BULLETIN
DE L’ACADEMIE
POLONAISEDESSCIENCES
sériedessciencesmath.,)astr.

et phys— Vol. XXIV, No. 10, 1976

MATHEMATICS(ALGEBRAIC TOPOLOGY)

On the Equivariant Chern Homomorphism
byJ. SEOMINSKA
Presentedby K. BORSUKon

January

14,

1976’Summary. A split coefficient system for the equivariant Bredon cohomology is defined. Its properties are used to show that Ks (X)® Q is iSomorphic to the Bredon cohomology of X with appropriate coefficients, provided G is a finite group and X is a compact G-CW complex. As a corollarywe obtain that Kg ® Q can be expressed in terms of the ordinary K-theory.

1. Split coefficient system.

Let G be a finite groupand DO; the category of

canonical G-orbits, i.e. G-sets of the form

G/H, where H is a subgroup

G-morphisms. Two orbits G/H and G/H'

are identified in Og; iff H and H' areof G, andconjugate in G. The category of contravariant functors from Og to the category Abof abelian groups is denoted by &¢. Objects of &; are called G-coefficient systems.

If H is a subgroup of G then there exists a functor (+)g: €s— Cy
anysuch that for

G-coefficient system M

My (H/H)=M
whenever

H/H'

Let Ogis an object(G/H')

of Dy.

be the full subcategoryof Og,

consisting of all orbits different from

G/G. If M is as above then for any canonical orbit ¢/H we will denote lim My by
M (G/H) and the structural morphisms of this limitOn
M (G/H)-»Myz (H/H)=M(G/H")

by p (H/H"). M (G/H) possess a natural structure of a WH=NH/H-module.
If ne NH

then the compositionnH

M(G/H)——

p(H/H')

M(G/H)———
M (G/H)=(M¢/n~* H' n)

is equal to p (¥/n~* H' n). Letm (G/H): M (G/H)—M (G/H)
be a WH-module morphism such that p (H/H") m(G/H) is the morphism My (*/H—~My (H/H") induced by the map H/H'— H/H, whenever H/H' is a canonicalH-orbit. Ker m (G/H) is denoted by M (G/H).

|909.

910

J. Stomihska

1.1. DEerFINITION. Aorbit

G/H

there

G-coefficient system AM is called split iff for any canonicalexistsaWH-modulemorphism’

7 (G/H): M (G/H)— M (G/H)

satisfyingy:m (GH) t (G/H)=id.
1.2. Examples.

Let Rg be a coefficient system defined for objects as Rg (G/H)==R(H)®Q, where R(H) is a unitary representation ring, and for G-maps G/H—
—G/H' as the composition of a restriction homomorphism and conjugation byelements of G. This follows from the proof of the Artin theorem (see Serre [4])that Rg is a split coefficient system. Furthermore, one can check that if M is anEE
1arbitrary Mackey functor(see Dress [2]) over Z a. System.

then M is a split coefficienttee

1.3. LEMMA.

If M

is a split G-coefficient systemthen‘Homg (N, M)=[]. Homyx (N(G/H), M (G/H)).G/He Og

whenever N is an object of C;.

Proof. Let {{e}}=90, =O,... ¢0,=D¢be a certain filtrationof Og,

such thatany Oy is a full subcategory of Og with k objects and if ¢/H is in O, and there exists
a morphism in Og from G/H' to G/H then ¢/H’ is in Oy, too. Let Ou \Dy_; ={G/H,}For any G-map f: G/H,— G/H, one can find a subgroup H; of H, conjugate
to H, in G and an element w of WH such that the morphism M (f) is equal to, thecomposition M (w) p (H;/H,). This yields a group isomorphism:
Hom, (N, M)=Homg,_ (N, M) ® Hompy, (N (G/H;), ker m (GH),where Homg (XN, M) denotes the group of all natural transformations from N lo,to Mg, The

statementof thelemma

follows

fromthe.above

formulabyinduction.

2. Bredon cohomology with the coefficient system R; and the K; ® O-theory.
If X 1s a G-CW complex then the equivariant Bredon cohomology of X with a coeflicient system M will be denoted by H* (X,M) (see [1]).
2.1. PROPOSITION.

theories

There exists a natural transformation of equivariant cohomology‘
che:aKo—~

k=0such that for any compact G-CW complex©

H?* (, Rg),.

X:

(ch? id) (X): Ks (X)Q Q—H® (X, Ro)
is an isomorphism.

ENProof. For any G-orbit G/H R; (G/H) is a divisible’ group and hence a WH-injective one. Because Rg is a split coefficient system then from 1.3. it follows thatOn

theEquivariant

Chern

Homomorphism011

Rs; is an injective object of Cg. Let fi, (X) denote the object of G; determined byh, (X) (G/H)=H, (X®). The injectivity of R¢ yields formulas (see [1], p. I-22).

H" (X, Rg) =Homg, (kh, (X), RG)= [|] Hom (H,(X*),*:

G/H€Dg= [1] B"@*, Q)®R(H).

!

WeRID® Q)=

G|HeOg:define L; and L, as the maps

[1 re o9r@MG/HeDg[]

H&E QRH)

f:G/H;—~GH,\ing

satisfying the conditions
S;

whereLy=(H"(f)®1id)

Sem,

»Sy

Sy and Sey are structural morphisms

L,=(d®R))

Sey,»of products.

It is easy to check that Homg_ (4, (X), Rs) is isomorphic to ker (L,—L,). Letgy (X) be the restriction|Kg (X)— Kg (X)—and letKg (X)|gX)= [] a:
G/H egX)~ [] Ka @®=3[] K@HQREH).

G/HeDgG/HeOgIf we let ch denote the ordinary Chern homomorphism then we can consider thecompositionache X)= [] (ch (XH®id) ¢(X): Ke X— [|] H*” (X%, Q)@ R(H):G/H egThe

G/H eginclusion im hg X<ker (IL, —L,)

followsfromthecommutativityof thediagrams,CH\Kig(Hy X")—m (xP)
Kg (X):

Ky (Hy XH)

anHr

———

,Kuy(X1)

ANed
Ku(x")andq

Kg (X)—— Ka (X)=K (X*) ® R (H)Jam
Ky (w),Kg (XB)———Kx (XX?)|EKw®id|\[1d QR (w)

K(X") @R(H)——————KX) QR (H)

.

X

912

J. Stominska

OO

whenever Hc H; are subgroups of G and w is an element of WH.

This

we have well-defined the equivariant Chern homomorphism

One can easilychs.

yields thatverify that for any orbit G/Hche (G/H) ®id: Ks (°/H) ® Q—H*(G/H, Rg)

is an isomorphism. Using the spectral sequence of the Atiyah—Hirzebruch type(see Matumoto [3]) we obtain the statement of the proposition.
Now, let Of be the full subcategory of Og consisting of all orbits G/H, suchthat H 1s a cyclic group.
2.2. COROLLARY. If X is a compact G-CW complex then Kg (X)QQ is isomorphic
to the direct sumo Do K(X"KX)
Furthermore,

if G is an abelian group© ReRs;(G/H).
(G/H)

then Kg (X) Q Q is isomorphicto®.
® KX,/HeOG (JH)
where¢@ denotes

the Euler function.

Proof. If Pis a G-module then H(X, P) denotes the cohomology of the cochain’complex Homg (C, (X), P). From Lemma(x,H*(X,1.3 it follows that(X¥,M(G/H)),M)=_ M)= BD HH,Hi (X7,M (G]H)

whenever M is a split coefficient system (see formulas 2.3 and 9.4, p. I-21 in [1]).
If H is a noncyclic subgroup of G then Rg (G/H) is a trivial group. From proposition XIL.2.5. in"[5] it follows thatHi (X7, Ro (G/HD)
1s isomorphic toH"(X",0)since Rg (G/H) is a-Q (WH)
Hom, (WH) (Zz (WH[F),©

Q (WH)

Rs(G/H)projective module andRg (G/H))=R¢’(G/H)"==Z Z(F)X) Re (G/H)=Z(WH|F)X

Z (WH)Rs (G/H)

whenever F is a subgroup of WH. Now it is sufficient to use Proposition 2.1.
INSTITUTE OF MATHEMATICS, NICHOLAS COPERNICUS UNIVERSITY, CHOPINA. 12-18,
87-100 TORUN’@NSTYTUT MATEMATYKI, UNIWERSYTET M. KOPERNIKA, TORUK)REFERENCES[1] G. E. Bredon, Equivariant cohomology theories, Springer Verlag, 1967.
[2] A. W.M., Dress, Contribution to the theory of induced representations,in Mathematic, 343, Springer Verlag, 1973.

LectureNotes

On the Equivariant Chern Homomorphism.

913[3] T. Matumoto, Equivariant cohomology theories on G-CW complexes, Osaka Journalof Mathematics, 10 (1973), 51—68.
[4] J. P. Serre, Représentations linéares des groupes finis, Collection Méthodes, Hermann,Paris 1967.
[5]H. Cartan, S. Eilenberg, Homological algebra, Princeton, 1956.

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