Equivariant $K$-groups of spheres with involutions

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EQUIVARIANT K-GROUPS OF SPHERES WITH INVOLUTIONS

JIN-HwAN CHot AND MIKIYA MASUDA

ABSTRACT. We calculate theR(G)-algebra structure on the reduced equivariant K-groups of two-dimensional spheres on which a compact Lie group G acts as a reflection. In particular, the reduced equivariant K-groups are trivialif Gisabelian, which shows that the previous Y. Yang's calculation in [8] is incorrect.

1. Introduction

Let G be a compact Lie group. As one of (stable) equivariant coho-:

mology theories, RO(G)-graded cohomology theory has been considered a fullequivariant generalization of ordinary singular cohomology, which is built in the interplay relating the Burnside ring, the real character ring RO(G), and G-homotopy theory (see [5] for more details). The main difference from the other equivariant cohomology theories is that the coefficient ringisindexed by all virtual real representations instead of integers, and equivariant K-theory is one of the adequate known ex- amples.

In 1995, Y. Yang [8] proved that, for any compact Lie group G, the coefficient groups K(;(8°)

==

Kc(8*) of RO(G)-graded equivariant K- theory can only have 2-torsion. He calculated the reduced equivariant

Received May 30, 2000.

2000 Mathematics Subject Classification: Primary 19147; Secondary 55N15, 55N25, 55N91.

Key words and phrases: RO(G)-graded cohomology theory, equivanantK-theory, equivariant vector bundle, representation sphere.

The manuscript was written based on the authors' talk at the 2nd Japan-Korea conference on Transformation Groups on August 8, 1999 at Okayama University of Science.

tThe first author was supported by postdoctoral fellowships program from Korea Science&Engineering Foundation (KOSEF).

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K-groups ofS1 andS2with a finite cyclic group G acting as involutions, and deduced his main result using the Bott periodicity theorem andJ.E.

McClure's results [7].

More precisely, if a compact G-space X has a base point

*

fixed by the G-action, then the reduced equivariant K-group KG(X) is defined to be the kernel of the restriction homomorphism KG(X) ^---t KG(*) induced from the inclusion map

*

<...t X. In fact, KG(X) and KG(X) are algebras over the complex character ringR(G) (although there is no identity element in KG(X)). Let ,\: G ^---t 0(1) = {±1} be a surjective homomorphism regarded as a one-dimensional real representation of G.

Denote by 1 the trivial one-dimensional real representation of G. Ac- tuallyY. Yang calculatedKG(SA) and K G(S1f9A) for finite cyclic groups G, whereSA andStEMdenote the one-point compactifications of the real representations Aand 1 EBA, respectively.

The authors proved recently in [2, Theorem 10.1] thatKG(SA)is isomorphic to the ideal inR(G) generated by(1-A)@C,which extends Y.

Yang's result[8,Theorem A] for G finite cyclic to any compact Lie group.

In this paper we apply the same technique to calculate theR(G)-algebra structure ofK a(S1f9A)^{when G} ^isacompact Lie group. In particular, we will prove thatK a(S1f9A) is trivial for any compact abelian Lie group G, which shows thatY. Yang's result [8, Theorem B] isnot true.

Inthe following our main results are stated. Denote byH the kernel of the real representationA. Given a character X of H and 9E G, a new character gx of H is defined by gX(h) ⁼ X(g-1hg) for h E H. Choose and fix an element bEG \ H. Since a character is a class function and G/H is of order two, ^bX⁼ gxfor all9E G \ H so that^bX is independent of the choice of the elementbEG \ H. Note that R(H) has a canonical R(G)-module structure given by cp. X= reSHCP @ X for cP E R(G) and X^E R(H).

THEOREM 1.1. Let G be a compact Lie group and let A: G ---t 0(1) beasurjectivehomomorphism. Denote by H the kernel of Aandchoose an element bEG \ H. Then

K

G(S1f9A) is isomorphic to the R(G)- module consisting of the elements X - ^bXⁱⁿR(H) for all charactersXof H. Moreover, the ring structure on KG(S1f9A) is given byaf3

=

0 for all elements a,f3 E Ka(S1ffiA).

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COROLLARY 1.2. Inparticular,

K

G ( SIfI!A) is trivial if there exists an element in G \ H commuting with all elements in H (for example, G is abelian).

2. Complex Z2-vector bundles over SIfI!A

We will begin by considering the structure of complex Z2-vector bundles over SIfI!A. In this case >.: ^Z2 ^-T ⁰⁽¹⁾

=

{±1} becomes an isomorphism.

LEMMA 2.1. Everycomplex~-vectorbundle over SIfI!A decomposes into the Whitney sum of Z2-invariant sub-line bundles.

Proof. The set of~-fixedpoints inSIfI!A constitutes a circle, denoted by S\ containing 0 and *. Then ^SI divides the sphere SIfI!A into two hemispheres, sayHI and H2 . Let E be a complex Z2-vector bundle over

SIfI!A. Note that the restriction ofE to the subspaceSI

c

^SIfI!A decomposes into the Whitney sum of Z2-invariant sub-line bundles. Choose a Z2-invariant sub-line bundle, say F, over SI. Then it is always pos- sible to extend F to a non-equivariant sub-line bundle over HI, since the restriction ofE to HI is non-equivariantly trivial and the set of one- dimensional subspaces of the fiber is homeomorphic to the Grassmann manifold Gdk,1) _~ Cpk-l, the fundamental group of which is trivial.

We now extend it over the other hemisphereH₂ using the Z2-action on E to get a resulting Z2-invariant sub-line bundle ofE. 0

LEMMA 2.2. Everycomplex Z2-vector bundle overSIfI!A is trivial.

Proof. It suffices to show that every complex Z2-line bundle E over

SIfI!A is trivial by Lemma 2.1. Choose two Z2-invariant hemispheres, denoted by Ho and H* of SIfI!A containing 0 and *, respectively, such that Ho UH*

=

^SIfI!A and Ho nH* is a Z2-invariant circle. Since Ho (resp. H*) is equivariantly contractible to 0 (resp. *), E restricted to Ho (resp. H*) is equivariantly isomorphic to the product bundle Ho x Eo (resp.H* xE*) whereEo (resp. E*) denotes the fiber of Eat0 (resp. *).

Then E induces a Z2-equivariant clutching map ofHonH* into the set Hom(Eo, E*) of linear maps between the two fibers Eo and E*. Note thatEo and E* are isomorphic as complex representations of Z2 since0

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and

*

are connected by Z2-fixed points, and thus the induced Z2-action on Hom{Eo, E*) is trivial. Since Z2 acts on the circle Ho nH* as a reflection, the clutching map is equivariantly null-homotopic, that is, E

is equivariantly trivial. 0

3. Induced vector bundles

In this section we review the notion of induced vector bundles defined in a way similar to the definition of induced representations.

Let G be a compact Lie group and let X be a G-space. The set of isomorphism classes of (real or complex) G-vector bundles over X, usually denoted by Vectc{X), is a semi-group under the Whitney sum operation. For a closed subgroup H of G, the inclusion map H ^<-t G in- duces a semi-group homomorphism reSH: Vectc{X) ---tVectH{X) called the restriction homomorphism. .On the other hand, there is another homomorphism ind~:VectH{X) ---t Vectc{X) called the induction ho- momorphism (see for instance [4] or [6]). In addition, if H has finite index in G, then the induction. homomorphism can be defined explicitly, which we will explain below..

LetE be an H-vector bundle overX. Choose a set ofrepresentatives {to, tt, ... , tl} of G/H. The identity element of G will alwaysbe selected asto. Define a group isomorphism 'tg : gHg-l ---t H by 'tg{ghg-1)

=

h for h E H and- 9 E G.

We first construct atiHti1-vector bundle tiE over X for each 0 :::;i :::;

t.

Consider the following pull-back diagram (ti¹)*E ^{- - - t} E

1 1

X

whereti1 denotes the action of ti 1 on X. Since the action of ti1 on X is 'tti-equivariant, Le., ti 1{hx)

==

'tti{h)ti 1{x) for h E tiHti1 and x E X, the pull-back bundle (ti 1)* E has a canonical tiHti1-action and it becomes atiHti1-vector bundle over X, simply denoted by tiE. Note that every element in the fiber oftiE at x E X is represented by tiv for some v in the fiber ofE at ti1X. We now define ind~E by the Whitney sum

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(without action) of all the bundlestiE and then give a G-action on it as follows.

Given9 E G, each element gti has a unique form ti,hi for some rep- resentative ti' E {to, ... , tl} and hi E H. So we define an action of9 on ind~E by

If

I:

tiVi is an element in the fiber ofind~E at x EX, then each Vi is contained in the fiber ofEat t;l x and ~Viis in the fiber of Eatt;;Igx since hit;1

=

t;/g. Thus the action of9 gives a linear map between the two fibers at

x

ând ^{gx for} âll ^{x EX.} Ît is easy to see that the definition gives actually an action of G on_ind~E. Moreover the action is compatible with the tiHt;l-action already defined on tiE.

We next show that the induced bundle is independent of the choice of the representatives {to, ... , tl} of G/ H. Choose another set ofrepr~

sentatives{so

=

e, SI, ... , SI}ofG/H. We may assume that eachs;lti is contained inH by arranging the order of representatives if necessary, and that every element in the induced bundle constructed with {so, ... ,SI}

is written by

I:

SiVi. It is easy to check that the bundle map defined

by

I:

tiVi ^{I - t}

I:

Si (S;lti)Vi gives an equivariant isomorphism between the

two induced bundles.

LEMMA3.1. H two H-vector bundles E and E' over X are isomor- phic, then so are the induced bundles_ind~E and_ind~E'. The converse is not true in general.

Proof. Given an H -vector bundle isomorphism <Po: E ^{- t}E', the map

<P: _ind~E ^{- t}_ind~E'

sending I:tivi to I:ti<PO(Vi) gives a G-vector bundle isomorphism. 0

LEMMA 3.2. Let W be a representation of H. Then the induced bundle of the product bundle X x W ^{- t}X isisomorphic to the product bundle X x _ind~W - t X.

Proof. The construction of induced bundles is the same as that of

induced representations. D

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LEMMA 3.3. Let L be a G-vector bundle and let E be an H-vector bundle over the same base space X. Then L_®ind~Eis isomorphic to ind~(resHL^®E).

Proof For 1ELand

:E

tiVi E_ind~E, the map

~; L ®_ind~E -+indk(resHL®E)

sending 1® :EtiVi to :Eti(ti11 ®Vi) gives a (non-equivariant) vector bundle map. Given9 E G we have gti

=

tifhifor some representative tif and hi EH. Then the equalities

~(g(l^®

L

^tiVi))

=

~(gl^®

L

^tif(hiVi))

= L

^tif(ti/g1^®~Vi)

= L

ti,hi(ti 11®Vi)

= L

^gti(ti11^®^Vi)

=

g~(l^®

L

tiVi)

imply that _~ is G-equivariant. On the other hand, the inverse bundle mapisgiven by the map sending

:E

ti(li®Vi) to :E{ti1i®tiVi) for li EL

andVi EE. 0

4. Decomposition of equivariant vector bundles

In this section we rephrase the relevant material from [2, Section 2]

to decompose equivariant vector bundles for the readers' convenience.

Let G be a compact Lie group and H a closed normal subgroup of G. Given a character X of H and 9 E G, a new character 9X of H is defined by 9X(h) = X(g-lhg) for h E H. This defines an action of G on the set Irr(H) of irreducible characters ofH. Since a character is a class function, H acts trivially on Irr(H). Therefore, the isotropy subgroup of G at X E Irr(H), denoted byG_x' containsH. We choose a representative from each G-orbit in Irr(H) and denote by Irr(H)jG the set of those representatives.

LetX be a connected G-space on whichH acts trivially. Then all the fibers of a G-vector bundle E over X are isomorphic as representations ofH. We call the unique (up to isomorphism) representation ofH the fiber H -representation of E.

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Asis well-known, E decomposes according to irreducible representations of H. For X E Irr(H), we denote by E(X) the X-isotypical component of E, that is, the largest H-subbundle of E with a multiple of X as the character of the fiber H-representation. Note that gE(X), that is E(X) mapped by 9 E G, is gx-isotypical component ofE. This means thatE(X) is actually a Gx-vector bundle and that EBXEG(X)E(X'), where G(X) denotes the G-orbit or X, is a G-subbundle of E. Since EBXEG(x)E(X') is nothing but the induced bundle indgxE(x), we have the following decomposition

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as G-vector bundles.

E =

EB

^indgxE(x)

XEIrr(H)/G

and KG(X)-+

IT

^KGx(X,X)

xElrr(H)/G

LEMMA 4.1. Two G-vector bundles E and E' are isomorphic ifand onlyifE(X) and E'(X) are isomorphic as Gx-vector bundles for each XE Irr(H)jG. In particular, E is trivialifandonlyifE(X) istrivialfor each XE Irr(H)jG.

Proof. The necessity is obvious since a G-vectorbundle isomorphism E -+ E' restricts to a Gx-vector bundle isomorphism E(X) -+ E'(X), and the sufficiency follows from the fact that indg_x isfunctorial. 0

COROLLARY 4.2. Two G-vector bundles E and E' are stably iso- morphic ifand onlyifE(X) and E'(X) are stably isomorphic for each

XEIrr(H)jG. 0

The observation above can be restated in K-theoryas follows. Denote byK Gx (X, X) the subgroup of_{K Gx (X)} generated by Gx-vector bundles overX with a multiple ofXas the character of fiber H-representations.

The reduced version KGx(X,X) can be defined accordingly. Then the map sendingE to IIxEIrr(H)/G E(X) gives group isomorphisms

KG(X) -+

IT

K Gx (X, X) XEIrr(H)/G

by Corollary 4.2.

LEMMA 4.3. Ifthere is a complex Gx-vector bundle L over X with X as the characterofthe fiber H -representation, then themap sending E

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toHomH(L, E) gives group isomorphisms

Kcx(X,X) ^{- t}KCx/H(X) and Kcx(X,X) ^{- t}KCx/H(X).

Proof. It is easy to check that the map KCx/H(X) - t Kcx(X, X)

sendingF to L (8)F gives the inverse of the map in the lemma. 0

REMARK. The lemma above does not hold in the real category in general, but it doesif Xisthe character of a real irreducible representation ofH with the endomorphism algebra isomorphic to the set of real numbers.

5. Proof of Theorem 1.1

We now return to our original setting to prove the main result. Here- after we omit the adjective "complex" for complex vector bundles and complex representations since we work in the complex category. At first consider the additive structure on

K

_c(8¹$>').

Let G be a compact Lie group. Denote byH the kernel of the silljec- tive homomorphism oX: G ^{- t}0(1) = {±1}. Choose and fix an element bEG \ H. Since Gx contains H and G j H is of order two, Gx is either H or G for each irreducible character X EIrr(H)jG. Note that, in each case, the condition of Lemma4.3holds, since there exists a Gx-extension ofX [Proposition 4.2][3] and it gives a trivial Gx-vector bundle with X as the character of the fiber H-representation. Therefore, according to the arguments in the previous. section, we have a group isomorphism (**) Kc(8¹$>') _~

IT

^K(8²⁾ ^X

IT

KC/H(8¹^$>').

xEIrr(H)/C XEIrr(H)/C s.t.Cx^{= H} ^s.t.^Cx=c

Itis well-known that K(8²)is infinite cyclic with generator(-1, where ( and 1, respectively, denote the dual bundle of the canonical line bundle and the trivial line bundle over 8 2~ Cpl (see for instance [Theo- rem 2.3.14][1]). Moreover,

K

C / H(8¹$>') is trivial since GjH ^rv Z2 and every Z2-vector bundle over 8¹^$>' is trivial by Lemma 2.2. Therefore the decomposition (*) and Lemma4.3in the previous section imply the following lemma.

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LEMMA 5.1. K C(Sltl3A) ^isa free abeliangroup generated byind~(X0 (( - 1)) for each X E Irr(H)jC such that G_x

=

H. Here, the same notation Xis used for the product H-vector bundle overSltl3).. with X as

the character of the fiber representation. 0

Proof of THEOREM 1.1. We now consider the map W: KC(Sltl3A) ~ R(H)

sending each generatorind~(X0((-1)) ofKC(Sltl3A)^toX-bX ER(H).

Since both KC(Sltl3A) ^and R(H) are free abelian groups, Wis a well- defined group homomorphism. Moreover it is injective, since the set of the elements X - bX^{for all X} ^E Irr(H)jC such that Gx ⁼ ^H can be extended to an additive basis ofR(H).

For each XE Irr(H), eitherC_x = H or C. In case that G_x= C, we have X- bX

= o.

Thus the image ofWis generated by the elements X- bX for all XE Irr(H), that is, the R(G)-submodule ofR(H) consisting of the elements X - bX for all characters X of H.

Given a character'PofC, as in Lemma 5.1, we use the same notation 'Pfor the product C-vector bundle over Sltl3A with'P as the character of the fiber representation. Then Lemma 5.3 implies that

'P0 ind~(x 0 ((-1)) = ind~(resH'P 0 X 0 (( - 1)).

Sinceb(resH'P) = resH'P, we have the equalities

W('P0 ind~(X 0 (( - 1))) = reSH'P0 X - b(resH'P)0 bX

= reSH'P0 (X - bX)

=

ep.W(ind~(X 0 (( - 1))) showing that Wis an R(G)-module homomorphism.

It remains to show the ring structure onKC(Sltl3A). ^Itsuffices to show that the tensor product of any two generators inKC(Sltl3A)is zero. Note that, given an induced bundle _ind~(x® (), the image ofX0 ( mapped by the b-action, that is the bX-isotypical component of ind~(x 0 (), is isomorphic to bX0 (* where (* denotes the dual bundle of (. In- deed, the pull-back bundle(b-¹)*(() is isomorphic to(*,since the action b-^{1 :} Sltl3A~ Sltl3).. is a reflection so that it induces the multiplication by

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o

-Ion the second cohomology level. Itfollows that

resHind~(X0 ( -1))

=

^X^{® ( -} ¹⁾

+

bX®(C -1)

=

^X^{® ( -} 1) - bX® ( - 1)

=

(X - bX)^{® ( -} 1),

since(* - 1

= -(( -

^{1) in} K(8²^). Therefore, by Lemma 3.3, we have the equalities

ind~(x ® ( - 1)) ® ind~(1] ® (( - 1))

= ind~(resHind~(x ® ( -1» ® 1] ® (( - 1))

=ind~(((X- bX)® 1] 0 ( _1)2)

=0,

since (( - 1)2= 0 in [((8²) (see [1, Theorem 2.3.14]).

Proof of COROLLARY 1.2. By assumption there exists an element bEG \ H such that bh

=

hb for all h E H. It follows that bX(h) = X(b-1hb)

=

^X(h) for any character XofH, which completes the proof.

o

References

[1) M. F. Atiyah,K-theory, Lecture notes by D. W. Anderson, W. A.Benjamin,Inc., New York-Amsterdam, 1967.

[2) J.-H. Cho, S. S. Kim, M. Masuda, and D. Y. Suh, Classification of equivariant complex vector bundles over a circle, preprint, 1999, arXiv:math.AT/991000l.

[3) J.-H. Cho, M. K. Kim, and D. Y. Suh, On extensions of representations for compact Lie groups, preprint, 1999, arXiv:math.RT/9912180.

[4] E.C. Cho andD.Y. Suh,Induction in equivariant K -theory and s-Smith equiva- lence of representations, Group Actions on Manifolds (R. Schultz, ed.), Contemp.

Math., vo!. 36, Amer. Math. Soc., 1985, pp. 311-315.

[5] G. Lewis, Jr., J.P.May, and J.E.McClure, Ordinary RO(G)-graded cohomology, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 208-212.

[6) M. Masuda and T.Petrie, Lectures on transformation groups and Smith equiv- alence, Group Actions on Manifolds (R. Schultz, 00.), Contemp. Math., vo!. 36, Amer. Math. Soc., 1985, pp. 191-242.

[7] J. E. McClure, Restriction maps in equivariant K-theory, Topology 25 (1986), no. 4, 399--409.

[8) Y. Yang, On the coefficient groups of equivariant K-theory, Trans. Amer. Math.

Soc.347(1995), no. 1, 77-98.

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Jin-Hwan Cho

School of Mathematics

Korea Institute for Advanced Study

207-43 Cheongryangri-dong, Dongdaemun-gu Seoul 130-012, Korea

E-mail: [email protected] Mikiya Masuda

Department of Mathematics Osaka City University

3-3-138 Sugimoto, Sumiyoshi-ku Osaka 558, Japan

E-mail: [email protected]