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. References [1]M. Atiyah, R. Bott, The moment map and equivariant cohomology, Topology 23 * *(1984), pp. 1-28. [2]M. Atiyah, F. Hirzebruch, Spin manifolds and group actions, in Essays in To* *pology and Related topics (M'emoires d'edi'es `a Georges de Rham), Springer, 1970, pp. 18-26. [3]C. Allday, V. 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Witten, The index of the Dirac operator in loop space, in Lecture Notes * *in Mathematics, vol. 1326, Springer Verlag, 1988, pp. 161-181. department of mathematics, m.i.t., cambridge, ma 02139 E-mail address: ioanid@math.mit.edu