EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY
IOANID ROSU
Abstract.Equivariant elliptic cohomology with complex coefficients was d*
*efined axiomati-
cally by Ginzburg, Kapranov and Vasserot [9] and constructed by Grojnows*
*ki [10]. We give
an invariant definition of complex S1-equivariant elliptic cohomology, a*
*nd use it to give an
entirely cohomological proof of the rigidity theorem of Witten for the e*
*lliptic genus. We also
state and prove a rigidity theorem for families of elliptic genera.
Contents
1. Introduction 1
2. Statement of results *
* 2
3. S1-equivariant elliptic cohomology *
* 4
4. S1-equivariant elliptic pushforwards *
* 9
5. Rigidity of the elliptic genus *
* 14
Appendix A. Equivariant characteristic classes *
* 22
References 25
1.Introduction
The classical level 2 elliptic genus is defined (see Landweber [14], p.56) as*
* the Hirzebruch
genus with exponential series the Jacobi sine1. It is intimately related with *
*the mysterious
field of elliptic cohomology (see Segal [19]), and with string theory (see Witt*
*en [22] and [23]).
A striking property of the elliptic genus is its rigidity with respect to group*
* actions. This was
conjectured by Ochanine in [18], and by Witten in [22], where he used string th*
*eory arguments
to support it.
Rigorous mathematical proofs for the rigidity of the elliptic genus were soon*
* given by
Taubes [21], Bott & Taubes [4], and Liu [15]. While Bott and Taubes's proof inv*
*olved the lo-
calization formula in equivariant K-theory, Liu's proof focused on the modulari*
*ty properties of
the elliptic genus. The question remained however whether one could find a dire*
*ct connection
between the rigidity theorem and elliptic cohomology.
Earlier on, Atiyah and Hirzebruch [2] had used pushforwards in equivariant K-*
*theory to
prove the rigidity of the A^-genus for spin manifolds. Following this idea, H.*
* Miller [16] in-
terpreted the equivariant elliptic genus as a pushforward in the completed Bore*
*l equivariant
cohomology, and posed the problem of developing and using a noncompleted S1-equ*
*ivariant
elliptic cohomology, to prove the rigidity theorem.
In 1994 Grojnowski [10] proposed a noncompleted equivariant elliptic cohomolo*
*gy theory
with complex coefficients. For G a compact connected Lie group he defined E *G(*
*-) as a co-
herent holomorphic sheaf over a certain variety XG constructed from a given ell*
*iptic curve.
___________
1For a definition of the Jacobi sine s(x) see the beginning of Section 4.
1
2 IOANID ROSU
Grojnowski also constructed pushforwards in this theory. At about the same time*
* and inde-
pendently, Ginzburg, Kapranov and Vasserot [9] gave an axiomatic description of*
* equivariant
elliptic cohomology.
Given Grojnowski's construction, it seemed natural to try to use S1-equivaria*
*nt elliptic
cohomology to prove the rigidity theorem. In doing so, we noticed that our proo*
*f relies on a
generalization of Bott and Taubes' "transfer formula" (see [4]). This generaliz*
*ation turns out
to be essentially equivalent to the existence of a Thom class (or orientation) *
*in S1-equivariant
elliptic cohomology.
We can generalize the results of this paper in several directions. One is to*
* extend the
rigidity theorem to families of elliptic genera, which we do in Theorem 5.6. An*
*other would be
to generalize from G = S1 to an arbitrary connected compact Lie group, or to re*
*place complex
coefficients with rational coefficients for all cohomology theories involved. S*
*uch generalizations
will be treated elsewhere.
2.Statement of results
All the cohomology theories involved in this paper have complex coefficients.*
* If X is a
finite S1-CW complex, H*S1(X) denotes its Borel S1-equivariant cohomology with *
*complex
coefficients (see Atiyah and Bott [1]). If X is a point *, H*S1(*) ~=C[u].
Let E be an elliptic curve over C. Let X be a finite S1-CW complex, e.g. a co*
*mpact S1-
manifold2. Then, following Grojnowski [10], we define E *S1(X), the S1-equivar*
*iant elliptic
cohomology of X. This is a coherent analytic sheaf of Z2-graded algebras over E*
*. We alter
his definition slightly, in order to show that the definition of E *S1(X) depen*
*ds only on X and
the elliptic curve E. Let ff be a point of E. We associate a subgroup H(ff) of *
*S1 as follows:
if ff is a torsion point of E of exact order n, H(ff) = Zn; otherwise, H(ff) = *
*S1. We define
Xff= XH(ff), the subspace of X fixed by H(ff). Then we will define a sheaf E *S*
*1(X) over E
whose stalk at ff is
E *S1(X)ff= H*S1(Xff) C[u]OC,0.
Here OC,0represents the local ring of germs of holomorphic functions at zero on*
* C = SpecC[u].
In particular, the stalk of E *S1(X) at zero is H*S1(X) C[u]OC,0.
THEOREM A. E *S1(X) only depends on X and the elliptic curve E. It extends to *
*an
S1-equivariant cohomology theory with values in the category of coherent analyt*
*ic sheaves of
Z2-graded algebras over E.
If f : X ! Y is a complex oriented map between compact S1-manifolds, Grojnow*
*ski
also defines equivariant elliptic pushforwards. They are maps of sheaves of OE-*
*modules fE!:
E *S1(X)[f]! E *S1(Y ) satisfying properties similar to those of the usual push*
*forward (see
Dyer [7]). E *S1(X)[f]has the same stalks as E *S1(X), but the gluing maps are *
*different.
If Y is a point, then fE!(1) on the stalks at zero is the S1-equivariant ell*
*iptic genus of
X (which is a power series in u). By analyzing in detail the construction of fE*
*!, we obtain
the following interesting result, which answers a question posed by H. Miller a*
*nd answered
independently by Dessai and Jung [6].
PROPOSITION B. The S1-equivariant elliptic genus of a compact S1-manifold is th*
*e Tay-
lor expansion at zero of a function on C which is holomorphic at zero and merom*
*orphic
everywhere.
___________
2A compact S1-manifold always has an S1-CW complex structure: see Alday and P*
*uppe [3].
EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 3
Grojnowski's construction raises a few natural questions. First, can we say *
*more about
E *S1(X)[f]? The answer is given in Proposition 5.7, where we show that, up to *
*an invertible
sheaf, E *S1(X)[f]is the S1-equivariant elliptic cohomology of the Thom space o*
*f the stable
normal bundle to f. (In fact, if we enlarge our category of equivariant CW -com*
*plexes to in-
clude equivariant spectra, we can show that E*S1(X)[f]is the reduced E*S1of a T*
*hom spectrum
X-Tf . See the discussion after Proposition 5.7 for details.)
This suggests looking for a Thom section (orientation) in E *S1(X)[f]. More g*
*enerally, given
a real oriented vector bundle V ! X, we can twist E*S1(X) in a similar way to o*
*btain a sheaf,
which we denote by E*S1(X)[V.]For the rest of this section we regard all the sh*
*eaves not on E,
but on a double cover ~Eof E. The reason for this is given in the beginning of *
*Subsection 5.2.
So when does a Thom section exist in E *S1(X)[V?]The answer is the following ke*
*y result.
THEOREM C. If V ! X is a spin S1-vector bundle over a finite S1-CW complex, th*
*en
the element 1 in the stalk of E*S1(X)[V ]at zero extends to a global section, c*
*alled the Thom
section.
The proof of Theorem C is essentially a generalization of Bott and Taubes' "t*
*ransfer for-
mula" (see [4]). Indeed, when we try to extend 1 to a global section, we see t*
*hat the only
points where we encounter difficulties are certain torsion points of E which we*
* call special
(as defined in the beginnning of Section 3). But extending our section at a spe*
*cial point ff
amounts to lifting a class from H*S1(XS1) C[u]OC,0to H*S1(Xff) C[u]OC,0via th*
*e restriction
map i* : H*S1(Xff) C[u]OC,0! H*S1(XS1) C[u]OC,0. This is not a problem, exce*
*pt when
we have two different connected components of XS1 inside one connected componen*
*t of Xff.
Then the two natural lifts differ up to a sign, which can be shown to disappear*
* if V is spin.
This observation is due to Bott and Taubes, and is the centerpiece of their "tr*
*ansfer formula."
Given Theorem C, the rigidity theorem of Witten follows easily: Let X be a c*
*ompact
spin S1-manifold. Then the S1-equivariant pushforward of f : X ! * is a map of*
* sheaves
fE!: E*S1(X)[f]! E*S1(*). From the discussion after Theorem A, we know that on *
*the stalks
at zero fE!(1) is the S1-equivariant elliptic genus of X, which is a priori a p*
*ower series in u.
Theorem C with V = T X says that 1 extends to a global section in E*S1(X)[f]= E*
**S1(X)[TX].
Therefore fE!(1) is the germ of a global section in E *S1(*) = OE. But any such*
* section is a
constant, so the S1-equivariant elliptic genus of X is a constant. This proves *
*the rigidity of
the elliptic genus (Corollary 5.5).
Now the greater level of generality of Theorem C allows us to extend the rigi*
*dity theorem
to families of elliptic genera. The question of stating and proving such a theo*
*rem was posed
by H. Miller in [17].
THEOREM D. (Rigidity for families) Let ß : E ! B be a spin oriented S1-equivari*
*ant
fibration. Then the elliptic genus of the family ßE!(1) is constant as a ration*
*al function, i.e.
when the generator u of H*S1(*) = C[u] is inverted.
4 IOANID ROSU
3. S1-equivariant elliptic cohomology
In this section we give the construction of S1-equivariant elliptic cohomolog*
*y with complex
coefficients. But in order to set up this functor, we need a few definitions.
3.1. Definitions.
Let E be an elliptic curve over C with structure sheaf OE. Let ` be a uniform*
*izer of E, i.e.
a generator of the maximal ideal of the local ring at zero OE,0. We say that ` *
*is an additive
uniformizer if for all x, y 2 V` such that x + y 2 V`, we have `(x + y) = `(x) *
*+ `(y). An
additive uniformizer always exists, because we can take for example ` to be the*
* local inverse
of the group map C ! E, where the universal cover of E is identified with C. No*
*tice that any
two additive uniformizers differ by a nonzero constant, because the only additi*
*ve continuous
functions on C are multiplications by a constant.
Let V` be a neighborhood of zero in E such that ` : V` ! C is a homeomorphism*
* on its
image. Denote by tfftranslation by ff on E. We say that a neighborhood V of ff *
*2 E is small
if t-ff(V ) V`.
Let ff 2 E. We say that ff is a torsion point of E if there exists n > 0 such*
* that nff = 0.
The smallest n with this property is called the exact order of ff.
Let X be a finite S1-CW complex. If H S1 is a subgroup, denote by XH the su*
*bmanifold
of X fixed by each element of H. Let Zn S1 be the cyclic subgroup of order n*
*. Define
a subgroup H(ff) of S1 by: H(ff) = Zn if ff is a torsion point of exact order n*
*; H(ff) = S1
otherwise. Then denote by
Xff= XH(ff).
Now suppose we are given an S1-equivariant map of S1-CW complexes f : X ! Y .*
* A
point ff 2 E is called special with respect to f if either Xff6= XS1 or Y ff6= *
*Y S1. When it is
clear what f is, we simply call ff special. A point ff 2 E is called special wi*
*th respect to X if
it is special with respect to the identity function id : X ! X.
An indexed open cover U = (Uff)ff2Eof E is said to be adapted (with respect t*
*o f) if it
satisfies the following conditions:
1. Uffis a small open neighborhood of ff;
2. If ff is not special, then Uffcontains no special point;
3. If ff 6= ff0are special points, Uff\ Uff0= ;.
Notice that, if X and Y are finite S1-CW complexes, then there exists an open c*
*over of E
which is adapted to f. Indeed, the set of special points is a finite subset of *
*E.
If X is a finite S1-CW complex, we define the holomorphic S1-equivariant coho*
*mology of
X to be
HO*S1(X) = H*S1(X) C[u]OC,0.
OC,0is the ring of germs of holomorphic functions at zero in the variable u, or*
* alternatively it
is the subring of C[[u]] of convergent power series with positive radius of con*
*vergence.
Notice that HO*S1is not Z-graded anymore, because we tensored with the inhomo*
*genous
object OC,0. However, it is Z2-graded, by the even and odd part, because C[u] a*
*nd OC,0are
concentrated in even degrees.
3.2. Construction of E *S1
We are going to define now a sheaf F = F`,Uover E whose stalk at ff 2 E is is*
*omorphic to
HO*S1(Xff). Recall that, in order to give a sheaf F over a topological space, i*
*t is enough to
give an open cover (Uff)ffof that space, and a sheaf Fffon each Ufftogether wit*
*h isomorphisms
EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 5
of sheaves OEfffi: Fff|Uff\Ufi-! Ffi|Uff\Ufi, such that OEffffis the identity f*
*unction, and the cocycle
condition OEfiflOEfffi= OEffflis satisfied on Uff\ Ufi\ Ufl.
Fix ` an additive uniformizer of E. Consider an adapted open cover U = (Uff)f*
*f2E.
Definition 3.1. Define a sheaf Fffon Uffby declaring for any open U Uff
Fff(U) := H*S1(Xff) C[u]OE(U - ff) .
The map C[u] ! OE(U - ff) is given by sending u to ` (the germ ` extends to U -*
* ff because
Uffis small). U - ff represents the translation of U by -ff, and OE(U - ff) is*
* the ring of
holomorphic functions on U - ff. The restriction maps of the sheaf are defined *
*so that they
come from those of the sheaf OE.
First we notice that we can make Fffinto a sheaf of OE |Uff-modules: if U U*
*ff, we want
an action of f 2 OE(U) on Fff(U). The translation map tff: U - ff ! U, which ta*
*kes u to
u + ff gives a translation t*ff: OE(U) ! OE(U - ff), which takes f(u) to f(u + *
*ff). Then we
take the result of the action of f 2 OE(U) on ~ g 2 Fff(U) = H*S1(Xff) C[u]O*
*E(U - ff) to be
~ (t*fff . g). Moreover, Fffis coherent because H*S1(Xff) is a finitely gener*
*ated C[u]-module.
Now for the second part of the definition of F, we have to glue the different*
* sheaves Fff
we have just constructed. If Uff\ Ufi6= ; we need to define an isomorphism of *
*sheaves
OEfffi: Fff|Uff\Ufi-! Ffi|Uff\Ufiwhich satisfies the cocycle condition. Recall *
*that we started with
an adapted open cover (Uff)ff2E. Because of the condition 3 in the definition o*
*f an adapted
cover, ff and fi cannot be both special, so we only have to define OEfffiwhen, *
*say, fi is not
special. In that case Xfi= XS1. Consider an arbitrary open set U Uff\ Ufi.
Definition 3.2. Define OEfffias the composite of the following maps:
Fff(U) = H*S1(Xff) C[u]OE(U - ff)
! H*S1(Xfi) C[u]OE(U - ff)
! (H*(Xfi) C C[u]) C[u]OE(U - ff)
*(Xfi) O (U - ff)
(*) ! H* fi C E
! H (X ) C OE(U - fi)
! (H*(Xfi) C C[u]) C[u]OE(U - fi)
! H*S1(Xfi) C[u]OE(U - fi)
= Ffi(U) .
The map on the second row is the natural map i* 1, where i : Xfi! Xffis the i*
*nclusion.
Lemma 3.3. OEfffiis an isomorphism.
Proof.The second and and the sixth maps are isomorphisms because Xfi= XS1, and *
*therefore
H*S1(Xfi) -~!H*(Xfi) C C[u]. The properties of the tensor product imply that t*
*he third and
the fifth maps are isomorphisms. The fourth map comes from translation by fi - *
*ff, so it is
also an isomorphism.
Finally, the second map i* 1 is an isomorphism because
a) If ff is not special, then Xff= XS1 = Xfi, so i* 1 is the identity.
b) If ff is special, then Xff6= Xfi. However, we have (Xff)S1 = XS1 = Xfi. *
*Then we can
use the Atiyah-Bott localization theorem in equivariant cohomology from *
*[1]. This
says that i* : H*S1(Xff) ! H*S1(Xfi) is an isomorphism after inverting u*
*. So it is
enough to show that ` is invertible in OE(U - ff), because this would im*
*ply that i*
becomes an isomorphism after tensoring with OE(U - ff) over C[u]. Now, b*
*ecause ff
is special, the condition 2 in the definition of an adapted cover says t*
*hat ff =2Ufi. But
6 IOANID ROSU
U Uff\ Ufi, so ff =2U, hence 0 =2U - ff. This is equivalent to ` being*
* invertible in
OE(U - ff).
Remark 3.4. To simplify notation, we can describe OEfffias the composite of the*
* following
two maps:
* * fi t*fi-ff* fi
H*S1(Xff) C[u]OE(U - ff) -i!HS1(X ) C[u]OE(U - ff) -! HS1(X ) C[u]OE(U -*
* fi) .
By the first map we really mean i* 1. The second map is not 1 t*fi-ff, becau*
*se t*fi-ffis not a
map of C[u]-modules. However, we use t*fi-ffas a shorthand for the correspondin*
*g composite
map specified in (*). Note that OEfffiis linear over OE(U), so we get a map of*
* sheaves of
Z2-graded OE(U)-algebras.
One checks easily now that OEfffisatisfies the cocycle condition: Suppose we *
*have three open
sets Uff, Ufiand Uflsuch that Uff\ Ufi\ Ufl6= ;. Because our cover was chosen t*
*o be adapted,
at least two out of the three spaces Xff, Xfiand Xflare equal to XS1. Thus the*
* cocycle
condition reduces essentially to t*fl-fit*fi-ff= t*fl-ff, which is clearly true.
Definition 3.5. Let U = (Uff)ff2Ebe an adapted cover of E, and ` an additive un*
*iformizer.
We define a sheaf F = F`,Uon E by gluing the sheaves Ffffrom Definition 3.1 via*
* the gluing
maps OEfffidefined in 3.2.
One can check now easily that F is a coherent analytic sheaf of algebras.
Notice that we can remove the dependence of F on the adapted cover U as follo*
*ws: Let
U and V be two covers adapted to (X, A). Then any common refinement W is going *
*to be
adapted as well, and the corresponding maps of sheaves F`,U! F`,W F`,Vare iso*
*morphisms
on stalks, hence isomorphisms of sheaves. Therefore we can omit the subscript U*
*, and write
F = F`. Next we want to show that F` is independent of the choice of the additi*
*ve uniformizer
`.
Proposition 3.6. If ` and `0are two additive uniformizers, then there exists an*
* isomorphism
of sheaves of OE-algebras f``0: F` ! F`0. If `00is a third additive uniformizer*
*, then f`0`00f``0=
f``00.
Proof.We modify slightly the notations used in Definition 3.1 to indicate the d*
*ependence
on `: F`ff(U) = H*S1(Xff) `C[u]OE(U - ff). Recall that u is sent to ` via the*
* algebra map
C[u] ! OE(U - ff). If `0 is another additive uniformizer, we saw at the beggin*
*ing of this
Section that there exists a nonzero constant a in OE,0such that ` = a`0. Choose*
* a square root
of a and denote it by a1=2. Define a map f``0,ff: F`ff(U) ! F`0ff(U) by x ` g *
*7! a|x|=2x `0g.
We have assumed that x is homogeneous in H*S1(Xff), and that |x| is the homogen*
*eous degree
of x. *
* 0
One can easily check that f``0,ffis a map of sheaves of OE-algebras. We also*
* have OE`fffiO
f``0,ff= f``0,fiO OE`fffi, which means that the maps f``0,ffglue to define a ma*
*p of sheaves f``0:
F` ! F`0. The equality f`0`00f``0= f``00comes from (`0=`00)1=2(`=`0)1=2= (`=`*
*00)1=2.
Definition 3.7. The S1-equivariant elliptic cohomology of the finite S1-CW comp*
*lex X is
the sheaf F = F`,Uconstructed above, which according to the previous results do*
*es not depend
on the adapted open cover U or on the additive uniformizer `. Denote this sheaf*
* by E*S1(X).
If X is a point, one can see that E *S1(X) is the structure sheaf OE.
EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 7
Theorem 3.8. E *S1(-) defines an S1-equivariant cohomology theory with values i*
*n the cate-
gory of coherent analytic sheaves of Z2-graded OE-algebras.
Proof.For E *S1(-) to be a cohomology theory, we need naturality. Let f : X ! *
*Y be
an S1-equivariant map of finite S1-CW complexes. We want to define a map of sh*
*eaves
f* : E *S1(Y ) ! E *S1(X) with the properties that 1*X= 1E*S1(X)and (fg)* = g*f*
**. Choose U
an open cover adapted to f, and ` an additive uniformizer of E. Since f is S1-e*
*quivariant,
for each ff we get by restriction a map fff: Xff! Y ff. This induces a map H*S1*
*(Y ff) C[u]
f*ff 1* ff *
OE(U - ff) -! HS1(X ) C[u]OE(U - ff). To get our global map f , we only have *
*to check
that f*ffglue well, i.e. that they commute with the gluing maps OEfffi. This f*
*ollows easily
from the naturality of ordinary equivariant cohomology, and from the naturality*
* in X of the
isomorphism H*S1(XS1) ~=H*(XS1) C C[u].
Also, we need to define E *S1for pairs. Let (X, A) be a pair of finite S1-CW *
*complexes, i.e.
A is a closed subspace of X, and the inclusion map A ! X is S1-equivariant. We *
*then define
E *S1(X, A) as the kernel of the map j* : E *S1(X=A) ! E *S1(*), where j : * = *
*A=A ! X=A
is the inclusion map. If f : (X, A) ! (Y, B) is a map of pairs of finite S1-CW*
* complexes,
then f* : E*S1(Y, B) ! E*S1(X, A) is defined as the unique map induced on the c*
*orresponding
kernels from f* : E*S1(Y ) ! E*S1(X).
Now we have to define the coboundary map ffi : E *S1(A) ! E *+1S1(X, A). This*
* is obtained
by gluing the maps H*S1(Aff) C[u]OE(U - ff) ffiff-1!H*+1S1(Xff, Aff) C[u]OE(U*
* - ff), where
ffiff: H*S1(Aff) ! H*+1S1(Xff, Aff) is the usual coboundary map. The maps ffif*
*f 1 glue well,
because ffiffis natural.
To check the usual axioms of a cohomology theory: naturality, exact sequence *
*of a pair, and
excision for E *S1(-), recall that this sheaf was obtained by gluing the sheave*
*s Fffalong the
maps OEfffi. Since Fffwere defined using H*S1(Xff), the properties of ordinary *
*S1-equivariant
cohomology pass on to E *S1(-), as long as tensoring with OE(U - ff) over C[u] *
*preserves
exactness. But this is a classical fact: see for example the appendix of Serre *
*[20].
This proves THEOREM A stated in Section 2.
Remark 3.9. Notice that we can arrange our functor E*S1(-) to take values in th*
*e category of
coherent algebraic sheaves over E rather than in the category of coherent analy*
*tic sheaves. This
follows from a theorem of Serre [20] which says that the the categories of cohe*
*rent holomorphic
sheaves and coherent algebraic sheaves over a projective variety are equivalent.
3.3. Alternative description of E *S1
For calculations with E*S1(-) we want a description which involves a finite o*
*pen cover of E.
Start with an adapted open cover (Uff)ff2E. Recall that the set of special poin*
*ts with respect
to X is finite. Denote this set by {ff1, . .,.ffn}. To simplify notation, denot*
*e for i = 1, . .,.n
Ui:= Uffi, and U0 := E \ {ff1, . .,.ffn} .
On each Ui, with 0 i n, we define a sheaf G as follows:
a) If 1 i n, then 8U Ui, Gi(U) := H*S1(Xffi) C[u]OE(U - ffi). The ma*
*p C[u] !
OE(U - ffi) was described in Definition 3.1.
b) If i = 0, then 8U U0, Gi(U) := H*(XS1) C OE(U).
8 IOANID ROSU
Now glue each Gi to G0 via the map of sheaves OEi0defined as the composite of t*
*he following
* 1 1 *
* ~=
isomomorphisms (U Ui\U0): H*S1(Xffi) C[u]OE(U -ffi) i-!H*S1(XS ) C[u]OE(U -ff*
*i) -!
t*-ffi 1
H*(XS1) C OE(U - ffi) -! H*(XS ) C OE(U).
Since there cannot be three distinct Ui with nonempty intersection, there is *
*no cocycle
condition to verify.
Proposition 3.10. The sheaf G we have just described is isomorphic to F, thus a*
*llowing an
alternative definition of E*S1(X).
Proof.One notices that U0 = [{Ufi| fi nonspecial}, because of the third conditi*
*on in the
definition of an adapted cover. If U [fiUfi, a global section in F(U) is a co*
*llection of sections
sfi2 F(U \ Ufi- fi) which glue, i.e. t*fi-fi0sfi= sfi0. So t*-fisfi= t*-fi0sfi0*
*in G(U \ Ufi\ Ufi0),
which means that we get an element in G(U), since the Ufi's cover U. So F|U0~=G*
*|U0. But
clearly F|Ui~=G|Uifor 1 i n, and the gluing maps are compatible. Therefore *
*F ~=G.
As it is the case with any coherent sheaf of OE-modules over an elliptic curv*
*e, E*S1(X) splits
(noncanonically) into a direct sum of a locally free sheaf, i.e. the sheaf of s*
*ections of some
holomorphic vector bundle, and a sum of skyscraper sheaves.
Given a particular X, we can be more specific: We know that H*S1(X) splits no*
*ncanonically
into a free and a torsion C[u]-module. Given such a splitting, we can speak of *
*the free part
of H*S1(X). Denote it by H*S1(X)free. The map i*H*S1(X)free! H*S1(XS1) is an in*
*jection
of finitely generated free C[u]-modules of the same rank, by the localization t*
*heorem. C[u] is
a p.i.d., so by choosing appropriate bases in H*S1(X)freeand H*S1(XS1), the map*
* i* can be
written as a diagonal matrix D(un1, . .,.unk), ni 0. Since i*1 = 1, we can cho*
*ose n1 = 0.
So at the special points ffi, we have the map i* : H*S1(Xffi)free! H*S1(XS1),*
* which in
appropriate bases can be written as a diagonal matrix D(1, un2, . .,.unk). Thi*
*s gives over
Ui\ U0 the transition functions u 7! D(1, un2, . .,.unk) 2 GL(n, C). However, w*
*e have to be
careful since the basis of H*S1(XS1) changes with each ffi, which means that th*
*e transition
functions are diagonal only up to a (change of base) matrix. But this matrix is*
* invertible over
C[u], so we get that the free part of E *S1(X) is a sheaf of sections of a holo*
*morphic vector
bundle.
An interesting question is what holomorphic vector bundles one gets if X vari*
*es. Recall
that holomorphic vector bundles over elliptic curves were classified by Atiyah *
*in 1957.
Example 3.11. Calculate E *S1(X) for X = S2(n) = the 2-sphere with the S1-actio*
*n which
rotates S2 n times around the north-south axis as we go once around S1. If ff i*
*s an n-torsion
point, then Xff= X. Otherwise, Xff= XS1, which consists of two points: {P+, P-*
*}, the
North and the South poles. Now H*S1(S2(n)) = H*(BS1 _ BS1) = C[u] xC C[u], on w*
*hich
C[u] acts diagonally. i* : H*S1(X) ! H*S1(XS1) is the inclusion C[u] xC C[u] ,!*
* C[u] x C[u].
Choose the bases
a) {(1, 1), (u, 0)} of C[u] xC C[u];
b) {(1, 1), (1, 0)} of C[u] x C[u].
Then H*S1(X) -~! C[u] C[u] by (P (u), Q(u)) 7! (P, Q-P_u), and H*S1(XS1) -~! *
*C[u] C[u]
by (P (u), Q(u)) 7! (P, Q - P ). Hence i* is given by the diagonal matrix D(1, *
*u). So E *S1(X)
looks locally like OCP1 OCP1(-1 . 0). This happens at all the n-torsion poin*
*ts of E, so
E *S1(X) ~=OE OE( ), where is the divisor which consists of all n-torsion p*
*oints of E, with
multiplicity 1.
EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 9
One can also check that the sum of all n-torsion points is zero, so by Abel's*
* theorem the
divisor is linearly equivalent to -n2 . 0. Thus E *S1(S2(n)) ~=OE OE(-n2 . *
*0). We stress
that the decomposition is only true as sheaves of OE-modules, not as sheaves of*
* OE-algebras.
Remark 3.12. Notice that S2(n) is the Thom space of the S1-vector space C(n), w*
*here z acts
on C by complex multiplication with zn. This means that the Thom isomorphism do*
*esn't hold
in S1-equivariant elliptic cohomology, because E*S1(*) = OE, while the reduced *
*S1-equivariant
* 2 2
elliptic cohomology of the Thom space is ~ES1(S (n)) = OE(-n . 0).
4.S1-equivariant elliptic pushforwards
While the construction of E *S1(X) depends only on the elliptic curve E, the *
*construction
of the elliptic pushforward fE!involves extra choices, namely that of a torsion*
* point of exact
order two on E, and a trivialization of the cotangent space of E at zero.
4.1. The Jacobi sine
Let (E, P, ~) be a triple formed with a nonsingular elliptic curve E over C, *
*a torsion point
P on E of exact order two, and a 1-form ~ which generates the cotangent space T*
*0*E. For
example, we can take E = C= , with = Z!1+ Z!2 a lattice in C, P = !1=2, and ~*
* = dz at
zero, where z is the usual complex coordinate on C.
As in Hirzerbruch, Berger and Jung ([12], Section 2.2), we can associate to t*
*his data a
function s(z) on C which is elliptic (doubly periodic) with respect to a sublat*
*tice ~ of index 2
in , namely ~ = Z!1+ 2Z!2. (This leads to a double covering ~E! E, and s can b*
*e regarded
as a rational function on the öd ubled" elliptic curve ~E.) Indeed, we can def*
*ine s up to a
constant by defining its divisor to be
D = (0) + (!1=2) - (!2) - (!1=2 + !2) .
Then we can make s unique by requiring that ds = dz at zero. We call this s th*
*e Jacobi
sine. It has the following properties (see [12]):
Proposition 4.1.
a) s(z) is odd, i.e. s(-z) = -s(z). Around zero, s can be expanded as a po*
*wer series
s(z) = z + a3z3 + a5z5 + . ...
b) s(z + !1) = s(z); s(z + !2) = -s(z).
c) s(z + !1=2) = a=s(z), a 6= 0 (this follows by looking at the divisor of *
*s(z + !1=2)).
We now show that the construction of s is canonical, i.e. it does not depend *
*on the identi-
fication E ~=C= .
Proposition 4.2. The definition of s only depends on the triple (E, P, ~).
Proof.First, we show that the construction of ~E= C=~ is canonical: Let E ~=C= *
*0be another
identification of E. We then have 0= Z!01+ Z!02, and P is identified with !01=*
*2. Since E is
also identified with C= , we get a group map ~ : C= -~!C= 0. This implies that*
* we have a
continuous group map ~ : C -~!C such that ~( ) = 0. Any such map must be multi*
*plication
by a nonzero constant ~ 2 C. Moreover, we know that ~!1=2 = !01=2. This implies*
* ~!1 = !01,
and since ~ takes isomorphically onto 0, it follows that ~!2 = !02+ m!01for*
* some integer
m. Multiplying this by 2, we get ~ . 2!2 = 2!02+ 2m!01. This, together with ~!*
*1 = !01, imply
that multiplication by ~ descends to a group map C=~ -~! C= ~0. But this precis*
*ely means
that the construction of ~Eis canonical.
10 IOANID ROSU
Notice that P can be thought canonically as a point on the öd ubled" ellptic *
*curve ~E. We
denote by P1 and P2 the other two points of exact order 2 on ~E. Then we form t*
*he divisor
D = (0) + (P ) - (P1) - (P2) .
Although the choice of P1 and P2 is noncanonical, the divisor D is canonical, i*
*.e. depends
only on P . Let s be an elliptic function on ~Eassociated to the divisor D. The*
* choice of s is
well-defined up to a constant which can be fixed if we require that ds = ß*~ at*
* zero, where
ß : ~E! E is the projection map.
Next, we start the construction of S1-equivariant elliptic pushforwards. Let *
*f : X ! Y be
an equivariant map between compact S1-manifolds such that the restrictions f : *
*Xff! Y ff
are oriented maps. Then we follow Grojnowski [10] and define the pushforward of*
* f to be a
map of sheaves fE!: E *S1(X)[f]! E *S1(Y ), where E *S1(X)[f]is the sheaf E *S1*
*(X) twisted by
a 1-cocycle to be defined later.
The main technical ingredient in the construction of the (global i.e. sheafwi*
*se) elliptic
pushforward fE!: E *S1(X)[f]! E *S1(Y ) ,is the (local i.e. stalkwise) elliptic*
* pushforward fE!:
HO*S1(Xff) ! HO*S1(Y ff).
In the following subsection, we construct elliptic Thom classes and elliptic *
*pushforwards in
HO*S1(-). The construction is standard, with the only problem that in order to *
*show that
something belongs to HO*S1(-), we need some holomorphicity results on character*
*istic classes.
4.2. Preliminaries on pushforwards
Let ß : V ! X be a 2n-dimensional oriented real S1-vector bundle over a finit*
*e S1-CW
complex X, i.e. a vector bundle with a linear action of S1, such that ß commute*
*s with the
S1 action. Now, for any space A with an S1 action, we can define its Borel con*
*struction
A xS1 ES1, where ES1 is the universal principal S1-bundle. This construction is*
* functorial,
so we get a vector bundle VS1 over XS1. This has a classifying map fV : XS1 ! B*
*SO(2n).
If Vunivis the universal orientable vector bundle over BSO(2n), we also have a *
*map of pairs,
also denoted by fV : (DVS1, SVS1) ! (DVuniv, SVuniv). As usual, DV and SV repre*
*sent the
disc and the sphere bundle of V , respectively.
But it is known that the pair (DVuniv, SVuniv) is homotopic to (BSO(2n), BSO(*
*2n - 1)).
Also, we know that H*BSO(2n) = C[p1, . .,.pn, e]=(e2 - pn), where pj is the uni*
*versal j'th
Pontrjagin class, and e is the universal Euler class. From the long exact seque*
*nce of the pair,
it follows that H*(BSO(2n), BSO(2n - 1)) can be regarded as the ideal generated*
* by e in
H*BSO(2n). The class e 2 H*(DVuniv, SVuniv) is the universal Thom class, which *
*we will
denote by OEuniv. Then the ordinary equivariant Thom class of V is defined as t*
*he pullback
class f*VOEuniv2 H*S1(DV, SV ), and we denote it by OES1(V ). Denote by H**S1(X*
*) the completion
of the module H*S1(X) with respect to the ideal generated by u in H*(BS1) = C[u*
*].
Consider the power series Q(x) = s(x)=x, where s(x) is the Jacobi sine. Since*
* Q(x) is even,
Definition A.8 gives a class ~Q (V )S1 2 H**S1(X). Then we define a class in H**
**S1(DV, SV ) by
OEES1(V ) = ~Q (V )S1. OES1(V ). One can also say that OEES1(V ) = s(x1) . .s.(*
*xn), while OES1(V ) =
x1. .x.n, where x1, . .,.xn are the equivariant Chern roots of V . We call OEES*
*1(V ) the elliptic
equivariant Thom class of V .
Also, we define eES1(V ), the equivariant elliptic Euler class of V , as the *
*image of OEES1(V ) via
the restriction map H**S1(DV, SV ) ! H**S1(X).
EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 11
Proposition 4.3. If V ! X is an even dimensional real oriented S1-vector bundle*
*, and X is
a finite S1-CW complex, then OEES1(V ) actually lies in HO*S1(DV, SV ). Cup pro*
*duct with the
elliptic Thom class
[ffiES1(V )
HO*S1(X) HO*S1(DV, SV ) ,
is an isomorphism, the Thom isomorphism in HO-theory.
Proof.The difficult part, namely that ~Q (V )S1 is holomorphic, is proved in th*
*e Appen-
dix, in Proposition A.6. Consider the usual cup product, which is a map [ : H**
*S1(X)
H*S1(DV, SV ) ! H*S1(DV, SV ), and extend it by tensoring with OC,0over C[u]. W*
*e obtain a
map [ : HO*S1(X) HO*S1(DV, SV ) ! HO*S1(DV, SV ). The equivariant elliptic Th*
*om class
of V is OEES1(V ) = ~Q (V )S1[ OES1(V ), so we have to show that both these cla*
*sses are holomor-
phic. But by Proposition A.6 in the Appendix, ~Q (V )S1 2 HO*S1(X). And the ord*
*inary Thom
class OES1(V ) belongs to H*S1(DV, SV ), so it also belongs to the larger ring *
*HO*S1(DV, SV ).
Now, cup product with OEES1(V ) gives an isomorphism because Q(x) = s(x)=x is*
* an invertible
power series around zero.
Corollary 4.4. If f : X ! Y is an S1-equivariant oriented map between compac*
*t S1-
manifolds, then there is an elliptic pushforward
fE!: HO*S1(X) ! HO*S1(Y ) ,
which is a map of HO*S1(Y )-modules. In the case when Y is a point, fE!(1) is*
* the S1-
equivariant elliptic genus of X.
Proof.Recall (Dyer [7]) that the ordinary pushforward is defined as the composi*
*tion of three
maps, two of which are Thom isomorphisms, and the third is a natural one. The e*
*xistence of
the elliptic pushforward follows therefore from the previous corollary. The pro*
*of that fE!is a
map of HO*S1(Y )-modules is the same as for the ordinary pushforward.
The last statement is an easy consequence of the topological Riemann-Roch the*
*orem (see
again [7]), and of the definition of the equivariant elliptic Thom class.
Notice that, if Y is point, HO*S1(Y ) ~=OC,0, so the S1-equivariant elliptic *
*genus of X is
holomorphic around zero. Also, if we replace HO*S1(-) = H*S1(-) C[u]OC,0by HM**
*S1(-) =
H*S1(-) C[u]M(C), where M(C) is the ring of global meromorphic functions on C,*
* the same
proof as above shows that the S1-equivariant elliptic genus of X is meromorphic*
* in C. This
proves the following result, which is PROPOSITION B stated in Section 2.
Proposition 4.5. The S1-equivariant elliptic genus of a compact S1-manifold is *
*the Taylor
expansion at zero of a function on C which is holomorphic at zero and meromorph*
*ic every-
where.
4.3. Construction of fE!
The local construction of elliptic pushforwards is completed. We want now to *
*assemble the
pushforwards in a map of sheaves. Let f : X ! Y be a map of compact S1-manifold*
*s which
commutes with the S1-action. We assume that either f is complex oriented or spi*
*n oriented,
i.e. that the stable normal bundle in the sense of Dyer [7] is complex oriented*
* or spin oriented,
respectively. (Grojnowski treats only the complex oriented case, but in order t*
*o understand
rigidity we also need the spin case.)
Let U be an open cover of E adapted to f. Let ff, fi 2 E be such that Uff\ Uf*
*i6= ;. This
implies that at least one point, say fi, is nonspecial, so Xfi= XS1 and Y fi= Y*
* S1. We specify
12 IOANID ROSU
now the orientations of the maps and vector bundles involved. Since Xfi= XS1, t*
*he normal
bundle of the embedding Xfi! Xffhas a complex structure, where all the weights *
*of the
S1-action on V are positive.
If f is complex oriented, it follows that the restriction maps fff: Xff! Y ff*
*and ffi:
Xfi! Y fiare also complex oriented, hence oriented. If f is spin oriented, this*
* means that the
stable normal bundle W of f is spin. If H is any subgroup of S1, we know that t*
*he vector
bundle W H ! XH is oriented: If H = S1, W splits as a direct sum of W H with a*
* bundle
corresponding to the nontrivial irreducible representations of S1; this latter *
*bundle is complex,
hence oriented, so the orientation of W induces one on W H. If H = Zn, Lemma 10*
*.3 of Bott
and Taubes [4] implies that W His oriented. In conclusion, both maps fffand ffi*
*are oriented.
According to Corollary 4.4, we can define elliptic pushforwards at the level *
*of stalks:
(fff)E!: HO*S1(Xff) ! HO*S1(Y ff) and (ffi)E!: HO*S1(Xfi) ! HO*S1(Y fi). The p*
*roblem is
that pushforwards do not commute with pullbacks, i.e. if i : Xfi! Xffand j : Y *
*fi! Y ffare
the inclusions, then it is not true in general that j*(fff)E!= (ffi)E!i*. Howev*
*er, by twisting
the maps with some appropriate Euler classes, the diagram becomes commutative. *
*Denote by
eES1(Xff=Xfi) the S1-equivariant Euler class of the normal bundle to the embedd*
*ing i, and by
eES1(Y ff=Y fi) the S1-equivariant Euler class of the normal bundle to j. Denot*
*e by
~[f]fffi= eES1(Xff=Xfi)-1 . (ffi)*eES1(Y ff=Y fi) .
A priori ~[f]fffibelongs to the ring HO*S1(Xfi)[____1____eE], ffbfiut we will s*
*ee later that we can
S1(X =X )
improve this.
Lemma 4.6. In the ring HO*S1(Xfi)[_1u, ____1____eE] wffefihave the following fo*
*rmula
S1(X =X )
j*(fff)E!~ff= (ffi)E!(i*~ff. ~[f]fffi) ,
Proof.From the hypothesis, we know that i*iE!is an isomorphism, because it is m*
*ultiplication
by the invertible class eES1(Xff=Xfi). Also, since u is invertible, the localiz*
*ation theorem implies
that i* is an isomorphism. Therefore iE!is an isomorphism. Start with a class*
* ~ffon Xff.
Because iE!is an isomorphism, ~ffcan be written as iE!~fi, where ~fiis a class *
*on Xfi.
Now look at the two sides of the equation to be proved:
1. The left hand side = j*(fff)E!iE!~fi= j*jE!(ffi)E!~fi= (ffi)E!~fi. eES1(*
*Y ff=Y fi), because
j*jE!= multiplication by eE (Y ff=Y fi).
2. The right hand side = (ffi)E![i*iE!~fi. eES1(Xff=Xfi)-1 . (ffi)*eES1(Y f*
*f=Y fi)] = (ffi)E![~fi.
(ffi)*eES1(Y ff=Y fi)] = (ffi)E!~fi. eES1(Y ff=Y fi), where the last equ*
*ality comes from the
fact that (ffi)E!is a map of HO*S1(Y fi)-modules.
Let f : X ! Y be a complex or spin oriented S1-map. Let U be an open cover ad*
*apted to
f, and ff, fi 2 E such that Uff\ Ufi6= ;. We know that ff and fi cannot be both*
* special, so
assume fi nonspecial. Let U Uff\ Ufi. Since U is adapted, ff =2U.
EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 13
Proposition 4.7. With these hypotheses, ~[f]fffibelongs to H*S1(Xfi) C[u]OE(U *
*- fi), and the
following diagram is commutative:
(fff)E!
H*S1(Xff) C[u]OE(U - ff) H*S1(Y ff) C[u]OE(U - ff)
~[f]fffi.i* j*
(ffi)E!
H*S1(Xfi) C[u]OE(U - ff) H*S1(Y fi) C[u]OE(U - ff)
t*fi-ff t*fi-ff
(ffi)E!
H*S1(Xfi) C[u]OE(U - fi) H*S1(Y fi) C[u]OE(U - fi)
Proof.Denote by W the normal bundle of the embedding Xfi= XS1 ! Xff. Let us sh*
*ow
that, if ff =2U, then eES1(W ) is invertible in H*S1(Xfi) C[u]OE(U - ff). Den*
*ote by wi the
nonequivariant Chern roots of W , and by mi the corresponding rotation numbers *
*of W (see
Proposition A.4 in the Appendix). Since Xfi= XS1, mi6= 0. Also, the S1-equivari*
*ant Euler
class of W is given by
eS1(W ) = (w1 + m1u) . .(.wr+ mru) = m1. .m.r(u + w1=m1) . .(.u + wr=mr)*
* .
But wiare nilpotent, so eS1(W ) is invertible as long as u is invertible. Now f*
*f =2U translates
to 0 =2U - ff, which implies that the image of u via the map C[u] ! OE(U - ff) *
*is indeed
invertible. To deduce now that eES1(W ), the elliptic S1-equivariant Euler clas*
*s of W , is also
invertible, recall that eES1(W ) and eS1(W ) differ by a class defined using th*
*e power series
s(x)=x = 1 + a3x2 + a5x4 + . .,.which is invertible for U small enough.
So ~[f]fffiexists, and by the previous Lemma, the upper part of our diagram i*
*s commutative.
The lower part is trivially commutative.
Now, since i* are essentially the gluing maps in the sheaf F = E *S1(X), we t*
*hink of the
maps ~[f]fffi. i* as giving the sheaf F twisted by the cocycle ~[f]fffi. Recall*
* from Definition 3.5 that
F was obtained by gluing the sheaves Fffover an adapted open cover (Uff)ff2E.
Definition 4.8. The twisted gluing functions OE[f]fffiare defined as the compos*
*ition of the fol-
* 1 .~[f]fffi
lowing three maps H*S1(Xff) C[u]OE(U -ff) i-!H*S1(Xfi) C[u]OE(U -ff) -! H*S1(X*
*fi) C[u]
t*fi-ff
OE(U - fi) -! H*S1(Xfi) C[u]OE(U - fi). The third map is defined as in Remark*
* 3.4.
As in the discussion after Remark 3.4, we can show easily that OE[f]fffisatis*
*fy the cocycle
condition.
Definition 4.9. Let f : X ! Y be an equivariant map of compact S1-manifolds, su*
*ch that it
is either complex or spin oriented. We denote by E*S1(X)[f]the sheaf obtained b*
*y gluing the
sheaves Fffdefined in 3.1, using the twisted gluing functions OE[f]fffi.
Also, we define the S1-equivariant elliptic pushforward of f to be the map of*
* coherent sheaves
over E
fE!: E*S1(X)[f]! E*S1(Y )
which comes from gluing the local elliptic pushforwards (fff)E!(as defined in 4*
*.4). We call fE!
the Grojnowski pushforward.
14 IOANID ROSU
The fact that (fff)E!glue well comes from the commutativity of the diagram in*
* Proposi-
tion 4.7. The Grojnowski pushforward is functorial: see [9] and [10].
5. Rigidity of the elliptic genus
In this section we discuss the rigidity phenomenon in the context of equivari*
*ant elliptic
cohomology. We start with a discussion about orientations.
5.1. Preliminaries on orientations
Let V ! X be an even dimensional spin S1-vector bundle over a finite S1-CW co*
*mplex
X (which means that the S1-action preserves the spin structure). Let n 2 N. We *
*think of
Zn S1 as the ring of n'th roots of unity in C. The invariants of V under the *
*actions of S1
and Zn are the S1-vector bundles V S1!1XS1 and V Zn! XZn. We have XS1 XZn.
Let N be a connected component of XS , and P a connected component of XZn whi*
*ch
contains N. From now on we think of V S1as a bundle over N, and V Znas a bundle*
* over P .
Define the vector bundles V=V S1and V Zn=V S1over N by
1 S1 Zn S1 Z S1
V|N = V S V=V ; V|N = V V n=V .
The decompositions of these two bundles come from the fact that S1 acts trivial*
*ly on the base
N, so fibers decompose into a trivial and nontrivial part.
Similarly, the action of Zn on P is trivial, so we get a fiberwise decomposit*
*ion of V|Pby the
different representations of Zn:
M n
V|P= V Zn V (k) V (__) .
0 0 such that nff 2 (notice that tor*
*sion points are
defined in terms of , and not ~). The smallest such n is called the exact orde*
*r of ff. From
Proposition 4.1 b), we know that if a 2 , s(x + a) = s(x). Since nff 2 , def*
*ine ffl = 1 by
s(x + nff) = ffls(x) .
EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 17
Now E*S1(X)[V ]was obtained by gluing the sheaves Fffalong the adapted open c*
*over (Uff)ff.
So to give a global section ~ of E*S1(X)[V ]is the same as to give global secti*
*ons ~ffof Fffsuch
that they glue, i.e. OE[Vf]ffi~ff= ~fifor any ff and fi with Uff\ Ufi6= ;. From*
* Definition 5.1, to
give ~ is the same as to give ~ff2 HO*S1(Xff) so that t*fi-ff(i*~ff. eES1(V ff=*
*V fi)-1) = ~fi, or
i*~ff.eES1(V ff=V fi)-1 = t*fi-ff~fi(i the inclusion Xfi,! Xff). Because ~ is s*
*upposed to globalize
1, we know that ~0 = 1. This implies that ~fi= t*fieES1(V=V fi)-1 for fi in a s*
*mall neighborhood
of 0 2 C.
In fact, we can show that this formula for ~fiis valid for all fi 2 C, as lon*
*g as fi is not
special. This means we have to check that ~fi= t*fieES1(V=V fi)-1 exists in HO**
*S1(Xfi) as long
as fi is not special. fi not special means Xfi= XS1. Then consider the bundle V*
*=V S1. We
saw in the previous subsection that according to the splitting principle, when *
*pulled back on
the flag manifold, V=V S1decomposes into a direct sum of line bundles L(m1) .*
* . .L(mr),
where mj are the rotation numbers. The complex structure on L(m) is such g 2 S1*
* acts on
L(m) by complex multiplication with gm .
Since XS1 is fixed by the S1 action, we can apply Proposition A.4 in the Appe*
*ndix: Let xj
be the equivariant Chern root of L(mj), and wj its usual (nonequivariant) Chern*
*Qroot. Then
xjQ= wj + mju, with u the generator of H*(BS1). Therefore t*fieES1(V=V fi) = *
*jt*fis(xj) =
* Q Q
jtfis(wj+ mju) = js(wj+ mju + mjfi) = js(xj+ mjfi).
So we have r
1-1 Y -1
~fi= t*fieES1(V=V S) = s(xj+ mjfi) .
j=1
We show that ~fibelongs to HO*S1(Xfi) as long as s(mjfi) 6= 0 for all j = 1, . *
*.,.r: Since
V=V S1has only nonzero rotation numbers, it has a complex structure. But chang*
*ing the
orientations of a vector bundle only changes the sign of the corresponding Eule*
*r class, so in
the formula above we can assume that V=V S1has a complex structure, for example*
* the one
for which all mj > 0. We group the mj which are equal, i.e. for eachPm > 0 we d*
*efine the set
of indices Jm = {j | mj = m}. Now we get a decomposition3 V=V S1= m>0 W (m), *
*where
W (m) is the complex S1-vectorQbundle on which g 2 S1 acts by multiplication wi*
*th gm . Now
we have to show that j2Jms(xj+ mfi)-1 gives an element of HO*S1(Xfi). This wo*
*uld follow
from Proposition A.6 applied to the power series Q(x) = s(x + mfi)-1 and the ve*
*ctor bundle
W (m), provided that Q(x) is convergent. But s(x + mfi)-1 is indeed convergent,*
* since s is
meromorphic on C and does not have a zero at mfi.
Now we show that if fi is nonspecial, s(mjfi) 6= 0 for all j = 1, . .,.r: Sup*
*pose s(mjfi) = 0.
Then mjfi 2 , so fi is a torsion point, say of exact order n. It follows that *
*n divides mj,
which implies XZn 6= XS1. But Xfi= XZn, since fi has exact order n, so Xfi6= XS*
*1 i.e. fi is
special, contradiction.
So we only need to analyze what happens at a special point ff 2 C, say of *
*ex-
act order n. We have to find a class ~ff 2 HO*S1(Xff) such that OE[Vf]ffi~ff*
* = ~fi, i.e.
t*fi-ff(i*~ff. eES1(V ff=V fi)-1) = t*fieES1(V=V fi)-1. Equivalently, we want *
*a class ~ffsuch that
i*~ff= t*ffeES1(V=V fi)-1.eES1(V ff=V fi), i.e. we want to lift the class t*ffe*
*ES1(V=V fi)-1.eES1(V ff=V fi)
from HO*S1(Xfi) to HO*S1(Xff). If we can do that, we are done, because the clas*
*s (~ff)ff2Cis
a global section in E *S1(X)[V,]and it extends ~0 = 1 in the stalk at zero. So *
*it only remains
___________
3This decomposition takes place on XS1, while the decomposition into line bun*
*dles L(mj) takes place only
on the flag manifold.
18 IOANID ROSU
to prove the following lemma, which is a generalization of the transfer formula*
* of Bott and
Taubes.
Lemma 5.3. Let ff be a special point of exact order n, and V ! X a spin S1-vect*
*or bundle.
Let i : XS1 ! XZn be the inclusion map. Then there exists a class ~ff2 HO*S1(XZ*
*n) such
that 1 1
i*~ff= t*ffeES1(V=V S)-1 . eES1(V Zn=V S) .
Proof.We first study the class t*ffeES1(V=V S1)-1 . eES1(V Zn=V S1) on each con*
*nected component
of XS1 in XZn. We will see that it lifts naturally to a class on XZn. The probl*
*em arises from
the fact that we can have two connected components of XS1 inside one connected *
*component
of XZn, and in that case the two lifts will differ by a sign. We then show that*
* the sign vanishes
if V has a spin structure.
As in the previous subsection, let N be a connected component of XS1, and P a*
* connected
component of XZn which contains N.
We now calculate t*ffeES1(V=V S1)-1, regarded as a class on N. From the decom*
*position (3)
V=V S1= V Zn=V S1 V (K)|N V (n_2)|N and from the table, we get the following*
* formula:
1-1 ff E S1-1
t*ffeES1(V=V S)= (-1) . eS1(V=V )cx
Y Y Y
(5) = (-1)ff. s(xj+ m*jff)-1 . s(xj+ m*jff)-1 . s(xj+*
* m*jff)-1
j2I0 j2IK j2In=2
Before we analyze each term in the above formula, recall that we defined the *
*number ffl = 1
by s(x + nff) = ffls(x).
a) j 2 I0: Here we chose the complex structure (V Zn=V S1)cxsuch that all m*j*
*> 0. Then,
* Q -*
*1 P q*
since s(xj + m*jff) = s(xj + q*jnff) = fflqjs(xj), we have: j2I0s(xj + mjff) *
* = ffl I0j .
Q -1 P Iq*j E Zn S1-1 P Iq*j ff(0) E Zn S1-1
I0s(xj) = ffl 0 . eS1(V =V )cx = ffl 0 . (-1) . eS1(V =V )or*
*. So we get
eventually
Y P q* 1
(6) s(xj+ m*jff)-1 = ffl I0j. (-1)ff(0). eES1(V Zn=V S)-1or.
j2I0
b) j 2 IK , i.e. j 2 Ik for some 0 < k < n_2. The complex structure on V (k) *
*is such that
g = e2ii=n2 Zn acts by complex multiplication with gk. Notice that in the previ*
*ous subsection
we defined the complex structure on V=V S1to come from the decompostion (3).*Th*
*is implies
that m*j= nq*j+ k, and therefore s(xj+ m*jff) = s(xj+ q*jnff + kff) = fflqjs(xj*
*+ kff).
Consider ~k the equivariant class on P corresponding to the complex vector bu*
*ndle V (k)
with its chosenQcomplex orientation, and the convergentQpower series Q(x) = s(x*
* + kff)-1.
Then i*~k = Iks(xj + kff)-1. Define ~K = 0 0. The
* n
rotation numbers satisfy m*j= q*jn + n_2, hence s(xj + m*jff) = fflqjs(xj + _2f*
*f). Consider the
power series Q(x) = s(x + n_2ff)-1. Q(x) satisfies Q(-x) = s(-x + n_2ff)-1 = -s*
*(x - n_2ff)-1 =
-ffls(x + n_2ff)-1 = (-ffl)Q(x), so Q(x) is either even or odd. According to De*
*finition A.8, since
V (n_2)oris a real oriented even dimensional vector bundle, Q(x) defines a clas*
*s ~n_2= ~Q (V (n_2)),
EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 19
*
* n_)
which is a clas on P . Now from the table, i*V (n_2)orand (i*V (n_2))cxdiffer b*
*y the sign (-1)ff(2,
n_)Q n -1
so Lemma A.9 (with fl = -ffl) implies that i*~n_2= (-ffl)ff(2 j2Iks(xj+ _2ff) *
* . Finally we
obtain
Y P q* n_
(8) s(xj+ m*jff)-1 = ffl In=2j. (-ffl)ff(2). i*~n_2.
j2In=2
Now, if we put together equations (5)-(8) and (4), and define ~P := ~K . ~n_2*
*, we have just
proved that t*ffeES1(V=V S1)-1 = fflff(N). eES1(V Zn=V S1)-1 . i*~P, or
1-1 E Z S1 ff(N) *
(9) t*ffeES1(V=V S) . eS1(V n=V ) = ffl . i ~P ,
where X X X n
oe(N) = q*j+ q*j+ q*j+ oe(K) + oe(__) .
I0 IK In=2 2
Now we want to describe oe(N) in terms of the correct rotation numbers mj of *
*V=V S1.
Recall that mj are the same as m*jup to sign and a permutation. Denote by eq*
*uality
modulo 2. We have the following cases:
a) jP2 I0. Suppose mj = -m*j. Then qj = -q*j, which implies q*j qj. The*
*refore
* P
I0qj I0qj.
b) j 2 IK . Let 0 < k < n_2. Suppose mj = -m*j= -q*jn - k = -(q*j+ 1)nP+ (n*
* - k).
Then qjP= -q*j- 1, which implies q*j+ 1 qj. So modulo 2, the sum IKq*
**jdiffers
from IKqj by the number of the sign differences mj = -m*j. But by defi*
*nition of
rotation numbers, the number of sign differences in two systems of rotat*
*ion numbers
is precisely the signPdifference oe(K)Pbetween the two corresponding ori*
*entations of
i*V (K). Therefore, IKq*j+ oe(K) IKqj.
c) j 2 In=2. Suppose mj = -m*j= -q*jn - n_2= -(q*j+P1)n + n_2. Then thisP*
*implies
q*j+ 1 qj, so by the same reasoning as in b) In=2q*j+ oe(n_2) In*
*=2qj.
We finally get the following formula for oe(N)
X X X
oe(N) qj+ qj+ qj .
I0 IK In=2
In the next lemma we will show that, for N and ~Ntwo different connected comp*
*onents of
XS1 inside P , oe(N) and oe(N~) are congruent modulo 2, so the class fflff(N). *
*~P is well-defined,
i.e. independent of N. Now recall that P is a connected component of XZn. The*
*refore
HO*S1(XZn) = PHO*S1(P ), so we can define
X
~ff:= fflff(N). ~P .
P
This is a well-defined class in HO*S1(XZn), so by equation (9), Lemma 5.3 is fi*
*nally proved.
Lemma 5.4. In the conditions of the previous lemma, oe(N) and oe(N~) are congru*
*ent modulo
2.
Proof.The proof follows Bott and Taubes [4]. Denote by S2(n) the 2-sphere with *
*the S1-action
which rotates S2 n times around the north-south axis as we go once around S1. D*
*enote by
N+ and N- its North and South poles, respectively. Consider a path in P which c*
*onnects N
with ~N, and touches N or ~Nonly at its endpoints. By rotating this path with t*
*he S1-action,
we obtain a subspace of P which is close to being an embedded S2(n). Even if it*
* is not, we
20 IOANID ROSU
can still map equivariantly S2(n) onto this rotated path. Now we can pull back *
*the bundles
from P to S2(n) (with their correct orientations). The rotation numbers are the*
* same, since
the North and the South poles are fixed by the S1-action, as are the endpoints *
*of the path.
Therefore we have translated the problem to the case when we have the 2-spher*
*e S2(n) and
corresponding bundles over it, and we are trying to prove that oe(N+ ) oe(N- *
*) modulo 2.
The only problem would be that we are not using the whole of V , but only V=V S*
*1. However,
the difference between these two bundles is V S1, whose rotation numbers are al*
*l zero, so they
do not influence the result.
Now Lemma 9.2 of [4] says that any even-dimensional oriented real vector bund*
*le W over
S2(n) has a complex structure. In particular, the pullbacks of V S1, V (K), an*
*d V (n_2) have
complex structure, and the rotation numbers can be chosen to be the mj describe*
*d above. Say
the rotation numbers at the South pole are ~mjwith the obvious notation convent*
*ions.PThenP
Lemma 9.1 of [4] says that, up to a permutation, mj - ~mj= n(qj - ~qj), and q*
*j ~qj
modulo 2. But this means that oe(N+ ) oe(N- ) modulo 2, i.e. oe(N) oe(N~) m*
*odulo 2.
Corollary 5.5. (The Rigidity theorem of Witten) If X is a spin manifold with an*
* S1-action,
then the equivariant elliptic genus of X is rigid i.e. it is a constant power s*
*eries.
Proof.By lifting the S1-action to a double cover of S1, we can make the S1-acti*
*on preserve
the spin structure. Then with this action X is a spin S1-manifold.
At the beginning of this Section, we say that if X is a compact spin S1-manif*
*old, i.e. the
map ß : X ! * is spin, then we have the Grojnowski pushforward, which is a map *
*of sheaves
ßE!: E*S1(X)[i]! E*S1(*) = OE .
The Grojnowski pushforward ßE!, if we consider it at the level of stalks at 0 2*
* E, is nothing
but the elliptic pushforward in HO*S1-theory, as described in Corollary 4.4. So*
* consider the
element 1 in the stalk at 0 of the sheaf E *S1(X)[i]= E*S1(X)[TX].
From Theorem 5.2, since T X is spin, 1 extends to a global section of E *S1(X*
*)[TX]. Denote
this global section by boldface 1. Because ßE!is a map of sheaves, it follows t*
*hat ßE!(1) is a
global section of E *S1(*) = OE, i.e. a global holomorphic function on the elli*
*ptic curve E. But
any such function has to be constant. This means that ßE!(1), which is the equi*
*variant elliptic
genus of X, extends to ßE!(1), which is constant. This is precisely equivalent *
*to the elliptic
genus being rigid.
The extra generality we had in Theorem 5.2 allows us now to extend the Rigidi*
*ty theorem
to families of elliptic genera. This was stated as THEOREM D in Section 2.
Theorem 5.6. (Rigidity for families) Let F ! E -i! B be an S1-equivariant fibra*
*tion such
that the fibers are spin in a compatible way, i.e. the projection map ß is spin*
* oriented. Then
the elliptic genus of the family, which is ßE!(1) 2 H**S1(B), is constant as a *
*rational function
in u, i.e. if we invert u.
Proof.We know that the map
ßE!: E*S1(E)[i]! E*S1(B)
when regarded at the level of stalks at zero is the usual equivariant elliptic *
*pushforward in
HO*S1(-). Now ßE!(1) 2 HO*S1(B) is the elliptic genus of the family. We have E *
**S1(E)[i]~=
E *S1(E)[fi(F)], where ø(F ) ! E is the bundle of tangents along the fiber.
Since ø(F ) is spin, Theorem 5.2 allows us to extend 1 to the Thom section 1.*
* Since ßE!is
a map of sheaves, it follows that ßE!(1), which is the elliptic genus of the fa*
*mily, extends to a
EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 21
global section in E *S1(B). So, if i : BS1 ,! B is the inclusion of the fixed p*
*oint submanifold
in B, i*ßE!(1) gives a global section in E *S1(BS1). Now this latter sheaf is *
*free as a sheaf
of OE-modules, so any global section is constant. But i* : HO*S1(B) ! HO*S1(BS*
*1) is an
isomorphism if we invert u.
We saw in the previous section that, if f : X ! Y is an S1-map of compact S1-*
*manifolds
such that the restrictions f : Xff! Y ffare oriented maps, we have the Grojnows*
*ki pushfor-
ward
fE!: E*S1(X)[f]! E*S1(Y ) .
Also, in some cases, for example when f is a spin S1-fibration, we saw that E **
*S1(X)[f]admits
a Thom section. This raises the question if we can describe E *S1(X)[f]as E *S*
*1of a Thom
space. It turns out that, up to a line bundle over E (which is itself E *S1of a*
* Thom space),
this indeed happens:
Let f : X ! Y be an S1-map as above. Embed X into an S1-representation W , i *
*: X ,!
W . (W can be also thought as an S1-vector bundle over a point.) Look at the em*
*bedding
f x i : X ,! Y x W . Denote by V = (f), the normal bundle of X in this embeddi*
*ng (if we
were not in the equivariant setup, (f) would be the stable normal bundle to th*
*e map f).
Proposition 5.7. With the previous notations,
E*S1(X)[f]~=E*S1(DV, SV ) E*S1(DW, SW )-1 ,
where DV , SV are the disk and the sphere bundles of V , respectively.
Proof.From the embedding X ,! Y xW , we have the following isomorphism of vecto*
*r bundles:
T X V ~=f*T Y W .
So, in terms of S1-equivariant elliptic Euler classes we have eES1(V ff=V fi) =*
* eES1(Xff=Xfi)-1 .
f*eES1(Y ff=Y fi) . eES1(W ff=W fi). Rewrite this as
~[f]fffi= eES1(V ff=V fi) . eES1(W ff=W fi)-1 ,
where ~[f]fffiis the twisted cocycle from Definition 4.8.
Notice that we can extend Definition 5.1 to virtual bundles as well. In othe*
*r words, we
can define E *S1(X)[-V ]to be E *S1(X) twisted by the cocycle ~[-Vf]ffi= eES1(V*
* ff=V fi). The above
formula then becomes
~[f]fffi= ~[-Vf]ffi. ~[W]fffi,
which implies that
(10) E *S1(X)[f]= E*S1(X)[-V ] E*S1(X)[W] .
So the proposition is finished if we can show that for a general vector bundle V
E *S1(DV, SV ) = E*S1(X)[-V ].
22 IOANID ROSU
Indeed, multiplication by the equivariant elliptic Thom classes on each stalk g*
*ives the following
commutative diagram, where the rows are isomorphisms:
.t*ffffiES1(V ff)
H*S1(Xff) C[u]OE(U - ff) H*S1(DV ff, SV ff) C[u]OE(U - ff)
eES1(V ff=V fi).i* i*
.t*ffffiES1(V fi)
H*S1(Xfi) C[u]OE(U - ff) H*S1(DV fi, SV fi) C[u]OE(U - ff)
t*fi-ff t*fi-ff
.t*fiffiES1(V fi)
H*S1(Xfi) C[u]OE(U - fi) H*S1(DV fi, SV fi) C[u]OE(U - fi) .
Notice that E*S1(DW, SW ) is an invertible sheaf, because it is the same as the*
* structure sheaf
E *S1(*) = OE twisted by the cocycle ~[W]fffi. In fact, we can identify it by t*
*he same method we
used in Proposition 3.11.
In the language of equivariant spectra (see Chapter 8 of [13]) we can say mor*
*e: With the
notation we used in Proposition 5.7, we define a virtual vector bundle T f, the*
* tangents along
the fiber, by
T X = T f f*T Y .
Using the formula T X V = f*T Y W , it follows that -T f = V W . From equ*
*ation (10)
it follows that *
E *S1(X)[f]= ~ES1(X-Tf ) ,
* -Tf 1
where ~ES1is reduced cohomology, and X is the S -equivariant spectrum obtain*
*ed by the
Thom space of V desuspended by W .
Appendix A. Equivariant characteristic classes
The results of this section are well-known, with the exception of the holomor*
*phicity result
Proposition A.6.
Let V be a complex n-dimensional S1-equivariant vector bundle over an S1-CW c*
*omplex
X. Then to any power series Q(x) 2 C[[x]] starting with 1 we are going to asso*
*ciate by
Hirzebruch's formalism (see [11]) a multiplicative characteristic class ~Q (V )*
*S1 2 H**S1(X).
(Recall that H**S1(X) is the completion of H*S1(X).)
Consider the Borel construction for both V and X: VS1 = V xS1ES1 ! X xS1ES1 =*
* XS1.
VS1 ! XS1 is a complex vector bundle over a paracompact space, hence we have a *
*classifying
map fV : XS1 ! BU(n). We define cj(V )S1, the equivariant j'th Chern class of V*
* , as the
image via f*Vof the universal j'th Chern class cj 2 H*BU(n) = C[c1, . .,.cn]. N*
*ow look at
the product Q(x1)Q(x2) . .Q.(xn). It is a power series in x1, . .,.xn which is *
*symmetric under
permutations of the xj's, hence it can be expressed as another power series in *
*the elementary
symmetric functions oej = oej(x1, . .,.xn):
Q(x1) . .Q.(xn) = PQ (oe1, . .,.oen) .
Notice that PQ (c1, . .,.cn) lies not in H*BU(n), but in its completion H**BU(n*
*). The map
f*Vextends to a map H**BU(n) ! H**(XS1).
EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 23
Definition A.1. Given the power series Q(x) 2 C[[x]] and the complex S1-vector *
*bundle V
over X, there is a canonical complex equivariant characteristic class ~Q (V )S1*
* 2 H**(XS1),
given by
~Q (V )S1 := PQ (c1(V )S1, . .,.cn(V )S1) = f*VPQ (c1, . .,.cn) .
Remark A.2. If T n,! BU(n) is a maximal torus, then then H*BT n= C[x1, . .,.xn]*
*, and
the xj's are called the universal Chern roots. The map H*BU(n) ! H*BT nis injec*
*tive, and
its image can be identified as the Weyl group invariants of H*BT n. The Weyl gr*
*oup of U(n) is
the symmetric group on n letters, so H*BU(n) can be identified as the subring o*
*f symmetric
polynomials in C[x1, . .,.xn]. Similarly, H**BU(n) is the subring of symmetric *
*power series
in C[[x1, . .,.xn]]. Under this interpretation, cj = oej(x1, . .,.xn). This a*
*llows us to identify
Q(x1) . .Q.(xn) with the element PQ (c1, . .,.cn) 2 H**BU(n).
Definition A.3. We can write formally ~Q (V )S1 = Q(x1) . .Q.(xn). x1, . .,.xn *
*are called the
equivariant Chern roots of V .
Here is a standard result about the equivariant Chern roots:
Proposition A.4. Let V (m) ! X be a complex S1-vector bundle such that the acti*
*on of S1
on X is trivial. Suppose that g 2 S1 acts on V (m) by complex multiplication wi*
*th gm . If xi
are the equivariant Chern roots of V (m), and wi are its usual (nonequivariant)*
* Chern roots,
then
xi= wi+ mu ,
where u is the generator of H*S1(*) = H*BS1.
We want now to show that the class we have just constructed, ~Q (V )S1, is ho*
*lomorphic in
a certain sense, provided Q(x) is the expansion of a holomorphic function aroun*
*d zero. But
first, let us state a classical lemma in the theory of symmetric functions.
Lemma A.5. Suppose Q(y1, . .,.yn) is a holomorphic (i.e. convergent) power seri*
*es, which
is symmetric under permutations of the yj's. Then the power series PQ such that
Q(y1, . .,.yn) = PQ (oe1(y1, . .,.yn), . .,.oen(y1, . .,.yn)) ,
is holomorphic.
We have mentioned above that ~Q (V )S1 belongs to H**S1(X). This ring is equi*
*variant coho-
mology tensored with power series. It contains HO*S1(X) as a subring, correspon*
*ding to the
holomorphic power series.
Proposition A.6. If Q(x) is a convergent power series, then ~Q (V )S1 is a holo*
*morphic class,
i.e. it belongs to the subring HO*S1(X) of H**S1(X).
Proof.We have ~Q (V )S1 = P (c1(V )S1, . .,.cn(V )S1), where we write P for PQ .
Assume X has a trivial S1-action. It is easy to see that H*S1(X) = (H0(X) C*
* C[u])
nilpotents. Hence we can write cj(E)S1 = fj+ ffj, with fj 2 H0(X) C C[u], and *
*ffj nilpotent
in H*S1(X). We expand ~Q (V )S1 in Taylor expansion in multiindex notation. We *
*make the
following notations: ~ = (~1, . .,.~n) 2 Nn, |~| = ~1 + . .+.~n, and ff~ = ff~1*
*1. .f.f~nn. Now
we consider the Taylor expansion of ~Q (V )S1 in multiindex notation:
X @|~|P
~Q (V )S1 = P (. .,.cj(V )S1, . .).= _____~(. .,.fj, . .).. f*
*f~ .
~ @c
24 IOANID ROSU
This is a finite sum, since ffj's are nilpotent. We want to show that ~Q (V )S1*
* 2 HO*S1(X). ff~
*
*|~|P
lies in HO*S1(X), since it lies even in H*S1(X). So we only have to show that @*
*___@c~(. .,.fj, . .).
lies in HO*S1(X).
But fj 2 H0(X) C C[u] = C[u] . . .C[u], with one C[u] for each connected c*
*omponent
of X. If we fix one such component N, then the corresponding component f(N)jlie*
*s in C[u].
|~|P
According to Lemma A.5, P is holomorphic around (0, . .,.0), hence so is @___@c*
*~. Therefore
@|~|P_(. .,.f(N)(u), . .).is holomorphic in u around 0, i.e. it lies in O . *
*Collecting the terms
@c~ j C,0
for the different connected components of X, we finally get
@|~|P_ 0
(. .,.fj, . .).2 OC,0 . . .OC,0= H (X) C OC,0.
@c~
But H0(X) C OC,0 H*(X) C OC,0= H*S1(X) C[u]OC,0= HO*S1(X), so we are done.
If the S1-action on X is not trivial, look at the following exact sequence as*
*sociated to the
pair (X, XS1):
* * S1 ffi*+1 S1
0 ! T ,! H*S1(X) -i!HS1(X ) -! HS1 (X, X ) ,
where T is the torsion submodule of H*S1(X). (The fact that T = keri* follows *
*from the
following arguments: on the one hand, keri* is torsion, because of the localiza*
*tion theorem;
on the other hand, H*S1(XS1) is free, hence all torsion in H*S1(X) maps to zero*
* via i*.) Also,
since T is a direct sum of torsion modules of the form C[u]=(un)
T C[u]OC,0~=T ~=T C[u]C[[u]] .
Now tensor the above exact sequence with OC,0and C[[u]] over C[u]:
* 1 ffi *+1 1
0 T HO*S1(X) i HO*S1(XS ) HOS1 (X, XS )
s t
* 1 ffi**+1 1
0 T H**S1(X) i H**S1(XS ) HS1 (X, XS ) .
We know ff := ~Q (V )S1 2 H**S1(X). Then fi := i*~Q (V )S1 = i*ff was shown pre*
*viously to be
in the image of t, i.e. fi = tf~i. ffifi = ffii*ff = 0, so ffitf~i= 0, hence ff*
*if~i= 0. Thus ~fi2 Im i*, so
there is an ~ff2 HO*S1(X) such that ~fi= i*~ff. s~ffmight not equal ff, but i*(*
*ff - ~ff) = 0, so
ff - ~ff2 T . Now, ~ff+ (ff - ~ff) 2 HO*S1(X), and s(~ff+ (ff - ~ff) = ff, whic*
*h shows that indeed
ff 2 Im s = HO*S1(X).
There is a similar story when V is an oriented 2n-dimensional real S1-vector *
*bundle over
a finite S1-CW complex X. We classify VS1 ! XS1 by a map fV : XS1 ! BSO(2n).
H*BSO(2n) = C[p1, . .,.pn]=(e2- pn), where pj and e are the universal Pontrjagi*
*n and Euler
classes, respectively. The only problem now is that in order to define charact*
*eristic classes
over BSO(2n) we need the initial power series Q(x) 2 C[[x]] to be either even o*
*r odd:
Remark A.7. As in Remark A.2, if T n,! BSO(2n) is a maximal torus, then the map
H*BSO(2n) ! H*BT nis injective, and its image can be identified as the Weyl gro*
*up invari-
ants of H*BT n. Therefore H*BSO(2n) can be thought of as the subring of symmetr*
*ic poly-
nomials in C[x1, . .,.xn] which are invariant under an even number of sign chan*
*ges of the xj's.
A similar statement holds for H**BSO(2n). Under this interpretation, pj = oej(x*
*21, . .,.x2n)
and e = x1. .x.n.
EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 25
So, if we want Q(x1) . .Q.(xn) to be interpreted as an element of H**BSO(2n),*
* we need to
make it invariant under an even number of sign changes. But this is clearly tru*
*e if Q(x) is
either an even or an odd power series.
Let us be more precise:
a) Q(x) is even, i.e. Q(-x) = Q(x). Then there is another power series S(x)*
* such that
Q(x) = S(x2), so Q(x1) . .Q.(xn) = S(x21) . .S.(x2n) = PS(. .,.oej(x21, *
*. .,.x2n), . .).=
PS(. .,.pj, . .)..
b) Q(x) is odd, i.e. Q(-x) = -Q(x). Then there is another power series*
* R(x)
such that Q(x) = xT (x2), so Q(x1) . .Q.(xn) = x1. .x.nT (x21) .*
* .T.(x2n) =
x1. .x.nPT(. .,.oej(x21, . .,.x2n), . .).= e . PT(. .,.pj, . .)..
Definition A.8. Given the power series Q(x) 2 C[[x]] which is either even or od*
*d, and the
real oriented S1-vector bundle V over X, there is a canonical real equivariant *
*characteristic
class ~Q (V )S1 2 H**S1(X), defined by pulling back the element Q(x1) . .Q.(xn)*
* 2 H**BSO(2n)
via the classifying map fV : XS1 ! BSO(2n).
Proposition A.6 can be adapted to show that, if Q(x) is a convergent power se*
*ries, ~Q (V )S1
actually lies in HO*S1(X).
The next result is used in the proof of Lemma 5.3.
Lemma A.9. Let V be an orientable S1-equivariant even dimensional real vector *
*bundle
over X. Suppose we are given two orientations of V , which we denote by Vor1an*
*d Vor2.
Define oe = 0 if Vor1= Vor2, and oe = 1 otherwise. Suppose Q(x) is a power seri*
*es such that
Q(-x) = flQ(x), where fl = 1. Then
~Q (Vor1) = flff~Q (Vor2) .
Proof. a) If Q(-x) = Q(x), ~Q (V ) is a power series in the equivariant Pont*
*rjagin classes
pj(V )S1. But Pontrjagin classes are independent of the orientation, so*
* ~Q (Vor1) =
~Q (Vor2).
b) If Q(-x) = -Q(x), then Q(x) = xQ~(x), with Q~(-x) = Q~(x). Hence ~Q (V *
*) =
eS1(V ) . ~Q~(V ). e(V )S1 changes sign when orientation changes sign, w*
*hile ~Q~(V ) is
invariant, because of a).
A.1. Acknowledgements. I thank Matthew Ando for suggesting that I study the rel*
*ation-
ship between rigidity and Thom classes in equivariant elliptic cohomology. I am*
* also indebted
to Mike Hopkins, Jack Morava, and an anonymous referee for helpful comments. Mo*
*st of all
I thank my advisor, Haynes Miller, who started me on this subject, and gave me *
*constant
guidance and support.
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department of mathematics, m.i.t., cambridge, ma 02139
E-mail address: ioanid@math.mit.edu