Induced character in equivariant K-theory and wreath products
aa r X i v : . [ m a t h . K T ] N ov INDUCED CHARACTER IN EQUIVARIANT K-THEORY,WREATH PRODUCTS AND PULLBACK OF GROUPS
GERMAN COMBARIZA, JUAN RODRIGUEZ, AND MARIO VEL ´ASQUEZ
Abstract.
Let G be a finite group, X be a compact G -space. In this note westudy the ( Z + × Z / Z )-graded algebra F qG ( X ) = M n ≥ q n · K G ≀ S n ( X n ) ⊗ C , defined in terms of equivariant K-theory with respect to wreath products asa symmetric algebra. More specifically, let H be another finite group and Y be a compact H -space, we give a decomposition of F qG × H ( X × Y ) in terms of F qG ( X ) and F qH ( Y ). For this, we need to study the representation theory ofpullbacks of groups. We discuss also some applications of the above result toequivariant connective K-homology. Notation
In this note we denote by S n the symmetric group in n letters. Let G be a finitegroup, let g, g ′ ∈ G , we say that g and g ′ are conjugated in G (denoted by g ∼ G g ′ )if there is s ∈ G such that g = sg ′ s − . We denote by[ g ] G = { g ′ ∈ G | g ∼ G g ′ } to the conjugacy class of g in G (or simply by [ g ] when G is clear from the context).We denote by G ∗ the set of conjugacy classes of G . We denote by C G ( g ) thecentralizer of g in G . Also R ( G ) will be the complex representation ring of G , withoperations given by direct sum and tensor product, and generated as abelian groupby the isomorphism classes of irreducible representations of G . The class functionring of G is the setClass( G ) = { f : G → C | f is constant in conjugacy classes } with the usual operations. 1. Introduction
Let X be a finite CW-complex. In [Seg96] Segal studied the vector spaces F ( X ) = M n ≥ K S n ( X n ) ⊗ C , these spaces carries several interesting structures, for example they admit a Hopfalgebra structure with the product defined using induction on vector bundles andthe coproduct defined using restriction. Date : November 21, 2019.
Later in [Wan00], Wang generalizes Segal’s work to an equivariant context. Let G be a finite group and X be a finite G -CW-complex, Wang defines the vectorspace F G ( X ) = M n ≥ K G ≀ S n ( X n ) ⊗ C , where G ≀ S n denotes the wreath product acting naturally over X n . Wang provesthat F G ( X ) admits similar structures as F ( X ). In particular F G ( X ) has a descrip-tion as a supersymmetric algebra in terms of K G ( X ) ⊗ C . In this paper we presenta proof of the above fact using a explicit description of the character of an inducedvector bundle. This description indicates that F G ( X ) has the size of the Fock spaceof a Heisenberg superalgebra.Following ideas of [Seg96], in [Vel15] appears another reason to study F G ( X ).When X is a G -spin c -manifold of even dimension, F G ( X ) is isomorphic to thehomology with complex coefficients of the G -fixed point set of a based configurationspace C ( X, x , G ) whose G -equivariant homotopy groups corresponds to the reduced G -equivariant connective K-homology groups of X . This description allows to relategenerators of F G ( X ) with some homological versions of the Chern classes.Let G and H be finite groups, X be a finite G -CW-complex and Y be a finite H -CW-complex, we also prove a K¨unneth formula for F G × H ( X × Y ), obtaining anisomorphism F G × H ( X × Y ) ∼ = F G ( X ) ⊗ F ( {•} ) F H ( Y )that is compatible with the decomposition as a supersymmetric algebra. In orderto do this, we need to study the representation theory of pullbacks of groups.Let Γ p / / p (cid:15) (cid:15) G π (cid:15) (cid:15) H π / / K be a pullback diagram of finite groups, with π and π surjective, in this case Γ canbe realized as a subgroup of G × H . We prove that when Γ is conjugacy-closed (seeDefinition 5.2) in G × H then we have a ring isomorphismClass(Γ) ∼ = Class( H ) ⊗ Class( K ) Class( G ) . Moreover we use that some conjugacy classes in ( G × H ) ≀ S n are closed in( G ≀ S n ) × ( H ≀ S n ) to prove the K¨unneth formula for the algebra F G × H ( X × Y ).This paper is organized as follows:In Section 2 we recall basic facts about equivariant K-theory, in particular werecall the construction for the character. Following ideas of [Ser77] we give anexplicit definition of the induced bundle and recall a formula (proved in [HKR00,Thm. D]) for a character of the induced bundle. In Section 3 we recall basic factsabout wreath products and its action over X n . In Section 4 we recall the definitionof F G ( X ) and give another way to obtain the description as a supersymmetricalgebra using the formula of the induced character. In Section 5 we study therepresentation theory of pullbacks. In Section 6 we recall some basic properties ofsemidrect products of direct products. In Section 7 we use results in Section 5 togive a K¨unneth formula for the Hopf algebra F G × H ( X × Y ). In Section 8 we dosome final remarks about the relation of F ( X ) and homological versions of Chernclasses. NDUCED CHARACTER IN EQUIVARIANT K-THEORY AND WREATH PRODUCTS 3 Induced character in equivariant K-theory
In this section we recall a decomposition theorem for equivariant K-theory withcomplex coefficients obtained by Atiyah and Segal in [AS89]. In the next sectionwe use that result to give a simple description of F qG ( X ). In this paper all CW-complexes (and G -CW-complexes) that we consider are finite. Definition 2.1.
Let X be a G -space. A G -vector bundle over X is a map p : E → X , where E is a G -space satisfying the following conditions(i) p : E → X is a vector bundle.(ii) p is a G -map(iii) For every g ∈ G the left translation E → E by g is bundle map.If p : E → X is a G -vector bundle we define the fiber over x ∈ X to the set p − ( x ) = { v ∈ E | p ( v ) = x } , when p is clear from the context we also denote this set by E x . Also if H ⊆ G is a subgroup, we can consider E as a H -vector bundle over X , we denote it byres GH ( E ). Definition 2.2.
Let X and Y be G -spaces. If p : E → Y is a G -vector bundle and f : X → Y is a G -map, then the pullback p ∗ E → X is a G -vector bundle over X defined as p ∗ E = { ( e, x ) ∈ E × X | p ( e ) = f ( x ) } . When i : X → Y is an inclusion we usually denote i ∗ ( E ) by E | Y .Details about G -vector bundles can be found in [Ati89]. Definition 2.3.
Let G be a group, let X be a finite G -CW-complex (see [tD87]), the equivariant K-theory group of X ,denoted by K G ( X ) is defined as the Grothendieckgroup of the monoid of isomorphism classes of G -equivariant vector bundles over X with the operation of direct sum. The functor K G ( − ) could be extended to anequivariant cohomology theory K ∗ ( − ), defining for n > K − nG ( X ) = ker (cid:16) K G ( X × S n ) i ∗ −→ K G ( X ) (cid:17) . And for any G -CW-pair ( X, A ), set K − nG ( X, A ) = ker (cid:18) K − nG ( X ∪ A X ) i ∗ −→ K − nG ( X ) (cid:19) . Finally for n < K − nG ( X ) = K nG ( X ) and K − nG ( X, A ) = K nG ( X, A ) . For more details about equivariant K-theory the reader can consult [Seg68].
Example 2.4.
If the action of G over X is free, then there is a canonical isomor-phism of abelian groups K G ( X ) ∼ = K ( X/G ) . Example 2.5.
If the action of G over X is trivial, then there is a canonical iso-morphism of abelian groups K G ( X ) ∼ = R ( G ) ⊗ Z K ( X ) , GERMAN COMBARIZA, JUAN RODRIGUEZ, AND MARIO VEL´ASQUEZ when R ( G ) denotes the (complex) representation ring of G . In particular when X = {•} we obtain K G ( {•} ) ∼ = R ( G ) . If Y is a finite G -CW-complex, we can define a G -action on K ( Y ). Let g ∈ G ,the pullback g ∗ : K ( Y ) → K ( Y ) , defines a G -action over K ( Y ). We will need the following lemma. Lemma 2.6.
Let Y be a finite G -CW-complex, then K ( Y /G ) ⊗ C ∼ = K ( Y ) G ⊗ C Proof.
It is a consequence of the Chern character and the analogous fact for singularcohomology. (cid:3)
In [AS89] a character for equivariant K-theory is constructed, that generalizesthe character of representations. We will recall this construction briefly. Let E bea G -vector bundle over X and g ∈ G . Note that X g is a C G ( g )-space, then if E is a G -vector bundle, E | X g is canonically a C G ( g )-vector bundle over X g . Consideringthe action given by pullback we have that the isomorphism class [( E | X g )] ∈ K ( X g )is a C G ( g )-fixed point. Then [( E | X g )] ∈ K ( X g ) C G ( g ) . Finally for every element λ ∈ S , we can form the vector bundle of λ -eigenvectors considering the action ofthe element g over π ( E | X g ) denoted by π ( E | X g ) λ . Then we can define a mapchar G : K G ( X ) ⊗ C → M [ g ] K ( X g ) C G ( g )) ⊗ C [ E ] M λ ∈ S [ π ( E | X g ) λ ] ⊗ λ ! [ g ] . Using the above Lemma we identify K ( X g ) C G ( g ) with K ( X g /C G ( g )). Theorem 2.7.
The map char G is an isomorphism of complex vector spaces. For a proof of the theorem see [AS89].2.1.
The induced bundle.
Now we will give an explicit construction of the in-duced vector bundle. It is a direct generalization of the induced representationdefined for example in Section 3.3 in [Ser77].Let H ⊆ G be a subgroup of G and E π −→ X an H -vector bundle over a G -space X . If we choose an element from each left coset of H , we obtain a subset R of G called a system of representatives of H (cid:31) G ; each g ∈ G can be written uniquely as g = sr , with r ∈ R = { r , . . . , r n } and s ∈ H , G = ` ni =1 Hr i , we suppose that r = e the identity of the group G . Consider the vector bundle F = L ni =1 ( r i ) ∗ E ,with projection π F : F → X and consider the following G -action defined over F :Let f ∈ F , then f = f r ⊕ · · · ⊕ f r n , where f r i ∈ ( r i ) ∗ E . If π F ( f ) = x then f r i = ( x, e ), where e ∈ E r i x . Let g ∈ G , notethat r i g − is in the same left coset of some r j , i.e. r i g − = sr j , for some s ∈ H .Define g ( f r i ) = ( gx, s − e ) ∈ ( r j ) ∗ E, NDUCED CHARACTER IN EQUIVARIANT K-THEORY AND WREATH PRODUCTS 5 and define the action of g on f by linearly.Now we will see that F does not depend on the set of representatives up toisomorphism. Let { r ′ , . . . , r ′ n } be another set of representatives of H (cid:31) G and let F ′ = L ni =1 ( r ′ i ) ∗ E . By reordering we can assume that r i and r ′ i are in the same leftcoclass, then r ′ i r − i ∈ H .We have an isomorphism of vector bundles over Xr ′ i r − i : ( r i ) ∗ E → ( r ′ i ) ∗ E ( x, e ) → ( x, r ′ i r − i e )inducing an isomorphism of G -vector bundles r ′ r − ⊕ . . . ⊕ r ′ n r − n : F → F ′ . We only need to verify that this map commutes with the action of G . To see this,let g ∈ G and f r i = ( x, e ) ∈ ( r i ) ∗ E , there exist s, s ′ ∈ H such that(2.8) r i g − = sr j and r ′ i g − = s ′ r ′ j . Note that gf r i ∈ ( r j ) ∗ E , then( r ′ j r − j ) g ( f r i ) = ( gx, r ′ j r − j s − e ) . On the other hand g ( r ′ i r − i f r i ) = ( gx, ( s ′ ) − ( r ′ i r − i e )) , but we know from (2.8) ( s ′ ) − r ′ i r − i = r ′ j r − j s − . Then the map r ′ r − ⊕ . . . ⊕ r ′ n r − n commutes with the G -action and then F and F ′ are isomorphic as G -vector bundles. We will denote the G -vector bundle F definedabove by Ind GH ( E ). Summarizing we have. Theorem 2.9.
Let G be a finite group, let H ⊆ G be a subgroup. Let X be a G -CW-complex, and let E be a H -vector bundle over X , there is a unique G -vectorbundle Ind GH ( E ) over X , up to isomorphism of G -vector bundles such that for every G -vector bundle F over X we have a natural identification Hom G (Ind GH ( E ) , F ) ∼ = Hom H ( E, res GH ( F )) . Proof.
Only remains to prove the identification. Let ξ ∈ Hom G (Ind GH ( E ) , F ), recallthat we have an inclusion of H -vector bundles E → Ind GH ( E ) v ∈ E x ( x, v ) . Define r ( ξ ) ∈ Hom H ( E, res GH ( F )) as follows, if v ∈ E x r ( ξ )( v ) = ξ ( x, v ) . It is clear that r ( ξ ) ∈ Hom H ( E, res GH ( F )). On the other hand if η ∈ Hom H ( E, res GH ( F )),define I ( η ) : Ind GH ( E ) → F as follows, if v i ∈ r ∗ i E , then v i = ( x, v ) with x ∈ X and v ∈ E r i x , then we define I ( η )( v i ) = r − i ( η ( v )) . Extending linearly I ( η ) to Ind GH ( E ).Now we will see that I ( η ) is G -equivariant. Let g ∈ G , let s ∈ H such that r i g − = sr j , GERMAN COMBARIZA, JUAN RODRIGUEZ, AND MARIO VEL´ASQUEZ then g · v i = ( gx, s − v ) . Now, I ( η )( g · v i ) = I ( η )( gx, s − v )= r − j ( η ( s − v ))= r − j s − η ( v )= gr − i η ( v )= gI ( η )( v i ) . Then I ( η ) ∈ Hom G (Ind GH ( E ) , F ). Now we will see that r and I are inverse of eachother. It is clear that r ( I ( η )) = η . On the other hand, I ( r ( ξ ))( v i ) = r − i ( r ( ξ )( v ))= r − i ξ ( r i x, v )= ξ ( x, v )= ξ ( v i ) . (cid:3) We have a formula for the character of an induced H -vector bundle, it is aparticular case of a formula for induced character of generalized cohomology theoriesin [HKR00] and [Kuh89]. We include a proof for completeness. Theorem 2.10 (Formula for the induced character) . Let X be a G -CW-complex,let H be a subgroup of G , let h be the order of H and E be a H -vector bundle,consider the map char G ◦ Ind GH : K H ( X ) ⊗ C → M [ g ] K ( X g ) C G ( g ) ⊗ C , let R be a system of representatives of H (cid:31) G . For each g ∈ G , we have char G ( g ) ◦ Ind GH ([ E ]) = M r ∈ R,r − gr ∈ H r ∗ (cid:16) char H ( r − gr )([ E ]) (cid:17) = 1 h M r ∈ G,r − gr ∈ H r ∗ (cid:16) char H ( r − gr )([ E ]) (cid:17) . Proof.
Our explicit definition of the induced bundle allows us to proof this resultjust by adapting the proof for representations contained in [Ser77]. The vectorbundle F = Ind GH ( E ) is the direct sum L ni =1 r ∗ i E , with R = { r , . . . , r n } . H (cid:31) G = { Hr , . . . , Hr n } . We know from the definition of the induced bundle that if we write r i g − in theform sr j with r j ∈ R and s ∈ H , then g sends r ∗ i E to r ∗ j E . Considering the actionof g in H (cid:31) G , we have thatchar G ( g ) (cid:16) Ind GH ( E ) (cid:17) = char G ( g ) M Hr i = Hr i g − r ∗ i E ⊕ M Hr i = Hr i g − r ∗ i E NDUCED CHARACTER IN EQUIVARIANT K-THEORY AND WREATH PRODUCTS 7
Note that g acts in each term of the direct sum on the right hand side, hence theright hand side of the above equation can be written aschar G ( g ) M Hr i = Hr i g − r ∗ i E ⊕ char G ( g ) M Hr i = Hr i g − r ∗ i E We will see that char G ( g ) (cid:16)L Hr i = Hr i g − r ∗ i E (cid:17) = 0. Because each 0-dimensionalbundle is trivial, it suffices to check that this condition holds on fibers, i.e.char G ( g ) M Hr i = Hr i g − r ∗ i E x = 0 , for all x ∈ X g . If we fix a basis for ( r ∗ i E ) x , the trace of the matrix representing theaction of g is zero because Hr i = Hr i g − .Now, if Hr i = Hr i g − we have that r i gr − i = s i with s i ∈ H . Thus as thecharacter is invariant under conjugationchar G ( g )( r ∗ i E ) = char H ( s i )( r ∗ i E ) . Finally as the character commutes with pullbacks we have that,char G ( g )(Ind GH ( E )) = M r ∈ R,rgr − ∈ H r ∗ i (cid:0) char H ( rgr − )( E ) (cid:1) = 1 h M s ∈ G,sgs − ∈ H r ∗ i (cid:0) char H ( s − hs )( E ) (cid:1) . (cid:3) Wreath product and its action on X n Let C be a set. There is a natural action of S n on C n defined as σ • ( c , . . . , c n ) = ( c σ − (1) , . . . , c σ − ( n ) )if G is a group, we define the wreath product as the semidirect product G n = G ≀ S n = G n ⋊ S n . This section is dedicated to describe the conjugacy classes and centralizers of ele-ments in G n , we follow [Mac15, Chapter 1, App. B] and [Wan00].First, we must recall the conjugacy classes in S n . Two elements s , s ∈ S n are conjugate if their cycle factorization correspond to the same partition of n . Forexample the elements (1 , , ,
5) and (1 , , ,
5) are conjugated and correspondto the partition 5 = 2 + 3. Note that every partition of n can be view as a function m : { , , · · · , n } → N as follows. If s ∈ S n then m s ( r ) is the number of r -cycesin s . Now in the general case, if x = (¯ g, s ) ∈ G n , then s can be decomposed asa product of disjoint cycles, if z = ( i i . . . i r ) is one of these cycles, the element g i r g i r − · · · g i is called the cycle product of x corresponding to z .Recall that G ∗ denotes the set of conjugacy classes of G . If x = (¯ g, s ) ∈ G n , let ρ ( x ) = m x ( r, c ) denote the number of r -cycles in s whose cycle product belongs to c , where c ∈ G ∗ and r ∈ { , , · · · n } . In this way every element x ∈ G n determinesa matrix ρ ( x ) = m x ( r, c ) of non-negative integers such that P r,c rm x ( r, c ) = n . GERMAN COMBARIZA, JUAN RODRIGUEZ, AND MARIO VEL´ASQUEZ
For example let G be the cyclic group, { g , g , g , g } , of 4 elements generatedby g and s = (1 , , , ∈ S . If x = ( g, g, g, g, g, s ) then ρ ( x ) = m x ( r, c ) lookslike g g g g r = 1 0 0 0 0 r = 2 0 0 1 0 r = 3 0 0 0 1 r = 4 0 0 0 0 r = 5 0 0 0 0the map ρ : G n → M n,v ( Z ) is called the type of x ∈ G n , where v = | G | . Proposition 3.1.
Two elements in G n are conjugate iff they have the same type.Proof. See [Mac15, Appendix 1.B] (cid:3)
By the above proposition we can assume that every element x ∈ G n is conjugatedto a product of elements of the form(( g, , . . . , , ( i u , . . . , i u r )) . Denote by g r ( c ) = (( g, , . . . , , (1 , . . . , r )) . Proposition 3.2.
The elements in the centralizer C G n ( g n ( c )) are of the form (( gz, . . . , z |{z} k +1 , . . . , gz ) , (1 , . . . , n ) k ) , with z ∈ C G ( g ) . Moreover C G n ( g n ( c )) ∼ = C G ( g ) ×h (1 , . . . n ) i .Proof. It follows from a direct computation. (cid:3)
We have described centralizers of elements in G n and in the next section we willuse this description to write char G n in terms of char G .Let X be a G -space, there is canonical G n -action over X n defined from the G -action over X G n × X n → X n ((¯ g, σ ) , ¯ x ) ¯ g ( σ • ¯ x )where ¯ g acts component-wise.In order to relate char G n with with char G we need to describe the fixed point setof a representative of each conjugacy class of G n . Let us start with the conjugacyclasses of elements (¯ g, σ ) where σ is an n -cycle. To this end we will need thefollowing result. Proposition 3.3.
Let ζ = (( g, , . . . , , σ ) whith g ∈ G and σ is a n -cycle. Thereis a canonical homeomorphism ( X n ) ζ /C G n ( ζ ) ∼ = X g /C G ( g ) . Proof.
We can assume σ = (1 , . . . , n ). Let ( x , . . . , x n ) ∈ ( X n ) ζ , then ζ ( x , . . . , x n ) = ( x , . . . , x n )it implies ( gx n , x , . . . , x n − ) = ( x , . . . , x n ) . NDUCED CHARACTER IN EQUIVARIANT K-THEORY AND WREATH PRODUCTS 9
Therefore x n = x n − = · · · = x , gx n = x , and then ( x , . . . , x n ) = ( y, . . . , y ) lies in the diagonal and y ∈ X g . This provesthat ( X n ) ζ ∼ = X g . On the other hand, if ¯ b ∈ C G n ( ζ ) then by Proposition 3.2¯ b = ((( gz, . . . , z |{z} k +1 , . . . , gz ) , σ k )where z ∈ C G ( g ). Then we obtain¯ b ( y, . . . , y ) = ((( gz, . . . , z |{z} k +1 , . . . , gz ) , σ k ) · ( y . . . , y ) = ( gzy, . . . , zy |{z} k +1 , . . . , gzy ) , showing that the orbit of ( y, . . . , y ) by C G n ( ζ ) is { ( gzy, . . . , zy |{z} k +1 , . . . , gzy ) : z ∈ C G ( g ) } . This proves the result. (cid:3) Fock space
Let X be a G -space, by the equivariant Bott periodicity theorem we know that K ∗ G ( X ) = K G ( X ) ⊕ K G ( X ) is a Z / Z -graded group. Denote by F G ( X ) = M n ≥ K ∗ G n ( X n ) ⊗ C , F qG ( X ) = M n ≥ q n K ∗ G n ( X n ) ⊗ C where q is formal variable giving a Z + -grading in F qG ( X ). They both have a nat-ural structure of abelian groups, we endow them with a product · , defined as thecomposition of the induced bundle and the K¨unneth isomorphism ⊠ (see [Min69]) K ∗ G n ( X n ) × K ∗ G m ( X m ) ⊠ −→ K ∗ G n × G m ( X n + m ) Ind −−→ K ∗ G n + m ( X n + m ) Proposition 4.1.
With the above operations F qG ( X ) is a commutative ( Z + × Z / Z ) -graded ring.Proof. The associativity follows from the following fact. Let [ E ] ∈ K ∗ G n ( X n ),[ E ] ∈ K ∗ G m ( X m ) and [ E ] ∈ K ∗ G k ( X k ), then( E · E ) · E ∼ = Ind G n + m + k G n + m × G k (cid:16) Ind G n + m G n × G n ( E ⊠ E ) ⊠ E (cid:17) ∼ = Ind G n + m + k G n × G m × G k ( E ⊠ E ⊠ E ) ∼ = Ind G n + m + k G n × G m + k (cid:16) E ⊠ Ind G m + k G m × G k ( E ⊠ E ) (cid:17) . For the graded commutativity, let [ E ] ∈ K ∗ G n ( X n ) and [ E ] ∈ K ∗ G m ( X m ), we willprove that E · E and E · E has the same character as G n + m -vector bundles over X n + m .Consider two inclusions of S n into S n + m . The first one is the inclusion bythe first n letters denoted by S n i n −→ S n + m , the second one is the inclusion bythe last n letters denoted by S n i m −−→ S n + m . Let x = (¯ g, σ ) ∈ G n + m and let r = (¯ h, τ ) ∈ G n + m , such that r − xr ∈ G n × G m , then there is η ∈ S n and η ∈ S m such that τ − στ = i ( η ) i ( η ), but i ( η ) i ( η ) is conjugated in S n + m to i ( η ) i ( η ), then there is γ ∈ S n + m such that γ − ( τ − στ ) γ = i ( η ) i ( η ).Then (( e, . . . , e ) , γ − )( r − xr )(( e, . . . , e ) , γ ) ∈ G m × G n . It implies that we have bijective correspondence between elements r ∈ G n + m suchthat r − xr ∈ G n × G m and elements s ∈ G n + m such that s − xs ∈ G m × G n . Thenif we apply Theorem 2.10, the number of terms in each direct sum computingInd G n + m G n × G m ( E ⊠ E ) and Ind G n + m G m × G n ( E ⊠ E )are the same. Moreover, for every ( α, β ) ∈ G n × G m ,char G n × G m ( α, β )( E ⊠ E ) ∼ = char G m × G n ( β, α )( E ⊠ E ) , then we have E · E and E · E have the same character, then the product iscommutative. The other properties follows directly. (cid:3) Definition 4.2.
Let R be a commutative ring, the graded-symmetric algebra of a Z -graded R -module M (denoted by S ( M )) is the quotient of the tensor algebra of M by the ideal I generated by elements of the form(i) x ⊗ y − ( − deg( x ) deg( y ) ( y ⊗ x )(ii) x ⊗ x , when deg( x ) is even.Now we will give another proof of the description of F qG ( X ) as a graded-symmetricalgebra given in [Seg96] or Theorem 3 in [Wan00]. Theorem 4.3.
There is an isomorphism of ( Z + × Z / Z ) -graded algebras Φ : F qG ( X ) → S ( ⊕ n ≥ q n K ∗ G ( X ) ⊗ C ) . Proof.
First note that using char G n we can define an injective group homomorphismin the following way. Consider the following sequence of maps K ∗ G ( X ) ⊗ C ∼ = −→ M g ∈ G ∗ K ∗ ( X g ) C G ( g ) ⊗ C λ −→ M x ∈ G n ∗ K ∗ (( X n ) x ) C Gn ( x ) ⊗ C ∼ = −→ K ∗ G n ( X n ) ⊗ C where the map λ is given by the assigning [(( g, , . . . , , (1 , . . . , n ))] G n to the conju-gacy class [ g ] G and using the identification in Proposition 3.3. This map is certainlyinjective. Define φ : K ∗ G ( X ) ⊗ C → K ∗ G n ( X n ) ⊗ C by the composition of the above sequence so that φ is injective and by the universalproperty of the graded-symmetric algebra we have a unique mapΦ : S M n ≥ K ∗ G ( X ) ⊗ C → F qG ( X )extending φ .Suppose inductively that im(Φ) contains K ∗ G k ( X k ) ⊗ C for k < n . Then byinduction we know that the image of the following composition S ( ⊕ n ≥ q n K ∗ G ( X ) ⊗ C ) × S ( ⊕ n ≥ q n K ∗ G ( X ) ⊗ C ) Φ × Φ −−−→ F qG ( X ) × F qG ( X ) · −→ F qG ( X )contains K ∗ G k ( X k ) ⊗ C · K ∗ G n − k ( X n − k ) ⊗ C ⊆ K ∗ G n ( X ) ⊗ C . Now we have that the image under char G n of NDUCED CHARACTER IN EQUIVARIANT K-THEORY AND WREATH PRODUCTS 11 n − M k =1 ( K ∗ G k ( X k ) ⊗ C ) · ( K ∗ G n − k ( X n − k ) ⊗ C )coincides with M x ∈ J K ∗ ((( X n ) x ) /C G n ( x )) ⊗ C , where J is the set of conjugacy classes in G n such that for every c , m • ( n, c ) = 0, inother words J is the set of conjugacy classes whose components in S n are not an n -cycle.On the other hand if x = (( g, , . . . , , (1 , . . . , n )) for some g ∈ G , then Propo-sition 3.3 gives us that im(char G n ◦ Φ) contains K ∗ (( X n ) x ) C Gn ( x ) . Finally, sincechar G n is an isomorphism we can conclude that Φ is surjective.To see that Φ is injective we can use the formula for the induced character,because this formula implies that if A ∈ S ( L k ≥ q k K ∗ G ( X ) ⊗ C ) is not zero thenthere exists n and x ∈ G n such that char G n ( x )(Φ( A )) = 0, then Φ( A ) = 0. (cid:3) Pullback of groups
Let Γ be a group fitting into the following pullback diagram(5.1) Γ p / / p (cid:15) (cid:15) G π (cid:15) (cid:15) H π / / K If the group Γ comes from a diagram 5.1 then it is isomorphic to a subgroup of G × H , namely Γ ∼ = { ( g, h ) ∈ G × H | π ( g ) = π ( h ) } . When maps π and π areclear from the context we denote Γ by G × K H . We suppose that π and π aresurjective.In this section we describe the class function ring of Γ in terms of the classfunction rings of G , H and K . In order to obtain this description we need that Γsatisfies the following condition. Definition 5.2.
Let G be a finite group and let H ⊆ G be a subgroup, let [ h ] H ∈ H ∗ , we say that [ h ] H is closed in G if,[ h ] H = [ h ] G ∩ H. We say that H is conjugacy-closed in G if, for every h ∈ H , [ h ] H is closed in G . Example 5.3.
The following are examples of conjugacy-closed subgroups: • The general linear groups over subfields are conjugacy-closed. • The symmetric group is conjugacy-closed in the general linear group. • The symmetric group on subsets are conjugacy-closed. • The orthogonal group is conjugacy-closed in the general linear group overreal numbers. • The unitary group is conjugacy-closed in the general linear group.
Remark 5.4.
Let G and H be groups • If H ⊆ G is conjugacy-closed in G , then the pullback of the inclusion i ∗ : Class( G ) → Class( H )is surjective. • If H is a retract in G , the pullback of the inclusion i ∗ : R ( G ) → R ( H )is surjective.When Γ is conjugacy-closed in G × H , we have a way to express the class functionring of Γ in terms of the class function rings of G , H and K . The same is true forthe representations ring when Γ is a retract of G × H .5.1. The class function ring of a pullback.
Consider a pullback diagram offinite groups such as (5.1). If we apply the representation ring functor we obtainthe following diagram(5.5) R (Γ) R ( G ) p ∗ o o R ( H ) p ∗ O O R ( K ) π ∗ o o π ∗ O O This diagram endows the rings R ( G ) and R ( H ) with a R ( K )-module structure.A similar statement is true changing the representation ring by the class functionring. We will prove that if Γ is a retract of G × H , then the diagram (5.5) is apushout. In fact, we have the following theorem. Theorem 5.6.
Let Γ , G , H and K be finite groups such as in the diagram (5.1).If Γ is conjugacy-closed in G × H , there is an isomorphism m : Class( G ) ⊗ Class( K ) Class( H ) → Class(Γ) of Class( K ) -modulesMoreover, if Γ is a retract of G × H , we have an isomorphism f : R ( G ) ⊗ R ( K ) R ( H ) → R (Γ) of R ( K ) -modules.Proof. In order to avoid confusion, in this proof we denote the product on Class(Γ),Class( G ) and Class( H ) by · and the generators of the tensor product by ρ ⊗ γ .The map f is defined as f : Class( G ) ⊗ Class( K ) Class( H ) → Class(Γ) ρ ⊗ γ p ∗ ( ρ ) · p ∗ ( γ )First we prove that the map f is well defined. Let ξ ∈ Class( K ), ρ ∈ Class( G )and γ ∈ Class( H ). Let ( g, h ) ∈ Γ f ( π ∗ ( ξ ) · ρ ⊗ γ ) ( g, h ) = ( p ∗ ( π ∗ ( ξ ) · ρ ) · p ∗ ( γ )) ( g, h )= ( π ∗ ( ξ ) · ρ ) ( g ) γ ( h )= ξ ( π ( g )) ρ ( g ) γ ( h )= ρ ( g ) ξ ( π ( h )) γ ( h )= f ( ρ ⊗ π ∗ ( ξ ) · γ )( g, h ) . NDUCED CHARACTER IN EQUIVARIANT K-THEORY AND WREATH PRODUCTS 13
Now we will prove that f is an isomorphism. Consider the following diagramwith exact rows0 / / ker( π ) / / f (cid:15) (cid:15) Class( G ) ⊗ C Class( H ) π / / f (cid:15) (cid:15) Class( G ) ⊗ Class( K ) Class( H ) / / f (cid:15) (cid:15) / / ker( i ∗ ) / / Class( G × H ) i ∗ / / Class(Γ) / / . Where map π is the quotient by the relations defining tensor product overClass( K ), map i ∗ is the pullback of the inclusion i : Γ → G × H , map f isthe natural isomorphism given by tensor product over C and the map f is therestriction of f to ker( π ). Note that as Γ is closed conjugacy in G × H the map i ∗ is surjective. We will prove that above diagram is commutative and that f is anisomorphism.First we need to verify that f (ker( π )) ⊆ ker( i ∗ ). Let ( g, h ) ∈ Γ, i ∗ [ f ( π ∗ ( ξ ) · ρ ⊗ γ − ρ ⊗ π ∗ ( ξ ) · γ )]( g, h ) =[ p ∗ ( π ∗ ( ξ )) · p ∗ ( ρ ) · p ∗ ( γ ) − p ∗ ( ρ ) · p ∗ ( π ∗ ( ξ )) · p ∗ ( γ )]( g, h ) = 0Now we prove that ker( i ∗ ) = f (ker( π )). For this we will prove that if f is aclass function in G × H such that i ∗ ( f ) ≡ f is orthogonal to every elementin f (ker( π )), then f has to be zero.Suppose that for every ξ ∈ Class( K ), ρ ∈ Class( G ) and γ ∈ Class( H ) we have X ( g,h ) ∈ G × H f ( g, h ) ρ ( g ) γ ( h )[ ξ ( π ( h )) − ξ ( π ( g ))] = 0 . Let us fix ρ ∈ Class( G ) and let η ( g ) = X h ∈ H f ( g, h ) γ ( h )[ ξ ( π ( h )) − ξ ( π ( g ))] . We observe that η is a class function on G that is orthogonal to every ρ in Class( G ),then η ≡ g, h ) ∈ G × H and ξ ∈ Class( K )(5.7) f ( g, h )[ ξ ( π ( h )) − ξ ( π ( g ))] = 0 . We already know that f ( g, h ) = 0 if ( g, h ) ∈ Γ, then let ( g, h ) / ∈ Γ, we have twocases. First suppose that π ( g ) is conjugate to π ( h ) in K , in this case there is¯ h ∈ H such that ( g, ¯ hh ¯ h − ) ∈ Γ and then f ( g, h ) = f ( g, ¯ hh ¯ h − ) = 0.Suppose now that π ( g ) is not conjugate to π ( h ) in K , in this case there is ξ ∈ Class( K ) such that ξ ( π ( g )) = ξ ( π ( g )) and equation 5.7 gives us that f ( g, h ) = 0.Then we conclude that ker( i ∗ ) = f (ker( π )). The map f is an isomorphism becauseit is the restriction of f and as the diagram is commutative we conclude that f isClass( K )-module isomorphism.When Γ is a retract in G × H , the same argument works changing characters byrepresentations, in particular the map i ∗ : R ( G × H ) → R (Γ) is surjective. (cid:3) Observe that the pullback is not always conjugacy-closed in the product as thefollowing example shows.
Example 5.8.
Consider the pullback of the symmetric groups S over the cyclicgroup C Γ / / (cid:15) (cid:15) S sgn (cid:15) (cid:15) S sgn / / C In this case the pullback Γ has 6 conjugacy classes, { γ , · · · , γ } . The prod-uct S × S has 9 conjugacy classes, { χ · · · , χ } . Observe that the elements((1 , , , (1 , , , , , (1 , , S × S by theelement ( e, (1 , S × S ) → Class(Γ) χ γ χ γ χ γ χ γ χ γ χ γ χ γ χ γ χ γ + γ . This map is not surjective.
Example 5.9.
Consider the following pullbackΓ / / (cid:15) (cid:15) D ψ (cid:15) (cid:15) C ⋉ C ψ / / S In this case the pullback Γ is isomorphic to the group C × ( C ⋉ C ) and itis conjugacy closed in the group D × ( C ⋉ C ). According with GAP[GAP18]the group D has group id (12,4) and generators d , d and d of orders 2 , ψ is given by ψ : D → S d (2 , d (1) d (1 , , . NDUCED CHARACTER IN EQUIVARIANT K-THEORY AND WREATH PRODUCTS 15
The group C ⋉ C has id (12,1) and it is generated by three elements g , g , g of orders 4 , ψ is given by ψ : C ⋉ C → S g (2 , g (1) g (1 , , . For these groups the pullback Γ is isomorphic to the group C × ( C ⋉ C ) withgroup id (24,7) and four generators f , f , f , f of orders 4 , , ∼ = Class( D ) ⊗ Class( S ) Class( C ⋉ C ) . For more examples please see https://sites.google.com/site/combariza/research/pullbacks-with-kernel-s3 .6.
Semidirect product of a direct product
Let A , A be groups with an action of a group G by automorphisms noted by a gi = g · a i , for a i ∈ A i and g ∈ G . Note that G acts also on the direct product A × A by acting on each component, i.e. ( a , a ) g := ( a g , a g ). In this section wedescribe the semidirect product of a direct product as a pullback of two semidirectproducts and then, we apply this for the wreath product of a direct product whichwill allow us to compute the Fock ring of a product.Consider the projections π i : A i ⋊ G → G and the pullback Γ associatedΓ (cid:15) (cid:15) / / A ⋊ G (cid:15) (cid:15) A ⋊ G / / G Proposition 6.1.
The pullback Γ is isomorphic to the semidirect product ( A × A ) ⋊ G. Proof.
Note that the pullback is the subgroup of ( A ⋊ G ) × ( A ⋊ G ) given byΓ = { ( a , g , a , g ) ∈ ( A ⋊ G ) × ( A ⋊ G ) : π ( a , g ) = π ( a , g ) , a ∈ A , g i ∈ G } that is, g = g . Consider the bijective function φ : Γ → ( A × A ) ⋊ G given by φ ( a , g, a , g ) = ( a , a , g ). On one hand φ [( a , g, a , g ) · ( b , h, b , h )] = φ ( a b g , gh, a b g , gh ) = ( a b g , a b g , gh ) . On the other hand ( a , a , g ) · ( b , b , h ) = ( a b g , a b g , gh ) which shows that φ is ahomomorphism of groups. (cid:3) Corollary 6.2.
Let
A, B be groups, there is an isomorphism ( A × B ) n ∼ = A n × S n B n . Now we proof that certain conjugacy classes in ( A × B ) n are closed in A n × B n . Proposition 6.3.
Let (¯ g, ¯ h, σ ) ∈ ( A × B ) n , where σ is an n -cycle. Then itsconjugacy class in A n × B n is closed. Proof.
Let x = (¯ g , ¯ h , σ ) and y = (¯ g , ¯ h , σ ) be elements in ( A × B ) n thatare conjugated in A n × B n , where σ and σ are n -cycles. We can suppose that σ = σ = (1 · · · n ).Note that (¯ g , σ ) ∼ A n (¯ g , σ ) and (¯ h , σ ) ∼ B n (¯ h , σ ) . Since as σ and σ are n -cycles, Q ni =1 g ,i ∼ A Q ni =1 g ,i , and Q ni =1 h ,i ∼ B Q ni =1 h ,i .On the other hand the type of x is given by m x ( r, c ) = ( r = n and ( Q ni =1 g ,i , Q ni =1 h ,i ) ∈ c y is given by m y ( r, c ) = ( r = n and ( Q ni =1 g ,i , Q ni =1 h ,i ) ∈ c x and y are equal, hence x and y are conjugated in ( A × B ) n . (cid:3) The Fock space of a product of spaces
In this section we apply results of Section 5 in order to obtain a decompositionof F G × H ( X × Y ) in terms of F G ( X ) and F H ( Y ). Let X be a G -space, we canendow to F G ( X ) with natural module structures as follows: • Consider the trivial G n -space {•} , and the unique G n -map π : X n → {•} ,then the pullback π ∗ : Class( G n ) → K ∗ G n ( X n ) ⊗ C induces a Class( G n )-module structure over K ∗ G n ( X ) ⊗ C , hence we have a F G ( {•} )-module structure over F G ( X ) defined componentwise. • Note that we have a quotient map s : G n → S n , then the pullback( π ◦ s ) ∗ : Class( S n ) → K G n ( X n ) ⊗ C induce a Class( S n )-module structure over K G n ( X ) ⊗ C , hence we have a F ( {•} )-module structure over F G ( X ) defined componentwise.As we observe in Section 5.1, ( G × H ) n is not closed conjugacy in G n × H n , then wecannot expect a decomposition of Class(( G × H ) n ) in terms of Class( G n ), Class( H n )and Class( S n ), but as the conjugacy classes with an n -cycle as component in S n are closed we have a decomposition of F G × H ( X × Y ) in terms of F G ( X ), F H ( Y )and F ( {•} ), with X a G -space and Y a H -space. Theorem 7.1.
There is an isomorphism of F ( {•} ) -modules F G × H ( X × Y ) K −→ F G ( X ) ⊗ F ( {•} ) F H ( Y ) . The map K is compatible with the symmetric algebra decomposition in Thm. 4.3.That means, we have a commutative diagram F G × H ( X × Y ) K / / d (cid:15) (cid:15) F G ( X ) ⊗ F ( {•} ) F H ( Y ) d ⊗ d (cid:15) (cid:15) S ( X × Y ) π G ⊗ π H / / S ( X ) ⊗ F ( {•} ) S ( Y ) . Where S ( X ) stands for S ( L n ≥ K G ( X ) ⊗ C ) , and similarly for S ( Y ) and S ( X × Y ) . NDUCED CHARACTER IN EQUIVARIANT K-THEORY AND WREATH PRODUCTS 17
Proof.
Let { E , . . . , E m } be a basis of K ∗ G ( X ) ⊗ C as complex vector space and let { F , . . . , F s } be a basis of K ∗ H ( Y ) ⊗ C as complex vector space. From the proof ofTheorem 4.3 we can conclude that { ∆ G,n,c,k ∈ K G n ( X n ) ⊗ C | n ≥ , c ∈ G ∗ , ≤ k ≤ m } is a basis of F G ( X ) as C -algebra, wherechar G n (∆ G,n,c,k )(( g , . . . , g n ) , σ ) = ( E k if Q ni =1 g σ i (1) ∈ c and σ is an n -cycle0 in any other case . In a similar way we define { ∆ H,n,d,l ∈ K H n ( Y n ) ⊗ C | n ≥ , d ∈ H ∗ , ≤ l ≤ s } a basis of F H ( Y ) as C -algebra, wherechar H n (∆ H,n,d,l )(( h , . . . , h n ) , σ ) = ( F l if Q ni =1 h σ i (1) ∈ d and σ is an n -cycle0 in any other case.Recall that we have an isomorphism K G × H ( X × Y ) ⊗ C ⊠ −→ ( K G ( X ) ⊗ K H ( Y )) ⊗ C , given by the external tensor product. It is proved in [Min69] or can be obtained(for complex coefficients) directly from the character.Using the above identification we have that { ∆ G × H,n,c × d, ( k,l ) | n ≥ , c ∈ G ∗ , d ∈ H ∗ , ≤ k ≤ m, ≤ l ≤ s } is a basis as C -algebra of F G × H ( X × Y ), wherechar ( G × H ) n (∆ G × H,n,c × d, ( k,l ) )(¯ g, ¯ h, σ ) = ( E k ⊠ F l if Q ni =1 g σ i (1) ∈ c, Q ni =1 h σ i (1) ∈ d and σ is an n -cycle0 in any other case.As the conjugacy classes when the character of the above elements is not zero isclosed in G n × H n we have∆ G × H,n,c × d, ( k,l ) = ∆ G,n,c,k · ∆ H,n,d,l , hence the map defined on generators as∆ G × H,n,c × d, ( k,l ) ∆ G,n,c,k ⊗ ∆ H,n,d,l is an isomorphism of F ( {•} )-modules satisfying the required conditions. (cid:3) final remarks In [Vel15] and [Vel19] a configuration space representing equivariant connectiveK-homology for finite groups was constructed. We recall the construction briefly.
Definition 8.1.
Let G be a finite group and ( X, x ) be a based G -connected, G -CW-complex. Let C ( X, x , G ) be the G -space of configurations of complex vectorspaces over ( X, x ), defined as the increasing union, with respect to the inclusions M n ( C [ G ]) → M n +1 ( C [ G ]) C ( X, x , G ) = [ n ≥ Hom ∗ ( C ( X ) , M n ( C [ G ])) , with the compact open topology. Notice that * refers to *-homomorphism, C ( X )denotes the C*-algebra of complex valued continuous maps vanishing at x and C [ G ] denotes the complex group ring.We endow C ( X, x , G ) with a continuous G -action as follows. If F ∈ C ( X, x , G ),we define g · F : C ( X ) −→ M n ( C [ G ]) f g · F ( g − · f ) . The space C ( X, x , G ) can be described as the configuration space whose elementsare formal sums n X i =1 ( x i , V i ) , when x i ∈ X − { x } and V i ⊆ C [ G ] ∞ such that if x i = x j then V i ⊥ V j , subject tosome relations, for details see [Vel19, Sec. 2.1]. We call the elements x i the points and to the vector spaces V i the labels . Remark 8.2.
When the based G -CW-complex ( X, x ) is not supposed to be G -connected, we define the configuration space C ( X, x , G ) = Ω C (Σ X, x , G ) , Where Ω denotes the based loop space and Σ denotes the reduced suspension.That description allow us to define a Hopf space structure on C ( X, x , G ) by putting together two configurations when labels in both of them are mutually or-thogonal.We have the following result: Theorem 8.3 (Thm. 5.2 in [Vel15]) . Let ( X, x ) be a based finite G -connected G -CW-complex. If we denote by k Gn ( X, x ) the n -th G -equivariant connective K-homology groups of the pair ( X, x ) , then there is a natural isomorphism π n ( C ( X, x , G ) G ) A n −−→ k Gn ( X, x ) . When a Hopf space Y is path-connected, consider the Hurewicz morphism λ : π ∗ ( Y ; C ) = M n ≥ π i ( Y ) ⊗ C → H ∗ ( Y ; C ) . We have the following result
Theorem 8.4 (Thm. of the Appendix in [MM65]) . If Y is a pathwise connectedhomotopy associative Hopf space with unit, and λ : π ∗ ( Y ; C ) → H ∗ ( Y ; C ) is theHurewicz morphism viewed as a morphism of Z -graded Lie algebras, then it inducesan isomorphism of Hopf algebras ¯ λ : S ( π ( Y ; C )) → H ∗ ( Y ; C ) . Applying the above theorem to C ( X, x , G ) we obtain. Corollary 8.5.
Let X be a finite G -CW-complex, if X is G -connected we have anisomorphism S ( k G ∗ ( X, x ) ⊗ C ) ∼ = H ∗ ( C ( X, x , G ) G ; C ) . NDUCED CHARACTER IN EQUIVARIANT K-THEORY AND WREATH PRODUCTS 19
In order to relate H ∗ ( C ( X, x , G ) G ; C ) with F G ( X ) we need to recall the followingresult proved in Theorem 6.13 in [Vel15] using the equivariant Chern characterobtained in [L¨uc02]. Theorem 8.6.
Let X be a G -CW-complex. There is a natural isomorphism of Z -graded complex vector spaces (here the graduation is given by q ) M q ≥ k Gn ( X ) ⊗ C ∼ = M n ≥ K Gn ( X ) ⊗ C [ q ] . Finally we can relate H ∗ ( C ( X, x , G ) G ; C ) with F qG ( X ) when X is an even di-mensional G -connected, G -Spin c -manifold. First we recall Poincar´e duality forequivariant K-theory.
Theorem 8.7. [BHS07]
Let M be a n -dimensional G -Spin c -manifold. Then thereexists an isomorphism D : K ∗ G ( M + ) −→ K Gn −∗ ( M ) . Applying Theorem 8.7 and Theorem 8.4 we can obtain the main result of thesection.
Theorem 8.8.
Let ( M, m ) be an even dimensional G -connected, G -Spin c -manifold.We have an isomorphism of Z -graded Hopf algebras H ∗ ( C ( M, m , G ) G ; C ) ∼ = F qG ( M ) . Proof.
Since M is a G -Spin c manifold we can use Theorem 8.7 and obtain thefollowing isomorphism of Z + × Z / Z -graded Hopf algebras S (cid:0) k G ∗ ( M, m ) ⊗ C (cid:1) ∼ = S M n ≥ q n K G ∗ ( M, m ) ⊗ C ∼ = S M n ≥ q n K ∗ G ( M + , +) ⊗ C ∼ = S M n ≥ q n K ∗ G ( M ) ⊗ C . Combining Corollary 8.5, Theorem 8.4 and Theorem 4.3 we obtain H ∗ ( C ( M, m , G ) G ; C ) ∼ = F qG ( M ) . (cid:3) For the case when M is not necessarily G -connected, we can obtain also a similarresult. For details consult [Vel15, Proposition 6.11]. Proposition 8.9.
Let X be a finite G -CW-complex, we have an isomorphism H ∗ (Ω C (Σ X, G ) G ; C ) ∼ = S ( k G ∗ ( X, x ) ⊗ C ) . In particular we have.
Example 8.10.
For X = S we haveΩ (cid:0) C (Σ( S ) , G ) (cid:1) ≃ BU G . Where BU G can be taken as the Grassmannian of finite dimensional vector sub-spaces of a complete G -universe. A complete G -universe is a countably infinite-dimensional representation of G with an inner product such that contains a copyof every irreducible representation of G , contains countably many copies of eachfinite-dimensional subrepresentation. Applying the above discussion to this Hopfspace we conclude that H ∗ (cid:0) ( BU G ) G ; C (cid:1) ∼ = R ( G ) ⊗ S (cid:0) π ∗ (( BU G ) G ) ⊗ C (cid:1) ∼ = R ( G ) ⊗ S M n ≥ R ( G n ) ⊗ C ∼ = S M n ≥ R ( G n ) ⊗ C . Summarizing, we have an isomorphism H ∗ (( BU G ) G ; C ) ∼ = F qG ( {•} ) = S M n ≥ R ( G n ) ⊗ C . We also have H ∗ (cid:0) ( BU G ) G ; C (cid:1) ∼ = S M n ≥ R ( G n ) ⊗ C ∼ = C [ σ , . . . , σ k , σ , . . . ]where { σ i , · · · , σ k i i } is a complete set of non isomorphic irreducible representationsof G i . We expect that the elements σ ki correspond in some sense with duals of G -equivariant Chern classes.Now suppose that M is a G -connected G -Spin c -manifold and N is a H -connected H -Spin c -manifold, then we have an isomorphism of Z -graded Hopf algebras H ∗ (cid:0) C ( M × N, ( m , n ) , G × H ) G × H ; C (cid:1) ∼ = F qG ( M ) ⊗ F ( {•} ) F qH ( N )In the case that M = N = S with trivial action we obtain H ∗ (( BU G × H ) G × H ; C ) ∼ = F G ( {•} ) ⊗ F ( {•} ) F H ( {•} ) . References [AS89] M. F. Atiyah and Graeme Segal. On equivariant Euler characteristics.
J. Geom. Phys. ,6(4):671–677, 1989.[Ati89] M. F. Atiyah. K -theory . Advanced Book Classics. Addison-Wesley Publishing CompanyAdvanced Book Program, Redwood City, CA, second edition, 1989. Notes by D. W.Anderson.[BHS07] Paul Baum, Nigel Higson, and Thomas Schick. On the equivalence of geometric andanalytic K -homology. Pure Appl. Math. Q. , 3(1, part 3):1–24, 2007.[GAP18] The GAP Group.
GAP – Groups, Algorithms, and Programming, Version 4.8.10 , 2018.[HKR00] Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel. Generalized groupcharacters and complex oriented cohomology theories.
J. Amer. Math. Soc. , 13(3):553–594, 2000.[Kuh89] Nicholas J. Kuhn. Character rings in algebraic topology. In
Advances in homotopy theory(Cortona, 1988) , volume 139 of
London Math. Soc. Lecture Note Ser. , pages 111–126.Cambridge Univ. Press, Cambridge, 1989.
NDUCED CHARACTER IN EQUIVARIANT K-THEORY AND WREATH PRODUCTS 21 [L¨uc02] Wolfgang L¨uck. Chern characters for proper equivariant homology theories and appli-cations to K - and L -theory. J. Reine Angew. Math. , 543:193–234, 2002.[Mac15] I. G. Macdonald.
Symmetric functions and Hall polynomials . Oxford Classic Texts inthe Physical Sciences. The Clarendon Press, Oxford University Press, New York, secondedition, 2015. With contribution by A. V. Zelevinsky and a foreword by Richard Stanley,Reprint of the 2008 paperback edition [ MR1354144].[Min69] Haruo Minami. A K¨unneth formula for equivariant K -theory. Osaka J. Math. , 6:143–146, 1969.[MM65] John W. Milnor and John C. Moore. On the structure of Hopf algebras.
Ann. of Math.(2) , 81:211–264, 1965.[Seg68] Graeme Segal. Equivariant K -theory. Inst. Hautes ´Etudes Sci. Publ. Math. , (34):129–151, 1968.[Seg96] Graeme Segal. Equivariant K -theory and symmetric products. Preprint, 1996.[Ser77] Jean-Pierre Serre. Linear representations of finite groups . Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott, Grad-uate Texts in Mathematics, Vol. 42.[tD87] Tammo tom Dieck.
Transformation groups , volume 8 of de Gruyter Studies in Mathe-matics . Walter de Gruyter & Co., Berlin, 1987.[Vel15] Mario Vel´asquez. A configuration space for equivariant connective K-homology.
J. Non-commut. Geom. , 9(4):1343–1382, 2015.[Vel19] Mario Vel´asquez. A description of the assembly map for the Baum-connes conjecturewith coefficients.
New York Journal of Mathematics , 29:668–686, 2019.[Wan00] Weiqiang Wang. Equivariant K -theory, wreath products, and Heisenberg algebra. DukeMath. J. , 103(1):1–23, 2000.
Departamento de Matem´aticas., Pontificia Universidad Javeriana, Cra. 7 No. 43-82- Edificio Carlos Ort´ız 5to piso, Bogot´a D.C, Colombia
E-mail address : [email protected] URL : https://sites.google.com/site/combariza/research UMPA L’unit´e de Math´ematiques Pures et Appliqu´ees, ENS de Lyon site MonodUMPA UMR 5669 CNRS 46, all´e d’Italie, 69364 Lyon Cedex 07, France
E-mail address : [email protected] Departamento de Matem´aticas., Pontificia Universidad Javeriana, Cra. 7 No. 43-82- Edificio Carlos Ort´ız 5to piso, Bogot´a D.C, Colombia
E-mail address : [email protected] URL ::