EQUIVARIANT K-THEORY AND EQUIVARIANT
COHOMOLOGY
By Ioanid Rosu
with an appendix by Allen Knutson and Ioanid Rosu
Abstract. For T an abelian compact Lie group, we give a description of T*
* -
equivariant K-theory with complex coefficients in terms of equivariant c*
*ohomol-
ogy. In the appendix we give applications of this by extending results o*
*f Chang-
Skjelbred and Goresky-Kottwitz-MacPherson from equivariant cohomology to
equivariant K-theory.
1. Introduction
Let T be a abelian compact Lie group, not necessarily connected. Let X be a
compact T -equivariant manifold, or more generally a finite T -CW complex. We
denote by H*T(X) the T -equivariant (Borel) cohomology of X, as described in
Atiyah and Bott [1], and by K*T(X) the T -equivariant K-theory of X, as describ*
*ed
in Segal [12]. All the cohomology theories in this paper have complex coefficie*
*nts,
unless otherwise noted. For example, K*T(X) = K*T(X; Z) Z C. Also, when X is
a point, we will write K*Tinstead of K*T(X), and similarly for H*T.
There were previous attempts to give a de Rham type of model for K*T(X).
The earliest version appeared in Baum, Brylinski and MacPherson [3]. The ideas
were further developed in Block and Getzler [4], and Duflo and Vergne [7]. The
present paper is inspired by Grojnowski's preprint [9], where he uses ideas fro*
*m the
papers mentioned above to define equivariant elliptic cohomology with complex
coefficients. This is a coherent analytic sheaf over a complex variety ET const*
*ructed
using an elliptic curve E. The stalk of this sheaf is defined in terms of equiv*
*ariant
cohomology. We therefore decided to see if a similar definition, adapted to K-
theory, can indeed produce K*T(X).
We will define a sheaf K*T(X) over the complex algebraic group CT = Spec K*T.
Denote by O the sheaf of algebraic functions on CT, and by Oh the sheaf of holo-
morphic functions. If ff is a point of CT, we will associate to ff a subgroup H*
*(ff)
of T . Denote by Xff= XH(ff), the subspace of X fixed by all elements of H(ff).
The stalk of K*T(X) at ff is then
K*T(X)ff= H *T(Xff) ,
where H *T(-) is an extension of the Borel equivariant cohomology H*T(-) by the
ring of germs at zero of holomorphic functions on tC, the complexification of t*
*he
Lie algebra of T . Now, if denotes the global sections functor, we will show t*
*hat
there exists an isomorphism
T : K*T(X) O Oh ~=K*T(X) .
1
2 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
This is the sense in which equivariant K-theory can be described in terms of
equivariant cohomology.
The map T essentially sheafifies the equivariant Chern character chT(V ) =
ch(V xT ET ), where V ! X is a complex equivariant T -vector bundle. However,
chT has to be twisted (translated) in an appropriate sense, to take into accoun*
*t the
point ff over which the germ of chT is taken. Notice that chT : K*T(X) ! H*T(X)
is not an isomorphism, unless T is a point, so our results show the sense in wh*
*ich
the equivariant Chern character becomes an isomorphism.
We should mention that both the idea of describing equivariant K-theory as
a sheaf and twisting the equivariant Chern character are present in Duflo and
Vergne [7]. The problem in their paper is that they cannot prove Mayer-Vietoris
for their cohomology theory, because they work with C1 functions. The advantage
of our approach is that we work with coherent holomorphic sheaves over the affi*
*ne
(Stein) manifold CT, and in this case the global section functor is exact.
The results of this paper can be extended in the several directions. Firstly,*
* we
can describe K*T(X) directly as an algebraic module instead of its holomorphic
extension K*T(X) O Oh. But in order to do that, one needs to define algebraic
sheaves Fff(see Definition 2.9) instead of holomorphic ones. And this can only
be done using completions, because the logarithm map is not algebraic. The
constructions therefore become more complicated, and we decided to relegate them
to another paper. Secondly, if G is a nonabelian connected group, then K*G(-) =
K*T(-)W , where W is the Weyl group, so one can also describe K*G(-) using
Borel equivariant cohomology. Thirdly, we can prove a similar result whenever
the coefficient ring R of the cohomology theories involved is an algebra over Q
adjoined the roots of unity. We need R to be a Q-algebra because the logarithm
map is only defined over Q, and we need to invert the roots of unity because we
want to split (delocalize) R[z]= into a direct sum of n copies of R.
While the details of the sheafifying process are somewhat technical, in princ*
*iple
the construction allows one to infer some results in equivariant K-theory from
the corresponding ones in equivariant cohomology. In the Appendix we give an
example of this, extending results of Chang-Skjelbred[6] and Goresky-Kottwitz-
MacPherson [8] from equivariant cohomology to equivariant K-theory.
2. Construction of the Sheaf
The purpose of this section is to define a sheaf valued T -equivariant cohomo*
*logy
theory, which we denote by K*T(-). This is a coherent analytic sheaf of algebras
over the algebraic variety CT = Spec K*T.
First, we want a simpler description of CT. Denote by ^Tthe Pontrjagin dual of
T , ^T= Hom (T; S1). For example, if T = (S1)p x G, where G is a finite abelian
group, then ^T= Zpx G. Denote by Cx the multiplicative algebraic group C \ {0}.
Proposition 2.1. There is an isomorphism of algebraic varieties
CT ~=Hom Z(T^; Cx ) .
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 3
If T is connected (i.e. a torus), and T is its integer lattice, then
CT ~=T Z Cx .
Proof.First, notice that K*T= R(T ) Z C, where R(T ) is the representation ring
of T . But we know that R(T ) = Z[T^], the group ring of T^, so we obtain that
K*T= C[T^], the group C-algebra of ^T. Define a map Hom Z(T^; Cx ) ! Spec C[T^]*
* =
CT as follows: Let ff : T^ ! Cx be a group homomorphism. By the universal
property of the group algebra, since Cx is the group of units of C, ff extends *
*to
a C-algebra map ff : C[T^] ! C. This is clearly not the zero map, so kerff is a
maximal ideal of C[T^], hence it belongs to Spec C[T^]. Choose an identificati*
*on
T ~= (S1)p x Zn1 x . .x.Znq. One can now check directly that the above map is
an isomorphism.
For the second description, recall that T is by definition the kernel of the
surjective map t ! T , where t is the Lie algebra of T . Now observe that, beca*
*use
T is connected, we can identify ^T~=*T. To see this, let O 2 ^T, i.e. O : T ! S*
*1 is a
continuous group homomorphism. The differential dO : t ! R satisfies dO(T)
Z. Then the restriction dO to T yields a map dO : T ! Z, i.e. an element of
*T. By choosing an identification T ~=(S1)p, it is easy to see that ^T~=*T. So *
*we
__
get CT = Hom Z(T^; Cx ) = Hom Z(*T; Cx ) = (*T)* Z Cx = T Z Cx . |__|
Suppose T ~=(S1)p x G, where G is a finite abelian group. Then one can apply
the previous proposition to obtain that CT ~=(Cx )px G. This formula shows that
CT is in fact an algebraic group (and it is the complexification of T ). In the*
* rest of
the paper, although it might generate some confusion, we are going to use addit*
*ive
notation for CT.
Let X be a finite T -CW complex. In order to define the sheaf K*T(X), we will
choose an appropriate open cover of U = (Uff)ff2CTof CT, indexed by the points *
*of
CT. On each Uffwe are going to define a coherent analytic sheaf of algebras Fff,
and on each nonempty intersection Uff\ Ufiwe define gluing maps OEfffi. The she*
*af
obtained by gluing the Fff's is denoted by K*T(X).
Let ff 2 CT. The construction of CT is functorial, so if H is any subgroup of*
* T ,
we get an inclusion map CH ! CT. If ff 2 Im(CH ! CT), we say that ff 2 CH .
Definition 2.2. Let ff 2 CT. Then denote by H(ff) the smallest subgroup H of T
such that ff 2 CH . If K is a subgroup of T and X is a space with a T -action, *
*denote
by XK X the subspace of points fixed by K. For ff 2 CT define Xff= XH(ff).
The fact that there exists a smallest subgroup H such that ff 2 CH is implied
by the formula CK \ CL = CK\L (which follows from an easy diagram chase). This
also implies that H(ff) is the intersection of all subgroups H such that ff 2 C*
*H .
Another immediate consequence is the following proposition which will be useful
later.
Proposition 2.3. H(ff) K () ff 2 CK .
In order to define the sheaf Fff, we also need an algebra map h : H*T! OhCT;0,
where OhCT;0denotes the ring of germs of holomorphic maps on CT at zero.
4 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
0
Denote by T the connected component of T which contains the identity. Then
we can identify OhCT;0= OhCT0;0. Also, let t*Cbe the complexification of the d*
*ual
of the Lie algebra of T . If W is a complex vector space, denote by S(W ) its
symmetric algebra. Then we can identify (see Atiyah and Bott [1]) H*T= S(t*C).
But T and T 0have the same Lie algebra t, so H*T= H*T0. Therefore, in order to
construct the map h : H*T! OhCT;0, we may assume that T = T 0, i.e. that T is
connected.
Now, if T denotes the integer lattice of T , Proposition 2.1 implies that we *
*can
write CT = T Z Cx . But tC = T Z C, so we have a local isomorphism between
tC and CT via the logarithm map 1 log. This gives an isomorphism of the local
rings OhtC;0~=OhCT;0, so it is enough to define a function h : H*T! OhtC;0. For
this, consider p 2 H*T= S(t*C). This can be thought of as a polynomial (global
algebraic function) on tC, hence as a global holomorphic function. Then define
h(p) as the germ of p at zero. We can summarize the above discussion in the
following definition.
Definition 2.4. h is the composite algebra homomorphism
H*T! OhtC;0! OhCT;0,
where the first map is taking the germ of a polynomial at zero and the second o*
*ne
comes from the logarithm map log : C ! Cx . We call a neighborhood U of zero
in CT small if for any p 2 H*Tthe germ h(p) extends to a holomorphic function
on U.
If U is a subset of CT, denote by U + ff = {x + ff | x 2 U} the translation o*
*f U
by ff.
Definition 2.5. Let A be a finite collection of subgroups of T . Define a relat*
*ion
on CT as follows: ff fi if, for any H 2 A, fi 2 CH implies ff 2 CH . The relat*
*ion
is reflexive and transitive, but not antisymmetric, so it is not an order rela*
*tion.
A cover U = (Uff)ff2CTindexed by all the points of CT is called adapted with
respect to the collection A
1. ff 2 Uff, and Uff- ff is small.
2. If Uff\ Ufi6= ;, then either ff fi or fi ff.
3. If Uff\ Ufi6= ;, and both ff and fi belong to CH for some H 2 A, then ff a*
*nd
fi belong to the same connected component of CH .
Proposition 2.6. For any finite collection A of subgroups of T there exists a
cover U of CT adapted to A. Any refinement of U is still adapted.
Proof.Define H = {CH | H 2 A}, and H0 = the set of all connected components
of the elements in H. Put a metric on CT which yields its usual topology. Denote
this metric by "dist".
Let ff 2 CT. If ff 2 C for all C 2 H0 (this is possible only when T is connec*
*ted),
then choose Uffsuch that ff 2 Uff, and Uff- ff is small. If, on the contrary, t*
*here
exists a connected component C 2 H0 such that ff =2C, then take Uffa ball of
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 5
center ff and radius d, with
1
d < __ min dist(ff; D) ,
2 D2H0;ff=2D
and such that Uff- ff is small.
We show that U = (Uff)ff2CTis adapted. Condition 1 in Definition 2.5 is trivi*
*ally
satifsfied. To prove Condition 2, let ff and fi be such that Uff\ Ufi6= ;. Supp*
*ose
we have neither ff fi, nor fi ff. Then by the definition of , there exist
two subgroups K and L of T such that ff 2 CK \ CL and fi 2 CL \ CK . But,
from the definition of Uffit follows that Uffis a ball of center ff and radius *
*d <
1_ 1_
21dist(ff; CL) 2dist(ff; fi). Similarly, Ufiis a ball of center fi and radius *
*less than
_ dist(ff; fi), so U and U cannot possibly intersect, contradiction.
2 ff fi
Finally, to show Condition 3, let ff; fi 2 CH be such that Uff\ Ufi6= ;. Supp*
*ose
ff and fi belong to different connected components of CH . Then by the same
type of reasoning as above, it follows that the radii of Uffand Ufiare smaller
than 1_2dist(ff; fi), so Uffand Uficannot possibly intersect, which again leads*
*_to a
contradiction. |__|
Now, let X be a finite T -CW complex. Then there exists a finite collection
A = (Hi)i of subgroups of T such that X has a equivariant cell decomposition of
the form X
X = Dnix T=Hi .
i
This is a disjoint union indexed by some finite set of i's, such that the subgr*
*oups
Hi might appear repeatedly. We denoted by Dn the open disk in dimension n; D0
is a point. T acts trivially on Dni, and by left multiplication on T=Hi.
Proposition 2.7. If K is a subgroup of T , then the subcomplex of X fixed by K
has a decomposition of the form
X
XK = Dnix T=Hi .
i:KHi
Proof.This follows from the fact that (T=H)K = T=H if K H, and (T=H)K = ; __
otherwise. |__|
By Proposition 2.6, there exists a cover U = (Uff)ff2CTof CT which is adapted
to the A.
Definition 2.8. We say that the cover U = (Uff)ff2CTof CT is adapted to a finite
T -CW complex X if U is adapted to the collection A of the isotropy groups Hi
appearing in a T -equivariant cell decomposition of X.
Fix U a cover adapted to X. For each ff 2 CT we are going to construct a
coherent holomorphic sheaf of algebras Fffover Uff.
Definition 2.9. For U Uffwe define
Fff(U) = H*T(X) H*TOhCT(U - ff) .
6 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
* h
The map HT ! OCT(U -ff) is the composition of the map h given by Definition 2.4
with the map OhCT;0! OhCT(U - ff). This last map is just extension of germs to
holomorphic functions on U - ff, which makes sense since U - ff is small. The
restriction maps of Fffare defined to be induced from those of the sheaf OhCT.
First, we notice that we can make Fffinto a sheaf of OhCT |Uff-modules: if U *
* Uff,
we want an action of f 2 OhCT(U) on Fff(U). The translation map tff: U - ff ! U,
which takes x to x + ff gives a map t*ff: OhCT(U) ! OhCT(U - ff), which sends
f(u) to f(u + ff). Then we take the result of the action of f 2 OhCT(U) on
g 2 Fff(U) = H*T(Xff) H*TOhCT(U - ff) to be (t*fff . g). Fffis coherent,
because H*T(Xff) is a finitely generated H*T-module.
Next, we want to glue the different sheaves Fffto obtain a sheaf K*T(X) on CT.
For this we have to define gluing maps, i.e. for every nonempty intersection Uf*
*f\Ufi
we have to define isomorphisms of sheaves OEfffi: Fff|Uff\Ufi! Ffi|Uff\Ufisatis*
*fying
the cocycle condition OEfiflO OEfffi= OEffflon every nonempty triple intersecti*
*on Uff\
Ufi\ Ufl.
Let ff; fi 2 CT such that Uff\Ufi6= ;. Since the cover U is adapted to A, it *
*follows
from Condition 2 of Definition 2.5 that either ff fi or fi ff. Without loss of
generality we can assume that fi ff. We use Proposition 2.3 (with K = H(ff))
to rephrase Proposition 2.7:
Proposition 2.10. X
Xff= Dnix T=Hi .
i:ff2CHi
In particular, if fi ff, then Xfi Xff. Denote by i : Xfi! Xffthe inclusion
map. Now we want to investigate conditions in which the restriction map i* :
H*T(Xff) ! H*T(Xfi) becomes an isomorphism. Let U Uff\ Ufi. The next
proposition is esentially Localization Theorem of Borel and Hsiang (see Atiyah
and Bott [1]).
Proposition 2.11. The restriction map
i* 1 : H*T(Xff) H*TOhCT(U - ff) ! H*T(Xfi) H*TOhCT(U - ff)
is an isomorphism.
Proof.First, notice that Proposition 2.10 implies that Xffis built from Xfiby
adding cells of the form Dn x T=L, with ff 2 CL and fi =2CL. So it is enough to
show that, when tensored with OhCT(U - ff) over H*T, H*T(Dn x T=L; Sn-1 x T=L)
becomes zero. Or, equivalently, it is enough to have H*T(T=L) = H*Lbecome zero.
So it is enough to prove that H*LH*TOhCT(U - ff) is the zero algebra.
ff 2 CL and fi =2CL imply that fi - ff 2= CL0, where L0 is the connected
component of L which contains the identity. Hence fi =2ff + CL0, and notice that
ff + CL0 is a connected component of CL, so by the definition of an adapted cov*
*er
Ufi\ff+CL0 = ;, i.e. Ufi-ff\CL0 = ;. But U Uff\Ufi Ufi, so U -ff\CL0 = ;.
Now, via the correspondence log : CT -~! tC, it is enough to show that, if lC
is the complexification of the Lie algebra of L, and if V is an open set in tC *
*such
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 7
* h
that V \ lC = ;, then HL H*TOtC(V ) = 0. Since this generates a sheaf if we let
V vary, it is enough to consider the stalks and show that for every x =2lC
H*LH*TOhtC;x= 0 .
Denote by IL the kernel of the surjective map of algebras H*T! H*L. It is
clear that IL kills H*Las an H*T-module. lC is the vanishing set of IL, so sin*
*ce
x =2lC, there exists a polynomial p in IL such that p(x) 6= 0. This means that p
is invertible in OhtC;x, hence 1 p is invertible in H*LH*TOhtC;x. By the balan*
*cing
property of the tensor product, 1 p = 0, because p goes to zero via the map
H*T! H*L. So we found an element in H*LH*TOhtC;xwhich is invertible and zero
__
at the same time. This can only happen if H*LH*TOhtC;xis the zero algebra. |_*
*_|
Let us now consider the translation map tfi-ff: U - fi ! U - ff. This induces
a map t*fi-ff: OhCT(U - ff) ! OhCT(U - fi). Eventually we want to produce a
translation map H*T(Xfi) H*TOhCT(U - ff) ! H*T(Xfi) H*TOhCT(U - fi). But the
problem is that t*fi-ffis not a map of H*T-modules, so we cannot simply tensor *
*it
with the identity map of H*T(Xfi). However, we hope to find some other group K
so that t*fi-ffis a map of H*K-modules.
Let K = T=H, with H = H(fi) as in Definition 2.2. The quotient map T ! K
induces a map of algebras H*K! H*T, so any module over H*Tcan be thought as
a module over H*K.
Proposition 2.12. The translation map t*fi-ff: OhCT(U - ff) ! OhCT(U - fi) is a
map of H*K-modules.
Proof.Recall that Uff\ Ufi6= ;. We assume as before that fi ff. Since fi 2 CH ,
it follows that ff 2 CH . Because both ff and fi belong to CH , Condition 3 of
Definition 2.5 implies then that ff and fi belong to the same connected compone*
*nt
of CH , or equivalently that fi - ff 2 CH0. Let hC be the complexification of *
*the
Lie algebra of H. The exponential map ss : hC ! CH0 is a surjective group
homomorphism, because H0 is a torus. Consider the element ss-1(fi - ff) 2 hC.
Denote it also by fi - ff. Let be the global sections functor. To prove the
proposition, it is enough to show that t*fi-ff: OhtC! OhtCis a map of H*K-
modules. But this is equivalent to showing that t*fi-ff(p) = p for all p 2 H*K.
Now, H*K = S(k*C), and since t*fi-ffis a ring map, it is enough to show that
t*fi-ff(p) = p for all p 2 k*C. Let p 2 k*C. Since K = T=H, it follows that p(a*
*) = 0
for all a 2 hC. But fi - ff 2 hC, so p(fi - ff) = 0. Let x 2 kC. Then t*fi-ff(p*
*)(x) =
p(x + fi - ff) = p(x) + p(fi - ff) = p(x), so indeed t*fi-ff(p) = p. This compl*
*etes_the
proof. |__|
Fact 2.13. Let Y be a finite T -CW complex which is fixed by a subgroup H of
T . Let K = T=H. Then there exists a natural isomorphism
H*T(Y ) ~=H*K(Y ) H*KH*T .
The proof of this is by induction over the T -equivariant cells of Y . (The c*
*ells
are all of the form Dn x T=L, with H L.)
8 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
Corollary 2.14. There exists a natural isomorphism
o*fi-ff: H*T(Xfi) H*TOhCT(U - ff) ! H*T(Xfi) H*TOhCT(U - fi) .
Proof.Xfi is fixed by H = H(fi), so Fact 2.13 implies that X*T(Xfi) *
*H*T
OhCT(U - ff) ~= X*K(Xfi) H*K OhCT(U - ff), where K = T=H. So define
o*fi-ff: H*T(Xfi) H*TOhCT(U - ff) ! H*T(Xfi) H*TOhCT(U - fi) as the compo-
sition of the maps X*T(Xfi) H*TOhCT(U - ff) ! X*K(Xfi) H*K OhCT(U - ff) !
X*K(Xfi)H*KOhCT(U -fi) ! H*T(Xfi)H*TOhCT(U -fi). The middle map is 1t*fi-ff,
which is well defined, because t*fi-ffwas shown in Proposition 2.12 to be a_map*
*_of
H*K-modules. |__|
Definition 2.15. Let U Uff\Ufi. Define OEfffi: Fff(U) ! Ffi(U) as the composite
*1 o*fi-ff
map H*T(Xff) H*TOhCT(U - ff) i-! H*T(Xfi) H*TOhCT(U - ff) -! H*T(Xfi) H*T
OhCT(U - fi). The first map is an isomorphism by Proposition 2.11, and it is ea*
*sy
to see that the second map is also an isomorphism.
Proposition 2.16. The maps OEfffiverify the cocycle condition, i.e. OEfiflO OE*
*fffi=
OEfffl, on every triple intersection Uff\ Ufi\ Ufl6= ;.
Proof.Without loss of generality, we can assume fl fi ff. We need to show
that the following diagram is commutative:
H*T(Xff) H*TOhCT(U - ff)
i*1
H*T(Xfi) H*TOhCT(U - ff)
*1
o*fi-ff i
H*T(Xfi) H*TOhCT(U - fi) H*T(Xfl) H*TOhCT(U - ff)
i*1 o*
fi-ff
H*T(Xfl) H*TOhCT(U - fi)
o*fl-fi
H*T(Xfl) H*TOhCT(U - fl)
In this diagram the only map which has not been defined is o*fi-ff: H*T(Xfl) H*T
OhCT(U - ff) ! H*T(Xfl) H*TOhCT(U - fi). But, since fl fi, it follows that H(f*
*i)
fixes Xfl, so we can replace Xfiby Xflin Corollary 2.14 and obtain a map which
we also denote by o*fi-ff.
Since H = H(fi) fixes Xfiand Xfl, we can use Fact 2.13 to replace for example
H*T(Xfi) H*TOhCT(U - fi) by H*K(Xfi) H*KOhCT(U - fi), where K = T=H. We have
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 9
to show that the following diagram is commutative:
H*K(Xfi) H*KOhCT(U - ff)
*1
1t*fi-ff i
H*K(Xfi) H*KOhCT(U - fi) H*K(Xfl) H*KOhCT(U - ff)
i*1 1t*
fi-ff
H*K(Xfl) H*KOhCT(U - fi)
*
* __
Now observe that both composites are equal to i* t*fi-ff. *
*|__|
Definition 2.17. Let X be a finite T -CW complex. Consider a cover U =
(Uff)ff2CTadapted to X, as in Definition 2.8. Then denote by K*T(X) the sheaf
obtained by gluing the sheaves Ffffrom Definition 2.9 via the gluing maps OEfff*
*ifrom
Definition 2.15.
Remark 2.18. Because the refinement of an adapted cover is adapted, the defi-
nition of K*T(X) does not depend on U, all such sheaves being canonically isomo*
*r-
phic.
Theorem 2.19. K*T(-) is a T -equivariant cohomology theory with values in the
category of coherent holomorphic sheaves of Z-graded OhCT-algebras.
Proof.First, let us explain the grading of K*T(X). The sheaves Fffwere obtained
by tensoring H*T(Xff), which is Z-graded, with a ring of holomorphic functions *
*over
the ring of coefficients H*T. This last ring is concentrated in even dimensions*
*, so
the resulting tensor product only has an odd part and an even part, i.e. it is *
*Z=2-
graded. We can therefore make it into a Z-graded sheaf by making it 2-periodic,
as in the case of K-theory.
Now, we show that the construction of K*T(X) is natural. Let X and Y be
two finite T -CW complexes, and f : X ! Y a T -equivariant map between them.
We want to define a map of sheaves f* : K*T(Y ) ! K*T(X) with the properties
that 1*X = 1K*T(X)and (fg)* = g*f*. Consider two T -cell decompositions of X
and Y , and let A be the collection of all subgroups L of T such that Dn x T=L
appears in the cell decomposition of either X or Y . Let U be an open cover
adapted to A. Since f is T -equivariant, for each ff we get by restriction a m*
*ap
fff: Xff! Y ff. This induces for all U Uffa map f*ff 1 : H*T(Y ff) H*TOhCT(U -
ff) ! H*T(Xff) H*TOhCT(U - ff), which commutes with restrictions. Therefore, we
obtain a sheaf map f*ff: Fff(Y ) ! Fff(X). To get our global map f*, we only ha*
*ve
to check that the maps f*ffglue well, i.e. that they commute with the gluing ma*
*ps
OEfffi. This follows easily from the naturality of ordinary equivariant cohomol*
*ogy,
and from the naturality in X of the isomorphism H*T(Xfi) ~= H*K(Xfi) H*KH*T.
Here Xfiis fixed by H = H(fi), and K = T=H (see Fact 2.13).
Let (X; A) be a pair of finite T -CW complexes, i.e. A is a closed subspace of
X and the inclusion map A ! X is T -equivariant. We then define K*T(X; A) as
the kernel of the map j* : K*T(X=A) ! K*T(point), where j : point = A=A ! X=A
10 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
is the inclusion map. If f : (X; A) ! (Y; B) is a map of pairs of finite T -CW
complexes, then f* : K*T(Y; B) ! K*T(X; A) is defined as the unique map induced
on the corresponding kernels from f* : K*T(Y ) ! K*T(X).
The last definition we need is of the coboundary map. If (X; A) is a pair of *
*finite
T -CW complexes, we want to define ffi : K*T(A) ! K*+1T(X; A). This is obtained
by gluing the maps
ffiff 1 : H*T(Aff) H*TOhCT(U - ff) ! H*+1T(Xff; Aff) H*TOhCT(U - ff) ,
where ffiff: H*T(Aff) ! H*+1T(Xff; Aff) is the usual coboundary map. The maps
ffiff 1 glue well, because ffiffis natural.
To check the usual axioms of a cohomology theory (naturality, exact sequence *
*of
a pair, and excision) for K*T(-), recall that it was obtained by gluing the she*
*aves
Fffalong the maps OEfffi. Since the sheaves Fffwere defined using H*T(Xff), the
properties of ordinary T -equivariant cohomology pass on to K*T(-), as long as
tensoring with OhCT(U - ff) over H*Tpreserves exactness. But this is implied_by
Proposition 3.3. |__|
Now, CT = Spec K*Tis a nonsingular affine complex variety, so it is Stein. By*
* a
generalization of Theorem B of Cartan, the sheaf cohomology vanishes on CT in
positive dimensions for any coherent sheaf. (See Gunning and Rossi [10]). This
implies that , the global sections functor, is exact, so we get the following
Corollary 2.20. K*T(-) is an T -equivariant cohomology theory with values in
the category of Z=2-graded algebras.
We want to investigate a few more useful properties of K*T(-).
Proposition 2.21. When X is a point, write K*T(X) = K*T. Then K*T~=OhCT.
Proof.Notice that if X is a point, translation by ff induces an isomorphism of *
*the
corresponding sheaf Fffon Uffto OhCT |Uff. Via this isomorphism, the gluing map
*
* __
OEfffiis the usual restriction of holomorphic functions. Therefore K*T~=OhCT. *
* |__|
Proposition 2.22. If X is a finite T -CW complex and ff 2 CT , then the stalk
of the sheaf K*T(X) at ff is
K*T(X)ff= H*T(Xff) H*TOhtC;0,
where OhtC;0is the ring of germs at zero of holomorphic functions on the comple*
*x-
ification of the Lie algebra of T .
Proof.The stalk of K*T(X) at ff is the same as the stalk of Fffat ff. But the s*
*talk
of Fffat ff is H*T(X) H*TOhCT;0. Now use the isomorphism OhCT;0~=OhtC;0given_by
the logarithm map. |__|
Proposition 2.23. Let A and B are compact abelian Lie groups, X is a finite
A-CW complex, and Y is a finite B-CW complex. Then
K*AxB(X x Y ) ~=K*A(X) C K*B(Y ) .
Proof.This follows from CAxB ~=CA x CB and from H*AxB(X x Y ) ~=H*A(X) C __
H*B(Y ). |__|
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 11
3. The Isomorphism and the Equivariant Chern Character
Let X be a finite T -CW complex. In this section we construct a natural iso-
morphism
T : K*T(X) K*TK*T! K*T(X) .
Here denotes the global section functor. OhCTis the ring of global sections of
OhCT, i.e. the ring of global holomorphic functions on CT.
Let us first construct a natural ring map
CHT : K0T(X; Z) ! K*T(X) .
Let E ! X be a complex T -vector bundle. Then one can form the Borel con-
struction E xT ET , where ET ! BT is the universal principal T -bundle over the
BT , the classifying space of T . Denote ET = E xT ET . ET is a complex vector
bundle over BT . Its ordinary Chern character chET belongs to the completion of
H*(X x BT ) = H*T(X) with respect to its augmentation ideal of H*T. Denote this
completion by H**T(X). We have thus obtained a ring map K*T(X; Z) ! H**T(X),
which takes E to chET. Denote this map by chT. Similarly, denote the n-th Chern
class of ET by cn(E)T. This belongs to the noncompleted ring H*T(X).
Proposition 3.1. If X is a finite T -CW complex, then H*T(X) is a finite H*T-
module, and H**T(X) ~=H*T(X) H*TH**T.
Proof.If we choose an identification T ~= (S1)p x G, where G is a finite abelian
group, it follows that H*T~= C[u1; : :;:up]. It is a standard result of commut*
*a-
tive algebra (see for example Section 2 of Matsumura[11]) that H*Tis noetherian.
Induction by the number of T -cells of X and the Mayer-Vietoris sequence show
that H*T(X) is a finite H*T-module. Using the above facts and Theorem 55 of [11*
*],_
it follows that H**T(X) ~=H*T(X) H*TH**T. |__|
Definition 3.2. If X is a finite T -CW complex, we define the holomorphic T -
equivariant cohomology of X to be
H *T(X) = H*T(X) H*TOhtC;0.
Notice that we can now rephrase Proposition 2.22 to imply that the stalk of
K*T(X) at ff is K*T(X)ff= H *T(Xff).
Proposition 3.3. H *Tand H**Tare flat over H*T.
Proof.H**Tis the completion of H*Twith respect to the augmentation ideal I =
ker(H*T! C), so by Corollary 1 of Theorem 55 in Matsumura [11] H**Tis flat over
H*T. The completion of H *Twith respect to its augmentation ideal is also H**T,*
* and
the map H *T! H**Tis local, so by Theorems 2 and 55 of [11], H**Tis faithfully *
*flat
over H *T. But H*T! H**Tflat and H *T! H**Tfaithfully flat imply that H*T! H_*T_
flat. |__|
As a corollary of this, the map H *T(X) ,! H**T(X) is injective.
Lemma 3.4. If E ! X be a complex T -vector bundle over a finite T -CW com-
plex, then chT(E), which a priori is an element of H**T(X) actually lies in H **
*T(X).
12 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
Proof.First, assume that E is a line bundle. Then chT(E) = ec1(E)T(by definition
of the Chern character). We know that c1(E)T 2 H*T(X). Proposition 3.1 says
that H*T(X) is a finite module over H*T. Let a1; : :;:am be a set of generators*
* for
H*T(X). Define fkij2 H*Tby
Xm
ai. aj = fkijak 8i; j; k 2 1; : :;:m .
k=1
Denote by c = c1(E)T. c is in H*T(X), so there exist gi2 H*Tsuch that
c = g1a1 + : :g:mam .
P P P P *
* p
We can also calculate c2 = k( i;jgigjfkij)ak, c3 = p( i;j;k;lgigjglfkij*
*fkl)ap, etc.
Choose some coordinates on tC such that H*T= C[u1; : :;:up]. We say that a
polynomial OE 2 C[u1; : :;:up] is dominated by another polynomial if all coef*
*fi-
cients of are positive and all the coefficients of OE are less in absolute va*
*lue than
the coefficients of . Also, if ; 2 H*T(X), then we can write = 1a1+: :+:m am
and = 1a1+ : :+:m am , with i; i2 H*T. We then say that is dominated by
if for all i i is dominated by i. Similar definitions of domination can be made
for the power series ring H**Tand for H**T(X).
Let 2 H*T= C[u1; : :;:up] be a polynomial such that for all i; j; k fkijand *
*gi
are dominated by . Then one can show by induction that cn is dominated by
(2n-2)2n-1(a1+: :+:amP), which in turn is dominated by 2n2n-1(a1+:P:+:am ). So
ec is dominated by 1+ n1 __2__(n-1)!2n-1(a1+: :+:am ) = 1+ n0 2_n!2n+1(a1+:*
* :+:
am ) = 1+2e2 (a1+: :+:am ). But this last element belongs to H*T(X)H*TOhtC;0=
H *T(X), so ec also belongs to H *T(X).
Second, assume E is a general complex T -vector bundle of rank n. Let SE be
the splitting space of E (or the flag variety of E _ see Bott and Tu [5]). SE c*
*omes
with an equivariant map ss : SE ! X. Then the splitting principle (see [5]) says
that the pull-back bundle ss*(E) decomposes as a sum of T -line bundles over SE*
* .
Say ss*(E) ~=L1 . . .Ln. Calculation using the Leray-Serre spectral sequence
shows that
H*T(SE ) = H*T(X)[x1; : :;:xn] = ,
where oei(x1; : :;:xn) is the i'th symmentric polynomial in the xj's. The xj =
c1(Lj)T are called the Chern roots of E. Moreover, we can identify H*T(X) as the
subring of H*T(SE ) generated by the polynomials in H*T(X)[x1; : :;:xn] which a*
*re
symmetric in the xj's. By tensoring with OhtC;0or H**Tthe same statement is true
about H *T(X) and H**T(X).
Now, consider chT(E) = ex1.. ...exn. Since Lj is a line bundle and xj = c1(Lj*
*)T,
the first part of the proof implies that exj 2 H *T(SE ) for all j. Therefore c*
*hT(E) 2
H *T(SE ), and since it is symmetric in the xj's it follows that chT(E) 2 H_*T(*
*X),_
which is what we wanted. |__|
Next, we define a multiplicative natural map
CHT : K0T(X; Z) ! K*T(X) .
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 13
Let E ! X be a complex T -vector bundle. Let ff 2 CT, and denote by H = H(ff).
Then ff 2 CH . By Proposition 2.1, CH ~= Hom Z(H^; Cx ), so we can think of ff *
*as a
group map ff : ^H ! Cx . Xffhas a trivial action of H, so the restriction E|Xff*
*of
E to Xffhas a fiberwise decomposition by irreducible characters of H:
E|Xff~=2H^ E() ,
where E() is the T -vector bundle where h 2 H acts by complex multiplication
with (h).
It would be tempting to define the germ of CHT(E) at ff to be chT(E|Xff), but
these germs would not glue well to give a global section of K*T(X). Instead, we*
* do
the following:
Definition 3.5. The germ of CHT(E) at ff is defined to be
X
CHT(E)ff= ff()chTE() ,
2H^
where H = H(ff).
Proposition 3.6. The germs CHT(E)ff glue to a global section CHT(E) 2
K*T(X).
Proof.We notice that by Proposition 3.4, CHT(E)ff does indeed belong to
H *T(Xff), which by Proposition 2.22 is the stalk of K*T(X) at ff.
Let ff; fi 2 CT with Uff\ Ufi6= ; and fi ff. This implies ff; fi 2 CH(fi)
and also H(ff) H(fi). Condition 3 of Definition 2.5 implies that fi - ff 2
CH(fi)0.P Now we have to proveP that OEfffiCHT(E)ff = CHT(E)fi, i.e. t*
*hat
o*fi-ff2H[(ff)ff()chTE()|Xfi= 2H[(fi)fi()chTE(). Consider the surjective
map ss : H[(fi)! H[(ff)inducedPby the inclusion H(ff) ,! H(fi). If 2 H[(ff),
we have E()|Xfi= 2ss-1() E(). Therefore it is enough to show that for all
2 [H(fi)o*fi-ff(ff()chTE()) = fi()chTE(), where = ss(). But this is equiv-
alent to o*fi-ffchTE() = (fi - ff)()chTE(). Denote by fl = fi - ff 2 CH(fi)0.
So it is enough to show that, for all fl 2 CH(fi)0and for all 2*
* H(fi),
o*flchTE() = fl()chTE().
The map H(fi)0 ,! H(fi) induces a Pontrjagin dual map [H(fi)! "H(fi)0. De-
note by 0 the image of via this map. Then it is easy to see that fl() = fl(0).
Also, because chT is natural in T and because H*T = H*T0= S(t*C), we have
chTE() = chT0E(). Therefore it is enough to show that for T and H(fi) con-
nected
o*flchTE() = fl()chTE() .
for all fl 2 CH(fi)and 2 H(fi).
To make notation less cumbersome, denote by H = H(fi). Choose an identi-
fication T = (S1)p x (S1)q, so that it induces an identification H = (S1)p x 1.
We calculate H^ = Zp; CH = (Cx )p. fl 2 CH , so fl = (fl1; : :;:fln), with fli *
*2 Cx .
2 ^H, so = (1; : :;:p; : :):. Then fl() = fl11. .f.lpp 2 Cx .
14 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
0 0 1 q
Let L = T =H . With our identifications, L = 1 x (S ) and T = K x L. Then
for any Y fixed by H we have
K*T(Y ) ~=K*L(Y ) K*LK*T~=K*L(Y ) C K*H .
The first isomorphism is the same statement as 2.13, but for equivariant K-theo*
*ry
instead of equivariant cohomology. The second isomorphism is given by the iden-
tification T = K x L.
Consider Y = Xfi. Via the identification above, we have E() = E() V (),
where the first E() is regarded as a T -bundle, and the second one as an L-bund*
*le.
V () 2 K*His the irreducible representation of H given by . Now we have
chTE() = chLE() chH V () 2 H *L(Y ) C H *H.
By construction, o*flacts only in the second argument, as the translation 1 t**
*fl.
We have to show now that
t*flchH V () = fl()chH V () .
Recall that = (1; : :;:p; : :):, with i 2 Z. Also, fl = (fl1; : :;:fln), with *
*fli 2
Cx . We can write fli= ei , for some i2 C. Since H = (S1)p, H*H= C[u1; : :;:up].
Then chH V () = e1u1 . .e.pup. Therefore t*flchH V () = e1(u1+1) . .e.p(up+p)=
*
*__
fl11. .f.lppchH V () = fl()chH V (). This completes the proof. |*
*__|
We have just finished constructing a natural map CHT : K0T(X; Z) ! K*T(X).
By taking the suspension of X instead of X, this induces a map CHT*
* :
K*T(X; Z) ! K*T(X).
Proposition 3.7. CHT is a ring map.
Proof.It is easy to chech that CHT is additive. Let us check that it is multipl*
*ica-
tive. Let E and F be two complex TP-vector bundles overPX. Let ff 2 CT, and H =
H(ff).PThen CHT(E)ff. CHT(F )ff= 2H^Pff()chTE()P. 2H^ ff()chTF () =
P ;2H^ ff() . ff()chTE() F () = 2H^ ff() += chTE() F () = __
2H^ ff()chT(E F )() = CHT(E F )ff. |__|
Because K*T(X) is a C-algebra and CHT is a ring map, we can extend CHT
to a natural map of C-algebras
CHT : K*T(X) ! K*T(X) .
Definition 3.8. Let X be a finite T -CW complex, and let ss : X ! point be the
map collapsing X to a point. Naturality of CHT yields the following commutative
diagram
CHT *
K*T KT
ss* ss*
K*T(X) CH K*T(X)
T
This defines a map T : K*T(X) K*TK*T! K*T(X). If we denote by K *T(X) =
K*T(X) K*TK*T, we can rewrite T : K *T(X) ! K*T(X).
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 15
Theorem 3.9. T is an isomorphism of T -equivariant cohomology theories.
Proof.Because of the Mayer-Vietoris sequence, it is enough to verify the isomo-
sphism for "equivariant points" of the form T=L, with L a subgroup of T . Choose
an identification r
Y
T = (S1)p x (S1)q x Zmi
i=1
such that via this identification
Yq Yr
L = (S1)p x Znix Zli.
j=1 i=1
Q q Q r
Then T=L = (1)p x j=1S1=Znix i=1Zmi=Zli.
We use now Proposition 2.23, which is also true if we replace K by K -theory.
Since the map commutes with the isomorphisms of Proposition 2.23, it is enough
to check that the following maps are isomorphisms:
(a) S1 : K *S1! K*S1;
(b) S1 : K *S1(S1=Zn) ! K*S1(S1=Zn);
(c) Zm : K *Zm(Zm =Zl) ! K*Zm(Zm =Zl).
To prove (a), notice that CS1 = Cx . K *S1= K*S1K*S1OhCx = OhCx; by
Proposition 2.21, K*S1= OhCx. By definition, the map S1 is the identity.
For (b), denote X = S1=Zn. K *S1(X) = K*S1(X) K*S1K*S1= K*ZnK*S1OhCx.
But we know that K*S1= C[z1 ] and K*Zn= C[z1 ]=. So we deduce
K *S1(X) = OhCx=. This last ring can be identified with C[z1 ]=,
since the condition zn = 1 makes all power series finite. In conclusion, K *S1(*
*X) =
K*Zn= C[z1 ]=.
Let us now describe the sheaf F = K*S1(X). Let ff 2 Cx . If ff =2Zn, Xff= ;,
so the stalk of F at ff is zero. If ff 2 Zn, Xff= X, and the stalk of F at ff *
*is
H*ZnH*S1OhC;0. But H*Zn= C, concentrated in degree zero (H*Znis Z-torsion in
higher degrees, so the components in higher degree disappear when we tensor with
C). It follows that F is a sheaf concentrated at the elements of Zn, where it h*
*as
stalk equal to C. Then the global sections of F are K*S1(X) = C : : :C, n
copies, one for each element of Zn.
The map S1 : K *S1(X) ! K*S1(X) comes from the ring map CHT :
C[z1 ]= ! C : : :C. Since z generates the domain of CHT, it is
enough to see where z is sent to. Let Zn = {1; ffl; ffl2; : :;:ffln-1}. Then *
*z repre-
sents the standard irreducible representation V = V (ffl) = C of Zn, where ffl *
*acts
on C by complex multiplication with ffl, which is regarded as an element of Cx .
Notice that V corresponds to the element = ffl 2 cZn= Zn. c1(V )S1 = 0, be-
cause c1(V )S1 lies in H2Zn= 0. Then chS1(V ) = ec1(V )S1= e0 = 1, and the stalk
of CHS1(V ) at ff 2 Zn is CHS1(V )ff= ff(ffl) = ff. Therefore CHS1 sends z to
(1; ffl; ffl2; : :;:ffln-1) 2 C : : :C, hence it is an isomorphism.
16 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
* * 1 l
For (c), denote X = Zm =Zl. As in (b), K Zm(X) = KZl = C[z ]=, and
K*Zm(X) = C : : :C, l copies. The proof that Zm = CHZm is an isomorphism __
is the same as above. |__|
Appendix A. Applications
We now give applications of the construction in this paper. First, we use the
Chang-Skjelbred theorem in equivariant cohomology to infer the corresponding
result for equivariant K-theory. Then, as a corollary we show how a result about
the equivariant cohomology of GKM manifolds can be extended to equivariant
K-theory. Along the way we need a natural splitting that does not seem to have
been noticed before in this area.
Definition A.1. Let X be a compact T -manifold, for T a compact abelian Lie
group. We say that X is equivariantly formal if the equvariant cohomology spect*
*ral
sequence collapses at the E2 term.
Many interesting T -spaces are equivariantly formal; for example any subvarie*
*ty
of complex projective space preserved by a linear action, or symplectic manifold
with a Hamiltonian action. Our reference for equivariantly formal spaces is [8]*
*. We
need three results about them: the first is that the map H*T(X) ! H*T(XT ) is an
injection, and the second is that for any H subgroup of T , XH is also equivari*
*antly
formal. The third is due to Chang and Skjelbred [6]:
Theorem A.2. Let X be an equivariantly formal T -manifold, and let X1 be its
equivariant 1-skeleton, i.e. X1 is the set of points in X with stabilizer of co*
*dimen-
sion at most one. We have inclusion maps i : XT ! X and j : XT ! X1.
Then the map i* : H*T(X) ! H*T(XT ) is injective, and the maps i* *
*and
j* : H*T(X1) ! H*T(XT ) have the same image.
The ring H*T(X1), in the notation of the above theorem, has not received much
study. It is typically much bigger than H*T(X), and though H*T(X) injects into *
*it,
it does not inject into H*T(XT ). These phenomena can be seen in the case of T 2
acting on X = CP2, where X1 is a cycle of three CP1's and therfore has H1, not
seen in either H*T(X) or H*T(XT ).
Lemma A.3. In the notation of Theorem A.2, there is a natural identification
H*T(X1) = H*T(X) kerj*.
Proof.By Theorem A.2 the images of the two maps i* : H*T(X) ! H*T(XT )
and j* : H*T(X1) ! H*T(XT ) are the same. But i* is injective, so we can identi*
*fy
H*T(X) with the image of i*. This implies that j* factors through a map H*T(X1)*
*_!
H*T(X), and this yields a splitting H*T(X1) = H*T(X) kerj*. |_*
*_|
This natural splitting will sheafify, allowing us to extend both results to K-
theory. First we give a lemma.
Lemma A.4. The map K*T! K *Tis faithfully flat.
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 17
*
Proof.Let O be the sheaf of algebraic functions on the complex variety Spec KT,
and Oh the sheaf of holomorphic functions. Then from the appendix of Serre [13]_
(see also Remark 4F of Matsumura [11]) the map O ! Oh is faithfully flat. |_*
*_|
Theorem A.5. We use the same notations as in Theorem A.2. Then i* :
K*T(X) ! K*T(XT ) is injective, and the maps i* and j* : K*T(X1) ! K*T(XT )
have the same image.
Proof.Let ff 2 CT. Any compact T -manifold admits a decompositionPas a finite
T -CW complex (see for example Allday and Puppe [2]). Let X = iDnix T=Hi
be such a cellPdecomposition. Then Proposition 2.7 says that if K is a subgroup
of T , XK = i:KHi Dnix T=Hi. In particular, this implies that (X1)ff= (Xff)1.
Let Y = Xff, again equivariantly formal. By Lemma A.3, there is a natural
identification H*T(Y1) = H*T(Y ) kerj*. By Proposition 3.3, the map H*T !
H *Tis flat, so tensoring with H *Tover H*Tyields a splitting H *T(Y1) = H *T(Y*
* )
kerj*. Now, we observed above that (Y1)ff= (Y ff)1. So we finally get a splitti*
*ng
H *T((X1)ff) = H *T(Xff) kerj*. This is compatible with the gluing maps of the
sheaf K*T(X), so we get K*T(X1) = K*T(X) kerj*.
The upshot of the above discussion is that i* : K*T(X) ! K*T(XT ) is injectiv*
*e (it
is injective on stalks), and K*T(X1) = K*T(X)kerj*. Take global sections. is l*
*eft
exact, so i* : K*T(X) ! K*T(XT ) is injective and K*T(X1) = K*T(X) kerj*.
This implies that i* and j* have the same image in K*T(XT ), namely K*T(X).
(Notice we couldn't have done this without using the splitting, because is not
right exact, so it doesn't commute with the image functor.)
Now recall that we have a natural isomorphism T : K *T(X) ! K*T(X). Trans-
lating the above results via T, we obtain that the maps i* : K *T(X) ! K *T(XT )
and j* : K *T(X1) ! K *T(XT ) have the same image. But K *T(X) = K*T(X)K*TK *T
and by Lemma A.4 the map K*T! K *Tis faithfully flat. So we obtain that i* and
j* have the same images in K*T(XT ), which is what we wanted. __
|__|
An alternate proof of this can be given by noticing that the sheaf maps i* :
K*T(X) ! K*T(XT ) and j* : K*T(X1) ! K*T(XT ) have the same image (because
they have the same image at the level of stalks). But the global section functo*
*r is
exact, since we work with coherent sheaves over Stein manifolds (see the comment
before Theorem 2.20). It follows that the maps i* : K *T(X) ! K *T(XT ) and
j* : K *T(X1) ! K *T(XT ) have the same image, and the proof proceeds as in the
previous theorem.
In [8] a special case of this is studied, in which XT is discrete and X1 is a*
* union
of S2's; these are called balloon manifolds or GKM manifolds. (An interesting
example of GKM manifolds are toric varieties.) In this case it is easy to calcu*
*late
the image of restriction from H*T(X1), by reducing it to the case of H*T(S2). T*
*he
theorem above lets us extend this result to K-theory.
Corollary A.6. Let X be a GKM manifold, and i* : K*T(X) ! K*T(XT ) the
restriction map. Then i* is an injection and a class ff 2 K*T(XT ) is in the im*
*age
if and for each 2-sphere B X1 with fixed points n and s, the difference ff|n -
18 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
ff|s 2 KT is a multiple of the K-theoretic Euler class of the tangent space TnB.
(Technically, we can only take the Euler class once we orient TnB, but either
orientation leads to the same condition on ff.)
If T = (S1)n, we can identify K*Twith Laurent polynomials, and this condition
says that the difference ff|s - ff|n of Laurent polynomials must be a multiple *
*of
1-w, where w is the weight of the action of T on TnB. (Again, we can only speak*
* of
the weight w once we orient this R2-bundle over the point n, but it doesn't mat*
*ter
because being a multiple of 1 - w is the same as being a multiple of 1 - w-1. In
most examples one has around a T -invariant almost complex structure with which
to orient all these tangent spaces simultaneously.)
Acknowledgements. We thank Victor Guillemin and Haynes Miller for valuable
suggestions and comments.
Massachusetts Institute of Technology, Cambridge, MA
E-mail address: ioanid@math.mit.edu
University of California at Berkeley, CA
E-mail address: allenk@math.berkeley.edu
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