain

Lemma 2.5. The homomorphism

Hp(K, Z) Hp(Tot(B,(K K, Z))) Hp(Tot(A,(K K, Z)))

x y1 x + y

is an isomorphism for allp > 0.

Proof. The short exact sequence of complexes

0 Tot(B,(K K, Z)) Tot(A,(K K, Z)) C>0(K, Z) 0

induces a long exact sequence in cohomology groups

Hp(Tot(B,(K K, Z))) Hp(Tot(A,(K K, Z))) K |A Hp(K, Z)

which splits by diagram (2.3).

Therefore we can conclude that there is a canonical splitting of the cohomologyofK K in terms of the cohomology ofKand of the cohomology of the doublecomplexB,(K K, Z).

Proposition 2.6. The homomorphism

Hp(K, Z) Hp(Tot(B,(K K, Z))) Hp(K, Z) =Hp(K K, Z)

x y z 1 x + y+ 2 z

is an isomorphism for all p > 0. Here 2 : K K K, (a, g) g denotesthe homomorphism induced by the projection on the second coordinate satisfying2 G= 1.


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3. Categorical definitions

3.1. 2-category of discrete G-sets with twist. We shall fix a discrete group G.

We define the 2-category of discrete G-sets with twist as follows:(1) Objects will be called discreteG-sets with twist, and they are pairs (X, ),

whereXis a discrete left G-set and Z2G(X, T) is a normalized 2-cocyclein the G-equivariant complex ofX, i.e. a map

: G G X T,

such that

[|||x] [|||x]1 [| ||x] [|||x]1 = 1

for all , , G, x X.Note that the previous equation is equivalent to the equation G = 1,when we see as element in C2,1(X G, T).

(2) Let (X, X ), (Y, Y) be discrete G-sets with twist. A 1-cell from (X, X )to (Y, Y), also called a G-equivariant map, is a pair (L, ),

(X, X ) (L,)(Y, Y)

where L: X Y is a morphism ofG-sets, C1G(X, T) is a normalized cochain such that G() = L

(Y)(X )1,

i.e., a map

: G X T

such that

[||x] [|x]1 [|| x] =Y

[|||L(x)] X

[|||x]1,

for all , G, x X.(3) Given two 1-cells (L, ), (L, ) : (X, X ) (Y, Y), a 2-cell : (L, )

(L, )

(X, X )

(L,)

(L,)

(Y, Y)

is 0-cochain C0G(X, T) such thatG() = 1, i.e., a map : X T,

such that

[x][x]1 =[||x] [||x]1

for all G, x X.

Let us define the composition on 1-cells and 2-cells in the following way. Let(F, F) : (X, X ) (Y, Y) and (G, G) : (Y, Y) (Z, Z) two 1-cells, definetheir composition as

(G, G) (F, F) = (G F, F(G)F) : (X, X ) (Z, Z),


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(X, X ) (F,F)

(GF,F(G)F)

(Y, Y)

(G,G)(Z, Z)

and if : (L, ) (L, ) and : (L, ) (L, ) are 2-cells, their compositionis the product of the maps, namely

=: : (L, ) (L, )

(X, X )(L,)

(L,)

(L,)

(Y, Y) = (X, X )

(L,)

(L,)

(Y, Y)

A straightforward calculation implies that

Lemma 3.1. The composition of 1-cells and 2-cells satisfy the axioms of a 2-category.

3.2. Pseudomonoids. A strict 2-monoidal 2-category is a triple (B,, I) whereBis a 2-category, : B B B is a 2-functor and I is an object in B, such that ( IdB) = (IdB ) and I X=X I= Xfor every object in B (for

more details see [6]).

Definition 3.2. Given a strict 2-monoidal 2-category (B,, I), a pseudomonoidin Bconsists of:

an object C B,

together with:

a multiplication 1-cell m : C C C, an identity-assigning1-cell I :I C,

together with the following 2-isomorphisms:

the associator:

C

C

C

C C C C

C

idm

mid

m

m

a (3.1)


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the left and right unit laws:

I C C C C I

C

Iid idI

m

=

=

r

(3.2)

such that the following diagrams commute:

the pentagon identityfor the associator:

(3.3) m(mid)(midid)

m(mid)(idmid)

m(idm)(ididm)

m(idm)(idmid)

m(mm)

m(aid)

a(midid)

a(idmid)

m(ida)

a(ididm)

where we use the equalities

(id m) (m id id) = m m

(m id) (id id m) = m m

in order to compose the upper 1-cellsa (m id id) anda (id idm),equalities which follow from the fact thatis a 2-functor.

the triangle identityfor the left and right unit laws:

(3.4) m(mid)(idIid)a(idIid)

m(rid)

m(idm)(idIid)

m(id)

m

where we use the fact thatCI =C = ICto make sense of the diagonalarrows.

Remark 3.3. The 2-category of G-sets with twist has a 2-monoidal structure,where the product of two objects (X, X ), (Y, Y) is given by

(X, X ) (Y, Y) = (X Y, X Y),

whereX Y is aG-set with the diagonalG-action andX Y :=1 (X )2 (Y).

In an analogous way we construct the product for 1-cells and 2-cells. A unit objectis any fixedG-set with one element and the constant function 1 for 2-cocycle.

3.3. Pseudomonoids in the 2-category of G-sets with twist. We are inter-ested in studying pseudomoniods in the 2-category of G-sets with twists, but inorder to get normalized cocycles we are forced to consider only pseudomonoidswhere the identity-assigning1-cell is strict in the sense that the cochain (the sec-ond component of the 1-cell) is trivial, and furthermore, that the unit constraintsin diagram (3.2) are identities, namely that the diagram (3.2) commutes. We willcall these pseudomonoids with strict unit.


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Proposition 3.4. A pseudomonoid with strict unit in the 2-category ofG-sets withtwists is equivalent to the following data:

A monoid(K,m, 1), whereK is aG-set, m isG-equivariant and1 K isaG-invariant element. C2,1(K G, T), C1,2(K G, T), C0,3(K G, T) such that

is a three cocycle in

Tot(A,(K G, T)), G (1)p

K

with

A,(K G, T) the double complex introduced in Definition 1.10.

Proof. A pseudomonoid in the 2-category ofG-sets is:

i) An object C = (K, ) whereK is a G-set and

: G G K T

such that is normalized in the components of the group G and thatG= 1.

ii) A multiplication 1-cell

m= (m, ) : (K K, ) (K, )

such thatm : K K K is aG-equivariant map, and a map

: G K K T

satisfying the equation

G= m ( )1.(3.5)

iii) An identity-assigning 1-cell

I = (1K, ) : ({}, 1) (K, )

where1K :{} Kis a map choosing aG-invariant element 1K :=1K()in K, and : G {} T is the constant map 1 because we are onlyconsidering pseudomonoids with strict unit. The cochain condition on

reads(G)[g1|g2||] =[g1|g2||1K]

and sinceis the constant function it follows that [g1|g2||1K] = 1, namelythat is normalized in the K variable.

iv) The left hand side of diagram (3.2) translates to the diagram

({} K, 1 )

(2,1)

(1Kid,1)(K K, )

(m,)

(K, )

where 2 is the projection on the second component. The composition ofthe 1 at the level of the G-sets implies the equation

m(1K, h) = h

for any element h K. Now, the composition of the 1-cells at the level ofthe cochains implies the equation

[g||] [g||1K|h] = 1

with g G and h K. Since the cochain is equal to the constantfunction 1, we have that [g||1K|h] = 1. Applying the same arguments as


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above we conclude that[g||h|1K] = 1 and therefore the left and right unitlaws imply that is normalized on the Kcomponents, and moreover that1

Kis a unit for the multiplication map m.

v) Diagram (3.1) translates to the diagram

(K K K, )(mid,1)

(idm,1)

(K K, )

(m,)

(K K, )

(m,)

(K, )

whose commutativity at the level ofG-sets implies that the multiplicationm : K K K is associative, and at the level of cochains the diagramimplies the equation

(G)[g||h1|h2|h3] =(3.6)

[g||h2|h3] [g||h1h2|h3]1 [g||h1|h2h3] [g||h1|h2]

1.

vi) Diagram (3.3) translates into the equation

[k1k2|k3|k4] [k1|k2|k3k4] =[k1|k2|k3] [k1|k2k3|k4] [k2|k3|k4](3.7)

for all elements k1, k2, k3, k4 in K.vii) Diagram (3.4) translates into the equality

(3.8) [k1|1K|k2] = 1

for all k1, k2 K, because the unit constraints are trivial.

Fromii)we have that the multiplication mapm : K K Kis a G-equivariantmap and v) tells us that the multiplication m is associative. From iii) we knowthatKis provided with aG invariant element 1Kandiv) tells us that this elementis a left and right unit for the multiplication m. We have then that (K,m, 1K) is amonoid endowed with a G-action compatible with m and 1K.

SinceK is aG-equivariant monoid with unit, we can use the notation of Defini-tion 1.6 to see that equation (3.5) can be written as

G()K() = 1,

equation (3.6) becomesG()K()

1 = 1

and equation (3.7) becomesK() = 1.

We furthermore have that G = 1 by the definition of an object in the category

ofG-sets with twist.Equation (3.8) implies that is normalized in the variable of the middle; thisfact, together with the fact that 2 = 1 implies that is normalized in all thevariables since

1 = K[1K|1K|k1|k2] =[1K|k1|k2]

1 = K[k1|k2|1K|1K] =[k1|k2|1K].

From iii) we know that is normalized in the K coordinate, and from iv) weknow that is normalized in the K coordinates. Therefore the maps , and


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are normalized in all variables since their normalization on coordinates ofG followfrom the definition of the 2-category ofG-sets with twists.

Summarizing we have that C2,1(K G, T), C1,2(K G, T) and

C0,3(KG, T) such that is a three cocycle in

Tot(A,(K G, T)), G (1)p

K

because we have that

(G (1)p

K )( ) =G() K()G() K()1G() K()

=1 1 1 1,

or in a diagram

4 1

3 1

2 1

1 1

0 1 2 3

KG

(K)1

G

KG

From the construction above, it is easy to see that if we are given the G equivari-ant monoid (K,m, 1K) with unit, plus the cocycle then one can constructin a unique way a pseudomonoid with strict unit in the 2-category ofG-sets withtwists. This finishes the proof.

Remark 3.5. For a fixed G equivariant monoid (K,m, 1K), the possible pseu-domonoid structures with strict unit in the 2-category of G-sets associated over(K,m, 1K) are classified by elements in

Z3(Tot(A,(K G, T))),

namely, 3-cocycles in the total complex ofA,(K G, T).

Definition 3.6. Given pseudomonoids (C, m, I, a) and (C, m, I, a) in a 2-categoryB, a morphism

F : (C, m, I, a)(C, m, I, a)

consists of:

1-cell F :C C

equipped with:

a 2-isomorphism

C

C

C C C

C

m

FF

F

m

F2


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a 2-isomorphism

C C

I

I

I

F

F0

(3.9)

such that diagrams expressing the following laws commute:

compatibility ofF2 with the associator:

m (m id) (F F F) m (F F) (m id) F m (m id)

m (id m) (F F F) m (F F) (id m) F m (id m)

m(F2F) F2(mid)

Fa

a(FFF)

m

(FF2)

F2(idm)

compatibility ofF0 with the left unit law:

m (I F) F

m (F F) (I id) F m (I id)

F

F

m(F0F)

F2(Iid)

compatibility ofF0 with the right unit law:

m (F I) F

m (F F) (id I) F m (id I)

Fr

Fr

m

(FF0)

F2(idI)

Definition 3.7. Given two pseudomonoids with strict unit K = (K,m, 1, , , )andK = (K, m, 1, , , ) in the 2-category ofG-sets with twists, a morphismF : K K is a morphism of pseudomonoids (as in Definition 3.6) such that the2-isomorphismF0 of diagram (3.9) is an identity.

Proposition 3.8. A morphism of pseudomonoids with strict unit in the 2-categoryofG-sets with twistsF :K K consists of the tripleF= (F ,,) with

F :K K

aG-equivariant morphism of monoids, and cochains C1,1(K G, T) and

C0,2(K G, T) such that

(G (1)p

K )( 1) = F/ F/ F/.

Proof. Following Definition 3.6 a morphism F :K K consists of:

i) A 1-cell (F, ) : (K, ) (K, ), i.e. a G-equivariant map F : K K

and a normalized cochain C1G(K, T) such that

(3.10) G= F/.


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ii) A 2-cell C0G(K K, T)

(K K,

)

(m,)

(FF,)

(K, )(F,)

(K K, )

(m,)

(K, )

such that

G= ( m)/( F).

Note that at the level of the G sets the diagram is commutative, thereforeF :K K preserves the multiplication; and since

K= (m)1,

we can rewrite the equation above as:

G K= /F.(3.11)

iii) The commutativity of the diagram (3.9)

(K, ) (K, )

({}, 1)

I

I

(F,)

implies that the map F :KK preserves the unit and that (g, 1) = 1for anyg G, namely that is normalized in the Kvariable and thereforewe could say that C1,1(K G, T).

iv) The commutativity of with the associator. This implies that Fpreservesthe associativity of the multiplications m and m, and that the followingequation is satisfied

[F(a1)|F(a2)|F(a3)] [a2|a3] [a1|a2a3] =[a1|a2] [a1a2|a3] [a1|a2|a3]

for all a1, a2, a3 K. Note that this last equation can be written as

K= /F.(3.12)

v) Compatibility with the left and right units, but as the 2-cellsF0, l ,r,l andr are identities, then this implies that [1|a] = 1 = [a|1] and thereforewe have that is normalized in the Kvariables and we can assume that C0,2(K G, T).

From the previous arguments it follows that F : K K is a G-equivariantmorphism of monoids, and moreover, calculating the differential, we have that

(G (1)p

K )( 1) = G() K()

1G()1 K()

1

= F/ F/ F/,

where the second equality follows from equations (3.10), (3.11) and (3.12). In adiagram


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3 F/

2 1 F/

1 F/

0 1 2

K

G

(K)1

G

This finishes the proof.

Definition 3.9. Given morphisms F = (F, F0, F2) and G = (G, G0, G2) from(C, m, I, a) to (C, m, I, a) pseudomonoids in B, a 2-morphism s: F G is a2-cells : F Gin Bsuch that the following diagrams commute:

compatibility with F2 andG2:

m (F F) m

(ss)

F2

m (G G)

G2

F m

sm G m

(3.13)

compatibility with F0 andG0:

F

I G

I

I

F0

G0

sI

(3.14)

Proposition 3.10. Given morphisms of pseudomonoids with strict unit in the 2-category ofG-sets with twistsF= (F ,,) andF = (F, , ), withF, F :K K, a 2-morphism: F F is a cochain: C0,1(K G, T) such that

(G (1)p

K )() = (/, /).

Proof. Let : (F, ) (F, ) be the 2-cell defined by the 2-morphism, then wehave that

G() = /.

Now, by diagram (3.13) we have that

K() = /,

and by diagram (3.14) we have that is a normalized cochain.

Remark 3.11. Propositions 3.4, 3.8 and 3.10 imply that the relevant informationencoded in cochains for the 2-category of pseudomonoids with strict unit ofG-setswith twists, is given by the cochains of the total complex

Tot(A,(K G, T)), G (1)p

K


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For a fixed G-equivariant monoid (K,m, 1K), Proposition 3.4 tells us that the3-cocyclesZ3(Tot(A,(K G, T))) are in one to one correspondence with the setof possible pseudomonoid structures with strict unit in the 2-category of G-setswith twist over K. If we only consider invertible morphisms of pseudomonoids asdefined in Proposition 3.8, we may define a groupoid which encodes the equivalenceclasses of pseudomonoid structures over K. Let us be more explicit.

3.4. Equivalence classes of pseudomonoid structures over a fixed monoid.

Definition 3.12. Fix a G-equivariant monoid (K,m, 1K). Define the groupoid

PsdmnG(K) whose set of objects is Z3(Tot(A,(K G, T))) and whose mor-phisms are invertible morphisms of pseudomonoids as defined in Proposition 3.8.The groupoid PsdmnG(K) encodes the information of all pseudomonoid strucures

overKand its coarse moduli space|PsdmnG(K)|, i.e. the set of equivalence classesdefined by the morphisms, is the set of equivalence classes of pseudomonoid struc-tures onK.

Note that a morphism in PsdmnG(K) consists of a triple (F ,,) where F :K Kmust be aG-equivariant automorphism. If we denote by

AutG(K) :={fAut(K) : f(gk) = gf(k) for all g G}

then we have that the group AutG(K) is isomorphic to the subgroup of Aut(KG)which leaves the G fixed; for a G-equivariant automorphismf AutG(K) we canassociate the automorphism fAut(K G) by the equation

f(k, g) := (f(k), g).

Since the automorphism f leavesG fixed, then the groups of automorphism act onthe double complex A,(K G, Z); we claim

Lemma 3.13. The set of equivalence classes of pseudomonoid structures onK is

isomorphic can be described by the quotient|PsdmnG(K)|=H3(Tot(A,(K G, T)))/AutG(K).

Proof. We first perform the quotient with the morphisms (F ,,) where F is theidentity on K; this quotient is precisely H3(Tot(A,(K G, T))). Then we seethat elements ofH3(Tot(A,(K G, T))) lying on the same orbit of the action ofAutG(K) define equivalent pseudomonoid structures.

In particular we may conclude that ifH3(Tot(A,(K G, T))) = 0, then allpseudomonoid structures with strict unit in the 2-category ofG-sets with twist overKare isomorphic to the trivial one. Since we are interested in finding pseudomonoidstructures with strict unit in the 2-category ofG-sets with twist over K non iso-morphic to the trivial one, we will calculate the group H3(Tot(A,(K G, T))),

the group AutG(K) and the set |PsdmnG(K)| for some particular examples.

The main tooll we will use in order to calculate the group H3(Tot(A,(K G, T))) will be the Lyndon-Hochschild-Serre spectral sequence. This spectral se-quence can be obtained if the complex Tot(A,(K G, T)) is filtered by thecomplexes

Fn := Tot(An,(K G, T))

thus defining a spectral sequence whose second page becomes

Ep,q2 =H

p(G, Hq(K, T))


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whereG acts onHq(K, T) through the induced action ofGonK; note in particularthat

E

0,q

2 =H

q

(K,T

)

G

andE

p,0

2 =H

p

(G,T

).On the other hand we have that

AutG(K)=CAut(K)((G))

where : G Aut(K) determines the action ofG, and

CAut(K)((G)) :={fAut(K) : [f, (g)] = 1 for allg G}

is the centralizer of(G) on Aut(K), consisting of the automorphisms ofKwhichcommute with the G-action. In particular when G= Aut(K) we have that AutG(K) =Z(Aut(K)). Let us see some examples:

3.4.1. K = Z/p for primep > 2, andG = Aut(Z/p) = Z/(p 1). Here we havethat AutG(K) = Z(Aut(K))= Z/(p 1) and that

H0(G, H3(K, T)) = H3(K, T)G=(Z/p)Z/(p1) = 0

H1(G, H2(K, T)) = Hom(Z/(p 1), 0) = 0

H2(G, H1(K, T)) = H2(Z/(p 1), Z/p) = 0

where the last equality follows from the fact that Hn(G, M) is anhilated by|G|forall n > 0 [7, III.10.2]. In this case H3(Tot(A,(K G, T))) = 0 and thereforeall pseudomonoid structures on K = Z/p are equivalent to the trivial one. Thisexample could be generalized as follows:

3.4.2. Groups with order relatively prime. Let us recall two facts. First, ifG is afinite group of order m, r is a positive integer with (m, r) = 1 and Ar = 0, thenHn(K, A) = 0 for all n and all subgroups K ofG, see [17, Proposition 1.3.1]. Andsecond, if e is the exponent ofH2(G, T) then e2 divides the order of G, see [17,Theorem 2.1.5].

Then for the case on which |G| is relatively prime to |K| we obtain that

H1(G, H2(K, T)) = 0 and H2(G, H1(K, T)) = 0.

Therefore we have that H3(Tot(A,(K G, T)))=H3(K, T)G and

|PsdmnG(K)|=H3(K, T)G/AutG(K).

3.4.3. The dihedral group Dn as a semi-direct product. The dihedral group is iso-morphic to Z/n Z/2 when Z/2 acts on Z/n by multiplication of 1. Sincethe induced action of Z/2 on the cohomology ring H(Z/n, Z) = Z[x]/nx mapsx x, we have that x2 x2 and thereforeH4(Z/n, Z)Z/2 = Z/nand

H2

(Z/2, H2

(Z/n, Z)) = H2

(Z/n, Z)Z/2

=Z/2 if n is even

0 if n is odd.

Since

H4(Dn, Z) =

Z/2 Z/2 Z/n if n is even

Z/2 Z/n if n is odd

we know that

H3(Tot(A,(Z/n Z/2, T))) =

Z/2 Z/n if n is even

Z/n if n is odd,


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and therefore

|PsdmnZ/2(Z/n)|= (Z/2 Z/n)/Aut(Z/n) if n is even(Z/n)/Aut(Z/n) if n is odd.

because in this case AutZ/2(Z/n) = Aut(Z/n).In particular, when n = 4 we have that Aut(Z/4) = Z/2. and therefore the

action of AutZ/2(Z/4) on H3(Tot(A,(Z/4 Z/2, T))) is trivial. Hence

|PsdmnZ/2(Z/4)|= Z/2 Z/4.

3.5. The case of the group acting on itself by conjugation. Perhaps themost known pseudomonoid with strict unit in the 2-category ofG-sets with twistwas introduced by Dijkgraaf, Pasquier and Roche in [9, Section 3.2] while definingthe quasi Hopf algebra Dw(G) with for w Z3(G; T). In the equations (3.2.5) and(3.2.6) of [9] they defined a 3-cocycle w w w Z

3(Tot(A,(G G, T))) bythe equations 1

w[g|h||x] :=w[g|h|x] w[ghxh1g1|g|h]

w[g|hxh1|h]

w[g||x|y] :=w[g|x|y] w[gxg1|gyg1|g]

w[gxg1|g|y]

w[x|y|z] :=w[x|y|z],

wherewwas used to define the algebra law,wto define the coalgebra law, andwencoded the fact that the coproduct is quasicoassociative (and other coherences).

This quasi Hopf algebra Dw(G) is known as the Twisted Drinfeld DoubleofGtwisted by w (cf. [25]). Firstly we claim the following.

Lemma 3.14. The 3-cocyclew w w equals1w, the image ofw under the

restricted shuffle homomorphis

1w defined in (2.2).Proof. From the following diagrams associated to (1w)[g|h|k]

g h

k

g

hkh1

h

ghkh1g1

hkh1 h

g

h

k

ghg1

gk

ghg1

gkg1

g

we obtain thatw w w = 1w.

Therefore by Lemma 1.11, Theorem 2.2 and Lemma 2.5 we get

Proposition 3.15. The cohomology class

[w w w] [w] H3(Tot(A,(G G, T))) H3(G, T)

1In order to get exactly the same formul it is necessary to change G by Gop.


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is equal to [w] where : G G G, (a, g) ag is the multiplication map.Moreover we obtain the isomorphism

H3

(G, T) H3

(Tot

(B,

(G G, T))) H3

(Tot

(A,

(G G, T)))[w] [x][w w w] + [x].

Now, sinceG G=G G, we have that the Lyndon-Hochschild-Serre spectralsequence collapses at the second page. And since the action of G on G is givenby conjugation, then the action ofG on H(G, T) is trivial. Hence we have thatH3(Tot(B,(G G, T))) sits in the middle of the short exact sequence

0H2(G, Hom(G, T)) H3(Tot(B,(G G, T))) Hom(G, H2(G, T)) 0.

Moreover, in the present situation we have (G) = Inn(G), and therefore

AutG(G) = CAut(G)(Inn(G)),

namely the group of automorphisms ofG which commute with all inner automor-phisms.

With the previous calculations at hand we can calculate the group H3(Tot(B,(GG, T))) and AutG(G) in some particular examples:

3.5.1. G simple and non abelian: When G is simple and nonabelian, its abelian-izationG/[G, G] is trivial. Therefore Hom(G, T) and Hom(G, H2(G, T)) are trivialand hence

H3(Tot(B,(G G; T))) = 0.

So we have that|PsdmnG(G)|=H3(G, T)/AutG(G).

When G is the alternating group An we have that AutAn(An) =CSn(An) = 1and therefore

|PsdmnAn(An)|=H3(An, T), forn = 6 and n >4.

In particular when n = 5 we have that H3(A5, T) = Z/120 and hence|PsdmnA5(A5)|= Z/120.

3.5.2. Binary icosahedral group. The binary icosahedral group A5 is a subgroup ofSU(2) that can be obtained as the pullback of the diagram

A5

SU(2)

A5 SO(3)

whereA5embeds inSO(3) as the group of isometries of an icosahedron. This groupsatisfiesH1(A5, Z) = 0, H2(A5, Z) = 0 and H3(A5, Z) = Z/120 (see [1, Page 279]),

and therefore H1

(A5, Z) = H2

(A5, Z) = H3

(A5, Z) = 0 and H4

(A5, Z) = Z/120.Hence H1(A5, T) = H2(A5, T) = 0, H

3(A5, T) = Z/120 and H3(Tot(B,(A5

A5; T))) = 0.In this case Inn(A5)=A5 and Aut(A5)= S5, therefore AutA5(G) = CS5(A5) =

1 and|PsdmnA5(A5)|= Z/120.

A similar argument applies to any superperfect group since by definition theyare the ones such that H1(G, Z) = 0 and H2(G, Z) = 0.


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3.5.3. The dihedral group G= Dn withn odd. In this case

Hom(Dn, T) = Z/2, H2(Dn; Z/2) = Z/2 and H

2(Dn; T) = 0,

hence

H3(Tot(B,(Dn Dn; T))) =H2(Dn, Hom(Dn, T) = Z/2.

Since H3(Dn; T) = Z/2 Z/n we have that the isomorphism classes of pseu-domonoid structures coming from the Twisted Drinfeld Double construction areZ/2 Z/n and that there is an independent pseudomonoid structure which comesfrom H2(Dn, Hom(Dn; T)) = Z/2. In this case we have that AutDn(Dn) = Z(Dn) =1 and therefore

|PsdmnDn(Dn)|=(Z/n Z/2 Z/2).

3.5.4. The symmetric group G = Sn forn 4. From [1, VI-5] we know that

Hom(Sn, T) = Z/2, H2(Sn; Z/2) = Z/2

H2(Sn; T) = Z/2 and H3(Sn; Z/2) = Z/2 Z/2

hence we get the exact sequence

0 Z/2H3(Tot(B,(Sn Sn; T))) Z/2 0.

In particular we could say that the nontrivial element in H2(Sn, Hom(Sn, T)) =Z/2 induces a pseudomonoid structure on Snwhich is not isomorphic to any struc-ture coming from the construction of the Twisted Drinfeld Double. This followsfrom the fact that for n = 2 and n = 6, Aut(Sn) = Inn(Sn) = Sn and thereforeAutSn(Sn) = Z(Sn) = 1 is the trivial group. Whenever n = 6 we know thatOut(S6) = Z/2; nevertheless AutS6(S6) = 1. Therefore we have that

|PsdmnSn(Sn)|= H3(Sn, T) H

3(Tot(B,(Sn Sn; T)))

with2

H3(Sn, T) =

Z/12 Z/2 if n= 4, 5

Z/12 Z/2 Z/2 if n 6

3.5.5. G cyclic group. When G = Z/n we get that

H3(Tot(B,(Z/n Z/n; T))) =H2(Z/n, Hom(Z/n, T)) = Z/n

and H3(Z/n, T) = Z/n. The action of AutZ/n(Z/n) = Z/n, which is the multi-

plicative group of units in Z/n, on H1(Z/n, T) = Z/n is given by multiplicationand while onH3(Z/n, T) = Z/nis given by the square of the multiplication; hencewe get that

|PsdmnZ/n(Z/n)|=(Z/n Z/n)/Z/n

where the action is given by

Z/n (Z/n Z/n) (Z/n Z/n), (a, (x, y))(ax,a2y).

For example whenn = 4 we have that

|PsdmnZ/4(Z/4)|=(Z/4)/(Z/4) Z/4.

2See Group Cohomology of Symmetric Groups in http://groupprops.subwiki.org


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4. The monoidal category of equivariant vector bundles on apesudomonoid

Let G be a group and K= (K,m, 1, , , ) a pseudomonoid with strict unit inthe 2-category ofG-sets with twists.

We define the category BunG(K) of G-equivariant finite dimensional bundlesover Kas follows:

An object is a K-graded finite dimensional Hilbert space H =

kKHk and atwistedG-action

:G U(H)

such that

Hk = Hk ( hk) = [|||k]() hk e h= h

for all , G, k K, hk Hk. Morphisms in the category are linear maps thatpreserve the grading and the twisted action, i.e., a linear map f : H H is amorphism if

f(Hk) Hk,

f( h) = f(h)

for all G, k Kandh H.We define a monoidal structure onB unG(K) as follows:

Proposition 4.1. LetH andH be objects inBunG(K), then the tensor productof Hilbert spacesH H is aG-equivariantK-bundle withK-grading(H H)k =

x,yK:xy=k Hx Hy and twistedG-action

(hx

hy

) :=[||x|y]( hx

hy

),

for allk K, G, hx Hx andh y Hy.

Proof. We just need to see that

( (hx hy)) = [|||xy]() (hx h

y)

for all , G, x, y K. In fact we have

((hx hy))

=[||x|y]

( hx hy)

=[||x|y] [|| x| y]

( hx) ( hy)

=[||x|y] [|| x| y] [|||x] [|||y]

()

hx ()

h

y

=[||x|y] [|| x| y] [|||x] [|||y]

[||x|y]

() (hx h

y)

=[|||xy]

() (hx hy)

,

where the last equality follows from the equation G()K() = 1 obtained in (3.5)in the proof of Proposition 3.4.


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Now, forH, H and H objects inB unG(K) the associativity constraint

: (H H) H H (H H),

for the monoidal structure is defined by

((hx hy) h

z ) = [x|y|z]

1hx (hy h

z )

for all x, , y, z K, hx Hx, hy Hy andh

z H

z .

In order to see that is natural isomorphism inBunG(K) we need to see that itpreserves the grading, which follows from the definition, and that is compatiblewith theG-action. The compatibility follows from the equalities

( [(hx hy) h

z ]) =[||xy|z]( (hx h

y) h

z )

=[||xy|z][||x|y](( hx hy) h

z )

=[||xy|z][||x|y]

[ x| y| z] hx ( h

y h

z)

=[||x|yz][||y|z]

[x|y|z] hx ( h

y h

z )

=[||x|yz]

[x|y|z] hx (h

y h

z )

= [x|y|z]1hx (hy h

z )

= ((hx hy) h

z ),

where the equality between the third and the fourth line follows from the equation1()2()1 = 1 obtained in (3.6) in the proof of Proposition 3.4.

The pentagonal axiom follows directly from the 3-cocycle condition of .Finally we define the unit object C as the one dimensional Hilbert space C

graded only at the unit element e K, endowed with trivial G-action. From thenormalization of all cochains, it follows that C is unit object in BunG(K) withrespect to the tensor product. All in all, we have just proved that

Proposition 4.2. For K = (K,m, 1, , , ) a pseudomonoid with strict unit inthe 2-category ofG-sets with twists, the triple(BunG(K), , C), endowed with thetensor product and the unit elementC is a monoidal category (or tensor category).

Remark 4.3. LetG be a group acting by the right over an abelian group H. ThenGalso acts by the left over Hthe abelian group of all characters ofH. In the casethat the 3-cocycle Z3(Tot(A,(H G, T))) is trivial, then the tensorcategory BunG(H) is just Rep(H G) the tensor category of finite dimensionalunitary representation of the semi-direct product H G, where G acts by the leftonHas g a:= ag1, for all g G, a H.

Remark 4.4. Consider the case of a group G acting on itself by conjugationand the pseudomonoid KwG = (G,m, 1, w, w, w) defined in section 3.5. In thecase that G is finite, the tensor category BunG(KwG) is exactly the category ofrepresentations Rep(Dw(G)) of the Twisted Drinfeld DoubleDw(G). Note that thequasi-Hopf algebraDw(G) is defined only forG finite, butBunG(K

wG) is defined for

an arbitrary discrete group. So, the tensor category BunG(KwG) is a generalization

of the Twisted Drinfeld Double of a finite group.


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The Twisted Drinfeld Double Dw(G) is important at least for the followingtwo reasons. First, Dw(G) is a quasi-triangular quasi-Hopf algebra [9] and thenRep(Dw(G)) is a braided tensor category and it defines representations of the braidgroup and knots and links invariants [24, Chapter 1]. Second, Rep(Dw(G)) is aModular Tensor Category (MTC), so it defines a 3D-TQFT using Reshetikhin-Turaev construction, see [3, Chapter 4] and [24, Chapter 4], .

For any group, the tensor category BunG(KwG) is a braided tensor category, so itdefines representations of the braid group and knots and links invariants. HoweverBunG(KG) is not a MTC ifG is infinite. In fact, the category of finite dimensionalunitary representation ofG is a full tensor subcategory ofBunG(KwG) and it hasinfinite many isomorphism classes of irreducible representations, so in particularBunG(K

wG) would also have infinitely many simple objects.

4.1. Morphism of pseudomonoids, monoidal functors and natural isomor-phisms. A morphism in the 2-category of pseudomonoids in the 2-category G-setswith twists induce a monoidal functor between the associated monoidal categories.

Proposition 4.5. LetF = (F ,,) : K K be a morphism of pseudomonoids.ThenFinduces a monoidal functor from the monoidal categories(BunG(K), , C)and(BunG(K), , C

).

Proof. Let F = (F ,,) :K K be a morphism of pseudomonoids as defined inProposition 3.8. We define a functor F : B unG(K) BunG(K) in the followingway: forH an object inB unG(K), theK-graded Hilbert space F(H) is the directsum

F(H)y =

{xK:F(x)=y}

Hx.

The twistedG-action onF(H) is defined as follows: takehy F(H)y defined bythe element hy = hx for some vector hx Hx with F(x) = y . Define the twisted

G-action onhy by hy :=[||x]( hx),

and note that

( hy) = ([|x] hx)

=[||x] [| x] ( hx)

=[||x] [| x] [|||x] hx

=[|||y] [||x] hx

=[|||y] hy

where the equality between the third and the fourth lines follows from the equationG = F/ obtained in (3.10) in the proof of Proposition 3.8. Therefore the

Hilbert space F(H) is an object in B unG(K

).Let us check that the functor F is monoidal: takeH, H two objects inBunG(K)

and consider the map

R: F(H) F(H) F(H H)

hx1 hx2 (x1, x2)

1hx1 hx2

where hx1 Hx1 and hx2 Hx2 but we see them both as elements in F(H)F(x1)

and F(H)F(x2) respectively, and the element hx1 hx2 we see it as an element in


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F(H H)F(x1x2). Let us show that the map R is a morphism in BunG(K), i.e.

thatR( (hx1 hx2)) =

R(hx1 hx2). For he left hand side we have:

R( (hx1 hx2))

=[||F(x1)|F(x2)][||x1] [||x2]R( hx1 hx2)

=[||F(x1)|F(x2)][||x1] [||x2] [x1|x2]1( hx1 hx2),

and for the right hand side we have:

R(hx1 hx2) =

[x1|x2]1(hx1 hx2)

=[||x1x2] [||x1|x2] [x1|x2]1( hx1 hx2);

and note that the two expressions coincide due to the equationG K= /F

that appeared in (3.11) under the proof of Proposition 3.8.The hexagonal axiom of the monoidal functor follows directly from the equation

K= /F

and we will not reproduce its proof.

Proposition 4.6. Given morphisms of pseudomonoids with strict unit in the 2-category ofG-sets with twistsF= (F ,,) andF = (F, , ), withF, F :K K, a 2-morphism : F F, induces a monoidal natural isomorphism betweenthe monoidal functorsF andF.

Proof. Using the notation defined above, we define the transformation between Fand F as follows:

F(H) F(H)

hx (x)1 hx.

Equations1= /and 2= / shown in the proof of Proposition 3.10 imply

that the transformation is natural and monoidal, respectively.

4.2. Automorphisms of pseudomonoids and their action on the monoidalcategory of equivariant vector bundles. Let us fix K = (K,m, 1, , , ) apseudomonoid with strict unit in the 2-category of G-sets with twists and let(BunG(K), , C) be the monoidal category ofG-equivariant bundles over K.

TakeF= (F ,,) : K K an invertible morphism of the pseudomonoid K andnote that Proposition 4.5 tells us that the induced monoidal functor

F: (BunG(K), , C) (BunG(K), , C)

becomes an automorphism of the monoidal category (BunG(K), , C).If we denote by

AutPsmnd(K)the 2-group of automorphisms of the pseudomonoid K, whose morphisms are invert-ible morphisms F :K K and whose 2-morphisms are the natural transformationsbetween functors : F F, and

Aut(BunG(K))

the 2-group of automorphisms of the monoidal category (BunG(K), , C), whosemorphisms are invertible monoidal functors and whose 2-morphisms are monoidal


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natural transformations, then we have that Propositions 4.5 and 4.6 imply thatthere is a 2-functor

AutPsmnd(K) Aut(BunG(K)): F F : F F

from the 2-group of automorphisms of the pseudomonoid K to the 2-group of au-tomorpshisms ofB unG(K).

To understand the previous action in more detail, let us start by studying thecategory AutPsmnd(K).

An automorphism F = (F ,,) : K K consists of a G-equivariant automor-phism F AutG(K), together with a degree 2 cochain 1 in Tot

(A,(K G, T)) such that

(G (1)p

K )( 1) = F/ F/ F/.

The automorphism Flies on the image of the forgetful functor

AutPsmnd(K) AutG(K)

F= (F ,,)F;

if and only if the cohomology classes [ ] and F[ ] are equal ascohomology classes in H2 (Tot(A,(K G, T))). If we define

AutG(K; [ ]) :={F AutG(K)|F[ ] = [ ]}

we have that the 2-group of automorphisms ofK sits in the exact sequence

0

Tot1(A,(K G, T)))

GK

Z2(Tot(A,(K G, T))) AutPsmnd(K) AutG(K; [ ]) 0

whereZ2(Tot(A,(K G, T))) denotes the degree 2-cocycles and Tot1(A,(KG, T)) parameterizes the 2-morphisms between the morphisms ofZ2(Tot(A,(KG, T))).

If we take equivalence classes of automorphisms in AutPsmnd(K) defined by the2-morphisms, we obtain a group which is usually denoted by

1(AutPsmnd(K)) ;

this group sits in the middle of the short exact sequence0 H2(Tot(A,(K G, T))) 1(AutPsmnd(K)) AutG(K; [ ]) 0;

and by the Lyndon-Hochschild-Serre spectral sequence we know that there is anexact sequence

0 H1(G, Hom(K, T)) H2(Tot(A,(KG, T))) H2(K, T)G d2H2(G, Hom(K, T)),

whered2 is the differential of the second page.


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defines a homomorphism

H2(Tot(A,(K G, T))) Aut(Groth(BunG(K))).

The previous morphism will be of interest when we compare it with the groupof automorphisms of the twisted equivariant K-theory ring in section 5.7.1.

5. The fusion product and the twisted G-equivariant K-theory ring

Whenever X is a finite G-CW complex with G a finite group, the elements inH3G(X; Z) classify the isomorphism classes of projective unitary stable and equivari-ant bundles over X, and these bundles provide the required information to defineequivariant Fredholm bundles over X; the homotopy groups of the space of sectionof a such bundle is one way to define the twisted G-equivariant K-theory groups ofX(see [4]). The homotopy classes of automorphisms of a projective unitary stableand equivariant bundle over X are in one to one correspondence with H2G(X; Z)and this group acts on the twisted G-equivariant K-theory groups.

Whenever the space Xis a discreteG-set, there is an equivalent but easier wayto define the twisted G-equivariant K-theory groups ofX. Let us review it.

5.1. TwistedG-equivariant K-theory. Take a normalized 2-cocycle : G GX Tand define an -twisted G-vector bundle over Xas a finite dimensional X-graded complex vector spaceE, which can alternatively seen as a finite dimensionalcomplex vector bundle p : E Xwith finite support, endowed with a G actionsuch that p is G equivariant, the action ofG on the fibers is complex linear, andsuch that the composition of the action on E satisfies the equation

g (h z) = (g, h||p(z))(gh z)

for allz in E. Two-twistedG-vector bundles overXare isomorphic if there existsa G equivariant map EE of complex vector bundles which is an isomorphismof vector bundles.

Definition 5.1. The -twisted G-equivariant K-theory ofX is the Grothendieckgroup

KUG(X; )

associated to the semi-group of isomorphism classes of-twistedG-vector bundlesover X endowed with the direct sum operation.

If we have aG-equivariant mapF :Y Xthen the pullback of bundles inducesa group homomorphism

F : KUG(X; ) KUG(Y; F).

5.1.1. For a normalized cochain C1G(X; T) with G = / then there is an

induced isomorphism of groups

: KUG(X; ) = KUG(X;

)

where(E) :=Eand the G-action onz (E) is given by the equation:

h z := [h||p(z)](h z).

Since cohomologous twistings induce isomorphic twisted K-theory groups, wehave that H2G(X; T) classifies the isomorphism classes of twistings for the G-equivariant

K-theory ofX. And since the isomorphisms and (G) are equal, we have


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as a cohomology class in H1G(K K K; T). We therefore have that if there existsa cochain C0G(K K K; T) such that

G= Kthen the product previously defined endows the group KUG(K; ) with a ringstructure. Summarizing:

Proposition 5.2. Consider Z2G(K; T) and C1,2(K G; T) satisfying the

equations

G K= 1, 2[] = 1.

Then the group KUG(K; ) endowed with the product structure becomes a ring.Let us denote this ring by

KUG(K; , ) := (KUG(K; ), )

and let us call the pair(, ) a multiplicative structure forK.

5.4.1. Many of the features of the twistedG-equivariant K-theory rings are betterunderstood if we work with the notation introduced in section 1.

Recall that the double complex A, := A,(K G; T) is the subcomplex ofC,(K G; T) disregarding the 0-th row. Consider the subcomplex A,>3 ofA,

defined as subcomplex ofC,(K G; T) where we disregard the first four rows.The double complexA,/A,>3 consists of the second, third and fourth rows of

the double complex C,(K G; T), and we have that if KUG(K; ) can be madeinto a ring is because there existsand such that the cochain becomesa 3-cocycle in the complex Tot(A,/A,>3) and the 3-cocycle can be seen in adiagram as follows

4

3 1

2 1

1 1

0 1 2 3

G

(K)1

G

KG

Proposition 5.3. If the cocycle Z2G(K; T)can be lifted to a 3-cocycle in the complex Tot(A,/A,>3) then the the group KUG(K; ) can be endowedwith the ring structure KUG(K; , ).

5.4.2. Let us see another way to understand the conditions under which the twist can define a multiplicative structure on KUG(K; ). Consider the filtration ofthe double complex A, given by the subcomplexes Fr := A,r. The spectralsequence that the filtration defines abuts to the cohomology of the total complexofA,

E, H(Tot(A,))

and has for first page

Ep,q1 =HpG(K

q; T)


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with differential

d1 : Ep,q1 E

p,q+11 , d1[x] := [(K)

(1)px].

If we have a twist, its cohomology class [] is an element in E2,11 . The elementd1[] is the first obstruction to lift to a 3-cocycle in Tot

(A,), that is, d1[] = 1if and only if there exists C1,2(K G; T) such that G K = 1. Notefurthermore that

d1[] = m[]

1 [] 2 []

and therefore we recover what we knew, namely that the twist may induce aproduct in KUG(K; ) if and only if the cohomology class [] is multiplicative.

If the cohomology class [] is multiplicative, then [] survives to the second page

of the spectral sequence with [] E2,12 . The second differential applied to [] isd2[] = [(2)

1]

3 (K)1

2 1

1

0 1 2

K

G(K)

1

3 [(K)1]

2

1 []

0 1 2

d2

and this is the second obstruction to liftto a 3-cocycle in Tot(A,), i.e. d2[] = 1if and only if there exists C0,3(K G, T) such that G(K)

1 = 1. Notethatd2[] measures the obstruction for the multiplication in KUG(K; ) to beassociative.

We have then that the equations d1[] = 1 = d2[] are the equations that needto be satisfied in order for the group KUG(K; ) to become a ring with respect tothe construction provided in this section.

If [] survives to the third page we have that d3[] = [K] and therefored3[] =1 implies that is a 3-cocycle in Tot(A,). So if [] survives to the fourthpage, and hence the page at infinity, then K = (K,m, 1, , , ) is a pseudomonoidwith strict unit in the 2-category ofG-sets with twists. Summarizing:

Proposition 5.4. Consider Z2G(K; T), then can be lifted to a 3-cocycle inTot(A,/A,>3) if and only ifd1[] = 1 = d2[], and this impliesthat KUG(K; , ) becomes a ring. If furthermored3[] = 1 then is a 3-cocycle inTot(A,) and this implies thatK= (K,m, 1, , , ) is a pseudomonoidwith strict unit in the 2-category ofG-sets with twists, and in this case

KUG(K; , )=Groth (BunG(K)) .

5.5. Isomorphism classes of Multiplicative structures. In this section wewant to determine some sufficient conditions under which the twisted G-equivariantK-theory rings KUG(K; , ) and KUG(K; , ) induced by the multiplicativestructures (, ) and (, ) become isomorphic.

Let us consider the double complex A,/A,>2 which consists of the first andthe second row of the double complex C,(K G, T). We claim


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Lemma 5.5. Consider(, )and(, )multiplicative structures onK. If and are cohomologous as 3-cocycles in Tot(A,/A,>2), then the ringsKU

G(K; , ) and KU

G(K; , ) are isomorphic.

Proof. If and are cohomologous as 3-cocycles in Tot(A,/A,>2),then there exists a 2-cochain 1 with A1,1 and A0,2 such that

G (1)p

K ( 1) = / /,

namely thatG= /and (K)

1(G)1 =/, or diagramatically

3

2 1 /

1 /

0 1 2

G

(K)1

G

The isomorphism : KUG(K; ) = KUG(K; ) induces an isomorphism of

rings

: KUG(K; , ) = KUG(K;

, )

where :=(K)1. Since (K)

1(G)1 =/, we have that =(G),

therefore the isomorphism and are equal and we obtain that KUG(K; , )=KUG(K; , ).

The short exact sequence of complexes

0 A,>2/A,>3 A,/A,>3 A,/A,>2 0

induces a long exact sequence in cohomology groups

H3(Tot(A,/A,>3)) H3(Tot(A,/A,>2)) H4(Tot(A,>2/A,>3))

[ ] [ ] [(2)1]

and we see that Lemma 5.5 and Proposition 5.4 imply that the subgroup

MSG(K) :={[ ] H3(Tot(A,/A,>2))|[K] = 1}

is precisely the group of equivalence classes of multiplicative structures associatedto the twisted G-equivariant K-theory of the monoid K. We define

Definition 5.6. The group

M SG(K) := {[ ] H3(Tot(A,/A,>2))|[2] = 1}

will be called the group ofmultiplicative structuresfor the G-equivariant K-theory

of the monoidK.And therefore we have that

Proposition 5.7. The elements of the group MSG(K) are in one to one corre-spondence with the set of isomorphism classes of ring structures (in the sense ofLemma 5.5) on the twistedG-equivariant K-theory to the monoidK.

In particular we have that there are at most #(M SG(K)) of different multiplica-tive structures in the twisted G-equivariant K-theory groups ofK.


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5.6. Automorphisms of the twisted equivariant K-theory ring. From sec-tion 5.1.1 we know that if C1G(K; T) satisfies G= 1 then the map inducesan isomorphism of groups

: KUG(K; ) = KUG(K; ).

Whenever (, ) is a multiplicative structure, it follows that the map induces anisomorphism of rings whenever the homomorphismKis the identity map, namelythat [K] = 1 as a cohomology class in H

1G(K K; T). If we define the group of

multiplicative elements by

H1G(K; T)mult:=

[] H1G(K; T)|K[] =1 []

2 [] m

[]1 = 1

we have then that the group of automorphisms of the twisted G-equivariant K-theory ring is equal to the multiplicative elements in H1G(K; T), i.e.

Aut(KUG(K; , )) = H1

G(K; T)mult

Using the spectral sequence of of section 5.4.2 we see that the multiplicativeterms appear in the second page of the spectral sequence

H1G(K; T)mult= E1,12

since d1[] = [K]1 is the obstruction of being multiplicative. If furthermore a

multiplicative element satisfies d2[] = 1, then we have that [] can be lifted to anelement [ ] in H2(Tot(A,)). This means that we have the exact sequence

0 H2(C(K; T)G) H2(Tot(A,)) H1G(K; T)multd2H3(C(K; T)G)(5.1)

where the G-invariant cochains come from the first page of the spectral sequence,

i.e. E0,q1 =Cq(K; T)G, and its cohomology appears in the second page, i.e. E0,q2 =Hq(C(K; T)G, K).

5.7. Relation between the Grothendieck ring associated to a monoidalcategory and the twisted equivariant K-theory ring. Let us consider a G-equivariant monoid with unit K. We have seen that to a pseudomoinoid with strictunit in the 2-category of G-sets with twist K = (K,m, 1, , , ) over K we canassociate the ring Groth(BunG(K)) of isormpism classes of objects in the monoidalcategoryBunG(K). At the same time, since Z

3(Tot(A,(K G, T))),then (, ) is a multiplicative structure and we get that

Groth(BunG(K))= KUG(K; , )

as rings. We have therefore a canonical map

H3(Tot(A,))MSG(K)(5.2)

[ ][ ]

from the isomorphism classes of pseudomonoid structures with strict unit ofG-setswith twist over K, to the group of multiplicative structures of the G-equivarianttwisted K-theory groups ofK. Let us understand this map in more detail.


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Consider the projection homomorphism between the exact sequences of com-plexes

0 A,>2

p

A,

A,/A,>2

=

0

0 A,>2/A,>3 A,/A,>3

A,/A,>2 0

and the homomorphism between the long exact sequences induced

H3(Tot(A,)

H3(Tot(A,/A,>2))

=

H4(Tot(A,>2)

p

H3(Tot(A,/A,>3)) H3(Tot(A,/A,>2))

H4(Tot(A,>2/A,>3))

whereand are the connection homomorphisms with [ ] =2[].By definition

M SG(K) = ker()

and furthermore note that the map is injective sinceH3(Tot(A,>3)) = 0. There-fore the canonical map defined in (5.2) is precisely the map

H3(Tot(A,)) M SG(K).

In complete generality it is difficult to give an explicit description of how the ker-

nel and the cokernel of the map H3(Tot(A,)) M SG(K) looks like, but for

calculations we can give the following description:

Theorem 5.8. The kernel of the homomorphismH3(Tot(A,)) M SG(K) is

isomorphic to

ker

H3(Tot(A,)) M SG(K)

=

H3(C(K, T)G)

and the cokernel is isomorphic to

coker

H3(Tot(A,)) M SG(K)

=ker

H4(C(K, T)G) H4(Tot(A,))

.

Proof. Let us start with the kernel of the map . From the long exact sequences ofcohomologies defined above we get that

ker

H3(Tot(A,)) M SG(K)

=

H3(Tot(A,>2))

where the groupH3(Tot(A,>2)) consists of elements in C0,3 that are closed underthe differentials G andKand therefore

H3(Tot(A,>2)) = Z3(K, T)G.

Now, since the elements K(C2(K, T)G) are all zero in H3(Tot(A,)) we have that

Z3(K, T)G

=

H3

C(K, T)G

and therefore

ker

H3(Tot(A,)) M SG(K)

=

H3(C(K, T)G)

.


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examples, and from them we will deduce interesting information with regard to thetwisted equivariant K-theory rings.

6. Examples

The main objective of this section is to use Proposition 5.9 to calculate the kerneland cokernel of the homomorphism

H3(Tot(A,)) M SG(K)

for different choices ofGandK, in order to show the different twisted G-equivariantK-theory rings over Kthat can appear.

6.1. Trivial action of G on K. In this case we have that the spectral sequence

defined in Proposition 5.9 collapses at the second page and moreover we have thatC(K, T) = C(K, T)G. Therefore we obtain the short exact sequence

0 H3(K, T) H3(Tot(A,)) M SG(K) 0,

thus implying that

MSG(K)=H3(Tot(B,(K G, T))),

and moreover that all multiplicative structures for the G-equivariant K-theory ofKcan be obtained from the ring structures defined by the Grothendieck rings ofthe monoidal categories B unG(K). We also obtain the short exact sequence

0 H2(K, T) H2(Tot(A,)) Hom(K, Hom(G, T)) 0

where in this case H1G(K, T)mult= Hom(K, Hom(G, T)).Furthermore, if [] H3(K, T) is non trivial then we can define a non-trivial

pseudomoinoid with strict unit in the 2-category ofG-sets with twist K= (K,m, 1, 0, 0, )with [0 0 ] non-zero in H3(Tot(A,)), such that

Groth(BunG(K))=R(G) Z Z[K]

whereR(G) is the Grothendieck ring of finite dimensional complex representationsofGand Z[K] is the group ring ofK, since we know thatR(G)ZZ[K] is isomorphicto the non-twisted ring structure on KUG(K).

6.1.1. G= Z/n andK= Z/m. In this case we have that H3(K, T) = Z/mand

H3(Tot(A,)) = Z/m Z/(n, m)

where (n, m) is the maximum common denominator of the pair n, m. ThereforeMSG(K) = Z/(n, m) and we have that all non trivial multiplicative structurescome from the group

Z/(n, m) = H3(Tot(B,)) H3(Tot(A,)).


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6.2. Adjoint action ofG on itself. From Lemma 2.5 we know that in this casewe have the split short exact sequence

0 H(Tot(B,)) H(Tot(A,)) H(G, T))

0

induced by the short exact sequence of complexes

0Tot(B,) Tot(A,) C>0(G, T) 0.

Since we have the inclusion H(Tot(B,)) H(Tot(A,)), we can deduce

that the groups E,02 of the 0-th column of the second page of the spectral se-quence converging to H(Tot(A,)) defined in Proposition 5.9, are unaffected bythe differentials di for i > 1; this follows from the injectivity between the spectralsequences associated to the filtrationsB,>q andA,>q of Tot(B,) and Tot(A,)respectively.

Therefore we have that E0,34 = E0,32 = H

3(C(G, T)G) and d2(E1,22 ) = 0 =

d3

(E2,1

3 ), and by Proposition 5.9 we get the short exact sequence

0 H3(C(G, T)G) H3(Tot(A,)) M SG(G) 0.(6.1)

Moreover, the cokernel of the inclusion

E0,q Hq(Tot(A,))

should match the cohomology groupHq(Tot(B,)) since this piece is built from thegroupsEr,s withr, s >1 andr+s= p, therefore we have the canonical isomorphism

Hq(C(G, T)G) Hq(Tot(B,)) =Hq(Tot(A,)), x yx + y

which in particular implies that

Hq(C(G, T)G) =Hq(G, T).

Then we can conclude that the composition of the maps

H3(Tot(B,)) H3(Tot(A,)) M SG(G)

is an isomorphism, and therefore we obtain the canonical isomorphism

(6.2) H3(Tot(B,)) =M SG(G), [ ][ ].

Also we obtain the short exact sequence

0 H2(C(G; T)G) H2(Tot(A,)) H1G(G; T)mult 0,

with H1G(G; T)mult=H2(Tot(B,))=Hom(G, Hom(G, T)).

Now we will study the multiplicative structures that define the pseudomonoidsconstructed via the formalism introduced in [9], and whose properties were outlinedin section 3.5. For this purpose we need to calculate the composition of the maps

H3(G, T) 1

H3(Tot(A,))

M SG(G)

where 1 is the induced map in cohomology which was defined at the chain levelin (2.2). This calculation will be carried out using the ring structure of the ringH(G, Z) together with the pullback map :H(G, Z) H(G G, Z) inducedby the multiplication : G G G as we have in Theorem 2.3. Since we havethe isomorphism

H3(Tot(A,)) H3(G, T) =H3(G G, T), x yx + 2 y


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where2 : G G G, (a, g)g is the projection, then the short exact sequenceof (6.1) implies that the following is also a short exact sequence

0 H3(C(K; T)G) H3(G, T) H3(G G, T) M SG(G) 0.

We conclude that the homomorphism

1 :H3(G, T) M SG(G)

is equivalent to the composition of homomorphisms

multiplicative structures subm 130626

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