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 cohomology. I*
*n the appen-
dix we give applications of this by extending results of Chang-Skjelbred*
* and Goresky-
Kottwitz-MacPherson from equivariant cohomology to equivariant K-theory.
1.Introduction
Let T be an abelian compact Lie group, not necessarily connected. Let X be a
compact T -equivariant manifold, or more generally a finite T -CW complex. We d*
*enote
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 described in Segal*
* [16]. All
the cohomology theories in this paper have complex coefficients, unless otherwi*
*se noted.
For example, K*T(X) = K*T(X, Z) Z C. Also, when X is a point, we write K*Tinst*
*ead
of K*T(X), and similarly for H*T.
The goal of this paper is to describe K*T(X) in terms of H*T(X). When T is th*
*e trivial
group, this is easy: it is a classical result that the Chern character ch : K*(*
*X) ! H*(X)
is an isomorphism (in this case one only needs to tensor with Q). In general ho*
*wever it
is not true that the equivariant version of the Chern character
chT : K*T(X) ! H**T(X)
is an isomorphism. (For the definition of chT see Lemma 3.1 and the discussion *
*before
it.)
The good news is that there is still a way in which one can describe K*T(X) i*
*n terms
of H*T(X). Details will be given later, but for now let us outline the main ste*
*ps of this
description. Let CT = Specm K*Tbe the complex algebraic group of the maximal id*
*eals
of K*T. The construction of CT is functorial in T , and if H ,! T is a compact *
*subgroup
of T , we can identify CH as a subgroup of CT via the map CH ! CT. If ff is a p*
*oint of
CT, denote by H(ff) the smallest compact subgroup H of T such that CH contains *
*ff.
Denote by Xff= XH(ff), the subspace of X fixed by all elements of H(ff). Denote*
* by O
the sheaf of algebraic functions on CT, and by Oh the sheaf of holomorphic func*
*tions.
Then we will define a sheaf, denoted by K*T(X), whose stalk at a point ff 2 CT *
*is
K*T(X)ff= H *T(Xff) ,
where H *T(-) is the extension of H*T(-) by the ring of holomorphic germs at ze*
*ro on
H*T. Moreover, the transition functions of K*T(X) will be also defined entirely*
* using the
equivariant cohomology of X. Now, if denotes the global sections functor, we*
* will
show that there exists an isomorphism
T : K*T(X) O Oh ~= K*T(X) .
This is the sense in which equivariant K-theory with complex coefficients can b*
*e de-
scribed in terms of equivariant cohomology.
1
2 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
We should say a few words about the isomorphism T. If we denote by chT the
equivariant Chern character (to be defined below), we will see that T is esent*
*ially a
sheaf version of chT. However, chT has to be twisted (translated) in an approp*
*riate
sense, to take into account the point ff over which the germ of chT is taken.
One may call this a de Rham model for equivariant K-theory, but it would be a*
* slight
misnomer, since we do not describe K*T(X) at the level of cocycles. As a matte*
*r of
fact, we do more: we describe the classes themselves, as sections in a sheaf bu*
*ild from
ordinary equivariant cohomology. But, if one were really intent on giving a de *
*Rham
model, one could use the sheaf model to define a cocycle in K-theory as a colle*
*ction of
germs of ordinary closed differential forms, and use a similar definition for c*
*oboundaries.
We do not purse this avenue because it would only obscure the purely topologica*
*l nature
of our description of K*T(X).
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 [4]. The ideas were*
* further
developed in Block and Getzler [5], and Duflo and Vergne [10]. In fact, both th*
*e idea of
describing equivariant K-theory as a sheaf and twisting the equivariant Chern c*
*haracter
are present in Duflo and Vergne. The problem in their paper is that they cannot*
* prove
the Mayer-Vietoris property for their cohomology theory, because they work with*
* C1
functions. The advantage of our approach is that we work with coherent analytic*
* sheaves
over the affine (Stein) manifold CT, and in this case the global section functo*
*r is exact.
The present paper is inspired mainly by Grojnowski's preprint [12]. In this *
*semi-
nal work, he uses ideas from the papers mentioned above to define equivariant e*
*lliptic
cohomology with complex coefficients. His model starts with an elliptic curve E*
*, and con-
structs for every torus T a complex variety ET and a coherent analytic sheaf El*
*l*T(X) over
ET, whose stalk at each point is defined in terms of equivariant cohomology. Th*
*e sheaf
Ell*T(X) is then defined by Grojnowski to be the ("delocalized") complex T -equ*
*ivariant
cohomology of X. Interestingly enough, equivariant K-theory is never explicitly*
* men-
tioned in Grojnowski's preprint, although he was most likely aware that an anal*
*ogous
construction to that of Ell*T(X) should lead to equivariant K-theory, if the el*
*liptic curve
E is replaced by the multiplicative group C = C \ {0}. The main contribution o*
*f the
present paper is to do exactly that: it starts with the multiplicative group C,*
* out of
which it defines the base complex variety CT, and constructs a coherent analyti*
*c sheaf
K*T(X) over CT. Then, the ring of global sections in K*T(X) turns out to be a f*
*aithfully
flat extension of equivariant K-theory1.
A simple exercise shows that if one starts instead with the additive group A *
*= C,
the resulting sheaf is nothing else but ordinary equivariant cohomology. An imp*
*ortant
conclusion is that, when working with complex coefficients, the difference betw*
*een equi-
variant cohomology, K-theory and elliptic cohomology stems mainly from the fact*
* that
these theories are associated to different complex groups of dimension one: the*
* additive,
the multiplicative, and the elliptic groups, respectively.
The results of this paper can be extended in several directions. First, we ca*
*n describe
K*T(X) directly, instead of describing its faithfully flat 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 loga*
*rithm
___________
1Besides its contribution to equivariant K-theory, one can regard the present*
* paper as giving a
rigorous definition for Grojnowski's equivariant elliptic cohomology: for this,*
* it is enough to change the
base manifold CT to Grojnowsi's ET. See also Rosu [15] for a definition of equi*
*variant elliptic cohomology
when T = S1.
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 3
map is not algebraic. The construction therefore becomes more complicated, and*
* we
decided to relegate it to another paper. Second, if G is a nonabelian connected*
* compact
Lie group, and T its maximal torus, then K*G(-) = K*T(-)W , where W is the Weyl
group, so one can also describe K*G(-) using Borel equivariant cohomology. Thir*
*d, we
can prove a similar result whenever the coefficient ring R of the cohomology th*
*eories
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 roo*
*ts of
unity because we want to split 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 *
*corre-
sponding ones in equivariant cohomology. In the Appendix we give examples of t*
*his,
extending results of Chang-Skjelbred [9] and Goresky-Kottwitz-MacPherson [11] f*
*rom
equivariant cohomology to equivariant K-theory.
2. A sheaf-valued cohomology theory
The purpose of this section is to define a sheaf valued T -equivariant cohomo*
*logy
theory, which we denote by K*T(-). In the next sections we are going to show t*
*hat
global sections of this sheaf are essentially equivariant K-theory. We already *
*knew that
K*T(X) can be regarded as a coherent sheaf over CT = Specm K*T(because it is a *
*K*T-
module). The novelty is that K*T(X) can be completely described in terms of ord*
*inary
equivariant cohomology (since we will show that K*T(X) is). Let us start with *
*a few
definitions.2.
2.1. Definitions
First we want a simpler description of CT = Specm K*T. Denote by ^Tthe Pontrj*
*agin
dual of T , i.e. ^T= Hom (T, S1). For example, if T = (S1)p x G, where G is a *
*finite
abelian group, then ^T~=ZpxG (the isomorphism is not natural, however). Denote *
*by C
the multiplicative algebraic group C \ {0}. Although it might generate some con*
*fusion,
we will use additive notation for C throughout the paper. The following straigh*
*forward
lemma gives two alternate descriptions of CT.
Proposition 2.1. There is a natural isomorphism of algebraic varieties
CT ~=Hom Z(T^, C) .
If T is connected (i.e. a torus), and T is its integer lattice, then there is *
*a natural
isomorphism
CT ~= T Z C .
Proof.It is easy to see that K*T= C[T^], the group algebra of ^T. We define a m*
*ap
: Hom Z(T^, C) ! Specm C[T^] = CT
by noting that ff 2 Hom Z(T^, C) extends to a non-zero C-algebra map ff0 : C[T^*
*] ! C.
We then take (ff) = ker(ff0), which is a maximal ideal of C[T ]. Since the dom*
*ain and
codomain of both take products of groups to products of varieties, it suffice*
*s to check
that is an isomorphism when T = S1 or T = Zn, which we leave to the reader.
For the second statement, notice that when T is a torus, there is a natural i*
*somorphism
^T-~! *T.
___________
2For a similar definition in the case of equivariant elliptic cohomology, see*
* Rosu [15]. The discussion
there is only for T = S1, but it generalizes easily with the same formalism as *
*here.
4 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
Suppose T ~=(S1)p x G, where G is a finite abelian group. Then one can apply *
*the
previous proposition to obtain that CT ~=Cpx G. This formula shows that CT is i*
*n fact
an algebraic group (and it is the complexification of T ). Just as we did for C*
*, in the rest
of the paper we are going to use additive notation for CT.
Definition 2.2. Let tC be the complexification of the Lie algebra of T . Then t*
*he expo-
nential map exp : C ! C extends to a complex algebraic group map
exp: tC ! CT .
To be more precise, this map is the composite map tC = T ZC ! T ZC = CT0 ! CT,
where T 0is the connected component of T containing the identity. (For the iden*
*tification
T Z C = CT0, use the previous proposition.) Note that when T is connected, t*
*he
exponential map is surjective.
Definition 2.3. We call a neighborhood U of zero in CT "small" if the above def*
*ined
exponential map, exp : tC ! CT, has a local inverse on U. We call a neigborhood*
* V of
zero in tC "small" if it is of the form exp-1(U), for U a small neighborhood of*
* zero in
CT.
Let A be a collection of compact subgroups of T . Define a relation on CT as *
*follows:
ff A fi if, for any H 2 A, fi 2 CH implies ff 2 CH . The relation A is reflex*
*ive and
transitive, but not antisymmetric, so it is not an order relation. When it is c*
*lear what A
is, we will omit it and write simply ff fi. The next definition singles out a*
* special class
of open covers of CT, called adapted covers. These will be used below in the de*
*finition
of the sheaf K*T(X).
Definition 2.4. Let A be a collection of compact subgroups of T , and let U = (*
*Uff)ff2CT
be an open cover indexed by the points of CT. Then U is called ä dapted to A" *
*if it
satisfies the following conditions:
1. ff 2 Uff, and Uff- ff is small.
2. If Uff\ Ufi6= ;, then either ff fi or fi ff.
3. If ff fi, and for some H 2 A ff 2 CH but fi =2CH , then Ufi\ CH = ;.
4. If Uff\ Ufi6= ;, and both ff and fi belong to CH for some H 2 A, then ff*
* and fi
belong to the same connected component of CH .
Proposition 2.5. If A is a finite collection of compact subgroups of T , then t*
*here 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 conne*
*cted),
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 center ff*
* and
radius d, with
d < 1_2 min dist(ff, D) ,
D2H0,ff=2D
and such that Uff- ff is small.
We show that U = (Uff)ff2CTis adapted: Condition 1 is trivially satifsfied. T*
*o prove
Condition 2, let ff and fi be such that Uff\ Ufi6= ;. Suppose we have neither f*
*f fi, nor
fi ff. Then by the definition of there exist two compact subgroups K and L *
*of T
such that ff 2 CK \ CL and fi 2 CL \ CK . But from the definition of Uffit foll*
*ows that Uff
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 5
is a ball of center ff and radius d < 1_2dist(ff, CL) 1_2dist(ff, fi). Simila*
*rly, Ufiis a ball
of center fi and radius less than 1_2dist(ff, fi), so Uffand Uficannot possibly*
* intersect,
contradiction.
Condition 3 is obviously satisfied, by construction.
Finally, to show Condition 4, 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 re*
*asoning
as above, it follows that the radii of Uffand Ufiare smaller than 1_2dist(ff, f*
*i), so Uffand
Uficannot possibly intersect, which again leads to a contradiction.
Let ff 2 CT. The construction of CT is functorial, so if H is any compact sub*
*group
of T , we get an inclusion map CH ! CT. If ff 2 Im(CH ! CT), we say that ff 2 C*
*H .
For ff 2 CH , denote by H(ff) the smallest compact subgroup H of T such that ff*
* 2 CH .
The fact that there exists a smallest H such that ff 2 CH is implied by the fo*
*rmula
CK \CL = CK\L , which follows from an easy diagram chase. This also implies tha*
*t H(ff)
is the intersection of all compact subgroups H such that ff 2 CH . Another imme*
*diate
consequence is the following formula, which will be useful later:
(1) H(ff) K () ff 2 CK .
Let X be a space with a T -action. If K is a compact subgroup of T , denote *
*by
XK X the subspace of points fixed by K. Also, if ff 2 CT, define
Xff= XH(ff).
Now we want to define the notion of a cover adapted to a finite T -CW complex. *
*So let
X be a finite T -CW complex. We know that there exists a finite collection A = *
*(Hi)i
of compactSsubgroups of T such that X has an equivariant cell decomposition of *
*the
form X = iDnix (T=Hi). Here we denoted by Dn the open disk in dimension n, and
by D0 a point. The group T acts trivially on Dni, and by left multiplication on*
* T=Hi.
Notice that if K is a compact subgroupSof T , then the subcomplex of X fixed by*
* K has
a decomposition of the form XK = i:K HiDnix (T=Hi). Taking K = H(ff), we get
[
(2) Xff= Dnix (T=Hi) .
i:ff2CHi
We say that the cover U = (Uff)ff2CTof CT is ä dapted to X" if U is adapted to *
*the
collection A of the isotropy groups Hi appearing in a T -equivariant cell decom*
*position
of X. Since this collection is finite, Proposition 2.5 implies that there alway*
*s exists a
cover adapted to X.
Next we discuss a few useful results in equivariant cohomology. We start with*
* a well-
known proposition which says that the ring of coefficients of (complex) T -equi*
*variant
cohomology is the polynomial algebra on tC, the complex Lie algebra of T .
Proposition 2.6. If T is an abelian compact Lie group, there is a natural isomo*
*rphism
S(t*C) -~! H*T ,
where S(-) denotes the symmetric algebra, and t*Cis the dual of tC.
Proof.T^= Hom (T, S1) is the group of irreducible characters of T , so for ~ 2 *
*^Tconsider
the complex vector bundle
V~ = ET xT C ,
over the classifying space BT , where the map T ! C is given by ~. Then the fir*
*st Chern
class of V~ gives a natural isomorphism c1 : ^T! H2(BT, Z).
6 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
If T is a torus, we saw in Proposition 2.1 that ^Tcan be identified with the *
*dual of the
integer lattice *T, so by tensoring the map c1 with C, we get the natural isom*
*orphism
c1 : t*C= *T C ! H2(BT, C). Taking symmetric products, we get the desired
isomorphism.
If T is a general (non-connected) compact abelian Lie group, the isomorphism *
*still
holds, since both domain and codomain depend only on the connected component of*
* T
containing the identity, which is a torus.
We now define an algebra homomorphism
h : H*T! OhtC,0
by taking a polynomial in H*T= S(t*C) and sending it to its germ at zero. The m*
*ap is
injective, so we can consider H*Tas a subring of OhtC,0. Let V be a small neigh*
*borhood of
zero in tC. Then, since the ring H*T OhtC,0consists of the germs of global hol*
*omorphic
functions on tC, the map h factors through the inclusion OhtC(V ) ,! OhtC,0, so*
* we can
define a map, also denoted by h,
h : H*T! OhtC(V ) .
Let U be a small neighborhood of zero in CT, and let V = exp-1(U), where exp: t*
*C ! CT
is the exponential map. Via the exponential, we have the following identificat*
*ions:
OhCT,0~-!OhtC,0and OhCT(U) -~! OhCT(V ).
Definition 2.7. Let U be a small neighborhood of zero in CT. Via the identifica*
*tions
above, we define the following two maps, and denote them also by h (the second *
*one is
the correstriction of the first):
h : H*T! OhCT,0 and h : H*T! OhCT(U) .
Now we define a few cohomology theories that we are going to use throughout t*
*he
paper. Let X be a finite T -CW complex. We define the holomorphic T -equivari*
*ant
cohomology of X to be
H *T(X) = H*T(X) H*TOhCT,0,
where the map h : H*T! OhCT,0is given in Definition 2.7. It is indeed a cohomo*
*logy
theory, because by Proposition 2.8 the map H*T! OhCT,0is flat. The theory is n*
*ot
Z-graded anymore; however, it can be thought of as Z=2-graded, by its even and *
*odd
part.
Let H**T(X) be the completion of H*T(X) with respect to the augmentation ideal
I = ker(H*T! C). Since H*T(X) is a finitely generated module over the Noetheri*
*an
ring H*T, a simple result on completions (see for example Matsumura [14], Theor*
*em 55)
implies that
H**T(X) ~=H*T(X) H*TH**T.
The ring H*Tis a polynomial ring, so we have the following well-known results f*
*rom
algebra (they are sometimes called GAGA results, since they first appeared in S*
*erre's
GAGA [17]).
Proposition 2.8. H *T= OhCT,0and H**Tare flat over H*T. If U is a small neighbo*
*rhood
of zero, then OhCT(U) is flat over H*T.
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 7
Proof.All the results we mention in this proof are from Matsumura [14]. Identif*
*y the
complex Lie algebra tC with X = Cn. We denote by O the algebraic structure she*
*af
of Cn, and by Oh the analytic structure sheaf. Let O^0be the completion of the *
*local
ring O0 with respect to its maximal ideal. It is sufficient to show that the n*
*atural
maps O(X) ! Oh0and O(X) ! O^0are flat, and that for any U open in X, the map
O(X) ! Oh(U) is flat. We start by noticing that O^0is the completion of O(X) wi*
*th
respect to its maximal ideal at zero, so by Corollary 1 of Theorem 55, we know *
*that O^0
is flat over O(X). The completion of Oh0with respect to its maximal ideal is al*
*so O^0,
so Oh0! O^0is flat. It is in fact faithfully flat, because it is local: see The*
*orem 3 (4.D).
Now one can check directly by the definition of flatness that having O(X) ! O^0*
*flat
and Oh0! O^0faithfully flat implies that O(X) ! Oh0is flat. Notice also that th*
*e same
proof can be used to show that O0 ! Oh0is flat.
Next let U X be an open set. By the local characterization of flatness, The*
*orem
(3.J), in order to show that O(X) ! Oh(U) is flat, we have to show that for any*
* x 2 U
the natural map Ox ! Ohxis flat. But we have already shown this when x = 0, and*
* the
proof for general x is the same.
Now in order to prove the proposition, just transfer the results we have prov*
*ed via
the exponential map, exp: Cn = tC ! CT. This is where we need U small.
In particular, it follows that H*T(X) and H *T(X) can be regarded as subrings*
* of
H**T(X).
2.2. Construction of K*T
Let X be a finite T -CW complex. Fix U a cover adapted to X, which exists bec*
*ause of
Proposition 2.5. We are going to define a sheaf F = FU over CT whose stalk at f*
*f 2 CT is
isomorphic to H *T(Xff). Recall that in order to give a sheaf F over a topologi*
*cal space, it
is enough to give an open cover (Uff)ffof that space, and a sheaf Fffon each Uf*
*ftogether
with isomorphisms of sheaves OEfffi: Fff|Uff\Ufi-! Ffi|Uff\Ufi, such that OEfff*
*fis the identity
function, and the cocycle condition OEfiflOEfffi= OEffflis satisfied on Uff\ Uf*
*i\ Ufl. If U is
a subset of CT, denote by U + ff = {x + ff | x 2 U} the translation of U by ff.
Definition 2.9. Define a presheaf Fffon Uffby taking, for any open U Uff,
Fff(U) = H*T(Xff) H*TOhCT(U - ff) ,
where the restriction maps are induced from those of OhCT. The ring map h : H**
*T!
OhCT(U - ff) is given in Definition 2.7.
Proposition 2.10. Fffis a coherent sheaf of OhCT-modules.
Proof.First we show that Fffis a sheaf of H*T-modules. If (Ui)i is an open cove*
*r of a
topological space Y , denote by Uij= Ui\ Uj, etc. Then a presheaf G is a sheaf *
*if and
only if for any m > 0 and any finite cover (Ui)i=1...mthe following sequence is*
* exact
Y r1 Y r2
0 -! G(Y ) -r0! G(Ui) -! G(Uij) -! . .-.! G(U1...m) -! 0 ,
i i ,
where oei(x1, . .,.xn) is the i'th symmetric polynomial in the xj's. The class*
*es xj =
c1(Lj)T are called the Chern roots of E. Moreover, we can identify H*T(X) as th*
*e subring
of H*T(SE ) generated by the polynomials in H*T(X)[x1, . .,.xn] which are symme*
*tric 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. .e.xn. Since Lj is a line bundle and xj = c1(Lj)T,*
* the
first part of the proof implies that exj2 H *T(SE ) for all j. Therefore chT(E)*
* 2 H *T(SE ),
and since it is symmetric in the xj's it follows that chT(E) 2 H *T(X), which i*
*s what we
wanted.
We have just proved that chT(E) is the germ of a holomorphic class, i.e. an e*
*lement
of H *T(X). By looking more carefully at the preceding proof, one can see in fa*
*ct that
we proved a stronger result:
Corollary 3.2. With the same notations as in Lemma 3.1, chT(E) is a global holo*
*mor-
phic class, i.e. an element of H*T(X) H*T OhCT.
Now, if we extend chT(E) on a small neighborhood U of 0 2 CT, we can regard it
as an element of H*T(X) H*TOhCT(U). It is important to see what happens to chT*
*(E)
when it is translated by the map ø*fi-fffrom Proposition 2.14.
The basic case is when X is a point and E is given by a representation V~ of *
*T , with
~ 2 ^T. Recall that CT = Hom (T^, C), and consider ff 2 CT. Then we translate c*
*hT(V~)
via the map ø*ff= t*ff: OhCT(U) ! OhCT(U + ff).
Lemma 3.3. Let T be a compact Abelian group, and T 0the connected component con-
taining the identity. Let ff 2 CT0 and ~ 2 ^T. Then, with the notations above,
t*ffchT(V~) = ff(~)chT(V~) .
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 15
Proof.The proof of this lemma is mostly formal, and just makes intensive use of*
* the
identifications and definitions we have made so far. Start by using the exponen*
*tial map
exp: tC = T Z C ! T Z C = CT0 ! CT .
The element ff is in the image, so pick a 2 tC such that exp(a) = ff. Denote by*
* l 2 *Tthe
element corresponding to ~ via the map ^T! *T. Then one can apply l to a 2 T *
* Z C
and get a complex number that we denote by l(a). Now it is easy to check the fo*
*rmula
ff(~) = exp(l(a)). We also know that chT(V~) = exp c1(V~) . So, via the exponen*
*tial
map, what we have to prove becomes
t*ac1(V~) = l(a) + c1(V~) ,
with the equality being now regarded in H*T. Let us look more closely at c1(V~)*
*. We saw
in the proof of Proposition 2.6 that there is an identification H*T= S(t*C), an*
*d that the
class c1(V~) 2 H*Tcan be identified to l if this is regarded in S(t*C) via *T *
* t*C S(t*C).
Denote by l(-) the polynomial function in S(t*C) corresponding to l. Then we ha*
*ve to
prove that
t*al(-) = l(a) + l(-) .
But this is obvious, it is just saying that l(-) is a linear function.
3.2. Construction of CHT
We define a multiplicative natural map
CHT : K0T(X, Z) ! K*T(X) .
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^, C), so we can think of ff as*
* a group
map ff : ^H! C. The space Xffhas a trivial action of H, so the restriction E|Xf*
*fof 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 w*
*ith
~(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 3.4. Let ff 2 CT and H = H(ff). Then the germ of CHT(E) at ff is def*
*ined
to be X
CHT(E)ff= ff(~)chTE(~) .
~2H^
Proposition 3.5. The germs CHT(E)ffglue to a global section CHT(E) 2 K*T(X).
Proof.We notice that, by Lemma 3.1, CHT(E)ffdoes indeed belong to H *T(Xff), wh*
*ich
by Proposition 2.23 is the stalk of K*T(X) at ff. Fix (Uff)ff2CTa cover of CT a*
*dapted to
X.
Let ff, fi 2 CT with Uff\ Ufi6= ; and ff fi. This implies ff, fi 2 CH(fi)a*
*nd also
H(ff) H(fi). Denote by L = H(ff) and H = H(fi). Condition 4 of Definition 2*
*.4
implies that fi - ff 2 CH0. Now we have to prove that OEfffiCHT(E)ff= CHT(E)fi,
16 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
P P
i.e. that ø*fi-ff~2^Lff(~)chTE(~)|Xfi= ~2H^fi(~)chTE(~). Consider the surj*
*ec-
tive map jP : ^H ! ^Linduced by the inclusion L ,! H. If ~ 2 ^L, we ha*
*ve
E(~)|Xfi= ~2j-1(~)E(~). Therefore it is enough to show that for all ~ 2 H^ we
have ø*fi-ffff(~)chTE(~) = fi(~)chTE(~), where ~ = j(~). But this is equivale*
*nt to
ø*fi-ffchTE(~) = (fi - ff)(~)chTE(~). Denote by fl = fi - ff 2 CH0. So it is en*
*ough to
show that, for all fl 2 CH0 and ~ 2 ^H,
ø*flchTE(~) = fl(~)chTE(~) .
Let K = T=H and Y = Xfi. Proposition 2.13 applied to equivariant K-theory giv*
*es
a natural isomorphism
K*K(Y ) K*KK*T-~! K*T(Y ) .
Via the identification above, we can think of E(~) as a tensor product F V (~*
*0), with
F a K-bundle and ~02 ^Tsome element in the preimage of ~ via the map ^T! ^H. (At
least, we know that E(~) is generated by such elements.) Via the same identific*
*ation,
translation by fl 2 CH0 becomes
ø*fl7! ø*i(fl) ø*fl,
where ß(ø) is the image of fl via the natural map ß : CT ! CK , and the second
fl is regarded in CT via the usual inclusion CH0 ! CT. But notice that ß(ø) = *
*0,
because of the exact sequence 0 ! CH ! CT ! CK ! 0. Also, chTE(~) becomes
chK F chTV (~0) 2 H *K(Y ) H *KH*T. So via the above correspondence we have
ø*flchTE(~) 7! chK F ø*flchTV (~0) .
Since fl 2 CH0, it follows that fl 2 CT0, and it is sufficient for us to show t*
*hat, for all
fl 2 CT0 and ~02 ^T,
ø*flchTV (~0) = fl(~)chTV (~0) .
But this is directly implied by Lemma 3.3, so we are done.
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). One can check easily that CHT is a ring map, since chT is. Because K**
*T(X)
is a C-algebra and CHT is a ring map, we can now extend CHT to a natural map of
C-algebras CHT : K*T(X) ! K*T(X). Finally, making X a point, we get a ring map
K*T,! K*T, so if we extend CHT by this, we obtain the desired natural map
CHT : K *T(X) ! K*T(X) .
Theorem 3.6. CHT is an isomorphism of T -equivariant cohomology theories.
Proof.Because of the Mayer-Vietoris sequence, it is enough to verify the isomor*
*phism
for "equivariant pointsö f the form T=L, with L a compact subgroup of T . Choo*
*se 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=1(S1=Zni) x i=1(Zmi=Zli).
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 17
We use now Proposition 2.24, which is also true if we replace K by K -theory*
* (be-
cause it is true for K-theory). Since the map CH commutes with the isomorphisms*
* of
Proposition 2.24, it is enough to check that the following maps are isomorphism*
*s:
(a) CHS1 : K *S1! K*S1;
(b) CHS1 : K *S1(S1=Zn) ! K*S1(S1=Zn);
(c) CHZm : K *Zm(Zm =Zl) ! K*Zm(Zm =Zl).
To prove (a), notice that CS1 = C. Then we have K *S1= K*S1 K*S1 OhC= OhC. By
Proposition 2.21, K*S1= OhC. Now notice that, by definition, the map CHS1 is *
*the
identity.
For (b), denote X = S1=Zn. Then we have K*S1(X) = K*S1(X) K*S1 K*S1= K*Zn K*S1
OhC. But we know that K*S1= C[z 1] and K*Zn= C[z 1]=. So we deduce
K *S1(X) = OhC=. This last ring can be identified with C[z 1]=*
*, since
the condition zn = 1 makes all power series finite. In conclusion, K *S1(X) = *
*K*Zn=
C[z 1]=.
Let us now describe the sheaf F = K*S1(X). Let ff 2 C. 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*Zn *
*H*S1OhC,0.
But H*Zn= C, concentrated in degree zero (H*Znis Z-torsion in higher degrees, s*
*o 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 has the stalk equal to C. Th*
*en the
global sections of F are K*S1(X) = C . . .C, n copies, one for each element *
*of Zn.
The map CHS1 : K *S1(X) ! K*S1(X) comes from the ring map CHT : C[z 1]= ! C . . .C. Since z generates the domain of CHT, it is enough to see where z*
* is sent.
Let Zn = {1, ffl, ffl2, . .,.ffln-1}. Then z represents the standard irreducibl*
*e representation
V = V (ffl) = C of Zn, where ffl acts on C by complex multiplication with ffl,*
* which is
regarded as an element of C. Notice that V corresponds to the element ~ = ffl 2*
* cZn= Zn.
c1(V )S1 = 0, because 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 se*
*nds z to
(1, ffl, ffl2, . .,.ffln-1) 2 C . . .C. One can easily check that this map is*
* an isomorphism.
For (c), denote X = Zm =Zl. As in (b), K *Zm(X) = K*Zl= C[z 1]=, and
K*Zm(X) = C . . .C, l copies. The proof that CHZm is an isomorphism is the s*
*ame
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 f*
*or equi-
variant K-theory. Then as a corollary we show how a result about the equivaria*
*nt
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 gro*
*up.
We say that X is equivariantly formal if the equvariant cohomology spectral seq*
*uence
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 w*
*ith a
18 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
Hamiltonian action. Our reference for equivariantly formal spaces is [11]. We*
* need
three results about them: The first is that the map H*T(X) ! H*T(XT ) is an inj*
*ection.
The second is that for any H compact subgroup of T , XH is also equivariantly *
*formal.
The third is due to Chang and Skjelbred [9] (see also [8] for a proof):
Theorem A.2. Let X be an equivariantly formal T -manifold, and let X1 be its eq*
*ui-
variant 1-skeleton, i.e. X1 is the set of points in X with stabilizer of codime*
*nsion 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 2acting on X*
* = CP2,
where X1 is a cycle of three CP1's and therefore 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 identi*
*fication
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 identify 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 sheafifies, allowing us to extend both results to K-th*
*eory.
Theorem A.4. 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 sam*
*e image.
Proof.Let ff 2 CT. Any compact T -manifold admits a decompositionSas a finite *
*T -
CW complex (see for example Allday and Puppe [2]). Let X = iDnix (T=Hi) be
suchPa cell decomposition. We saw that if K is a compact subgroup of T , XK =
ni ff ff
i:K HiD x (T=Hi). In particular, this implies that (X1) = (X )1.
Let Y = Xff, which is again equivariantly formal. By Lemma A.3 there is a nat*
*ural
identification H*T(Y1) = H*T(Y ) kerj*. By Proposition 2.8 the map H*T! H *T*
*is
flat, so tensoring with H *Tover H*Tyields a splitting H *T(Y1) = H *T(Y ) ke*
*rj*. Now,
we observed above that (Y1)ff= (Y ff)1. So we finally get a splitting 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 (since
it is injective on stalks), and K*T(X1) = K*T(X) kerj*. The global section fu*
*nctor is
left exact, so i* : K*T(X) ! K*T(XT ) is injective and K*T(X1) = K*T(X) k*
*erj*.
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 CHT : K *T(X) ! K*T(X). Trans-
lating the above results via CHT, we obtain that the maps i* : K *T(X) ! K *T(X*
*T ) and
j* : K *T(X1) ! K *T(XT ) have the same image. But K *T(X) = K*T(X) K*TK *Tand*
* by
Lemma 2.22 the map K*T! K *Tis faithfully flat. So we obtain that i* and j* hav*
*e the
same images in K*T(XT ), which is what we wanted.
EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY 19
An alternative proof of Theorem A.4 can be given by noticing that the sheaf m*
*aps
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
Corollary 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 before.
In [11] 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 e*
*xample
of GKM manifolds are toric varieties.) In this case it is easy to calculate the*
* image of
restriction from H*T(X1), by reducing it to the case of H*T(S2). The theorem ab*
*ove lets
us extend this result to K-theory.
Corollary A.5. Let X be a GKM manifold, and i* : K*T(X) ! K*T(XT ) the restrict*
*ion
map. Then i* is an injection; and a class ff 2 K*T(XT ) is in the image if for*
* each
2-sphere B X1 with fixed points N and S, the difference ff|N - ff|S 2 KT is a*
* multiple
of the K-theoretic Euler class of the tangent space TN B. (Technically, we can *
*only take
the Euler class once we orient TN B, but either orientation leads to the same c*
*ondition
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 TN B. (Again, we can only speak of the we*
*ight w
once we orient this R2-bundle over the point N, but it doesn't matter because b*
*eing 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 t*
*angent
spaces simultaneously.)
After finishing this paper, similar results in the algebraic case have appear*
*ed in [18],
particularly with regard to the extension of Chang-Skjelbred's results to K-the*
*ory. More
specifically, their Corollary 5.10 is the algebraic analogue of our Theorem A.4*
*. Notice
also that their results, remarkably, hold over Z, while ours only hold over C. *
*We thank
Angelo Vistoli for explaining his results to us.
We also found out that Atiyah [3] proved a Chang-Skjelbred lemma in the conte*
*xt
of equivariant K-theory. He not only did it more or less at the same time as C*
*hang
and Skjelbred, but his results are stronger: Let G be a torus and X a compact *
*G-
manifold. Denote by Xi the equivariant i-skeleton of X. Then Atiyah shows that *
*the
long exact sequence of the pair (X \ Xi, X \ Xi+1) splits into short exact sequ*
*ences
0 -! K*-1G(X \ Xi+1) -ffi!K*G(Xi+1\ Xi) -! K*G(X \ Xi) -! 0. This in turn is
equivalent to the having a long exact sequence
0 -! K*G(X) -! K*G(X0) -! K*+1G(X1 \ X0) -! K*+2G(X2 \ X1) -! . . .
As noted by Bredon [7], Atiyah's argument carries over to equivariant cohomolog*
*y with
compact support and rational coefficients. In this context, the exactness up to*
* the term
K*+1G(X1 \ X0) is just the assertion of the Chang-Skjelbred lemma. In cohomolog*
*ical
setting, the above sequence may be thought of as the E1 term of the spectral se*
*quence
coming from the filtration of the Borel construction XG by the subspaces (Xi)G *
*. More-
over, its exactness is in fact equivalent to the equivariant formality of X, i.*
*e. to the
freeness of H*G(X). (The direction Atiyah proved is the harder one.) We thank M*
*atthias
Franz for pointing out these results to us.
20 EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
A.1. Acknowledgements. We thank Victor Guillemin for persuading us to write this
paper, and Haynes Miller for constant guidance and support throughout this proj*
*ect. We
also thank Michele Vergne for going carefully through the paper and suggesting *
*several
corrections and improvements. Finally, we thank Lars Hesselholt, Payman Kassaei*
* and
Behrang Noohi for helpful discussions.
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
References
[1]M. Atiyah, R. Bott, The moment map and equivariant cohomology, Topology 23 *
*(1984), 1-28.
[2]C. Allday, V. Puppe, Cohomological methods in transformation groups, Cambri*
*dge University Press,
1993.
[3]M. Atiyah, Elliptic operators and compact groups, Lecture Notes in Mathemat*
*ics 401 (1974).
[4]P. Baum, J.L. Brylinski, R. MacPherson, Cohomologie 'equivariante d'elocali*
*s'ee, C.R. Acad. Sci.
Paris, Serie I 300 (1985), 605-608.
[5]J. Block, E. Getzler, Equivariant cyclic homology and equivariant different*
*ial forms, Ann. Sci. 'Ec.
Norm. Sup. 27 (1994), 493-527.
[6]R. Bott, L. Tu, Differential forms in algebraic topology, Springer, 1982.
[7]G. Bredon, The free part of a torus action and related numerical equalities*
*, Duke Math. J. 41
(1974), 843-854.
[8]M. Brion, M. Vergne, On the localization theorem in equivariant cohomology,*
* Mathematics ArXiv
(1997), http: //xxx.lanl.gov, dg-ga/9711003.
[9]T. Chang, T. Skjelbred, The topological Schur lemma and related results, An*
*n. of Math. 100 (1974),
307-321.
[10]M. Duflo, M. Vergne, Cohomologie 'equivariante et descente, Ast'erisque 215*
* (1993), 3-108.
[11]M. Goresky, R. Kottwitz, R. MacPherson, Equivariant cohomology, Koszul dual*
*ity, and the local-
ization theorem, Invent. Math. 131 (1998), 25-83.
[12]I. Grojnowski, Delocalized equivariant elliptic cohomology, Yale University*
* preprint, 1994.
[13]R. Gunning, H. Rossi, Analytic functions of several complex variables, Pren*
*tice-Hall, 1965.
[14]H. Matsumura, Commutative Algebra, Benjamin/Cummings, 1980.
[15]I. Rosu, Equivariant elliptic cohomology and rigidity, Amer. J. of Math. 12*
*3 (2001), 647-677.
[16]G. Segal, Equivariant K-theory, Publ. Math. I.H.E.S. 34 (1968), 129-151.
[17]J.-P. Serre, G'eom'etrie alg'ebrique et g'eom'etrie analytique, Anal. Inst.*
* Fourier 6 (1956), 1-42.
[18]G. Vezzosi, A. Vistoli, Higher algebraic K-theory for actions of diagonaliz*
*able groups, Mathematics
ArXiv (2001), http: //xxx.lanl.gov, math.AG/0107174.
[19]C. Weibel, An introduction to homological algebra, Cambridge University Pre*
*ss, 1994.