BORDISM OF SEMI-FREE S1-ACTIONS DEV SINHA Abstract.We calculate the geometric and homotopical (or stable) bordism r* *ings associated to semi- free S1 actions on complex manifolds, giving explicit generators for the * *geometric theory. To calculate the geometric theory, we prove a case of the geometric realization conjec* *ture, which in general would determine the geometric theory in terms of the homotopical. The determina* *tion of semi-free actions with isolated fixed points up to cobordism complements similar results from sy* *mplectic geometry. 1.Introduction In this paper we describe both the geometric and homotopical bordism rings as* *sociated to S1-actions in which only the two simplest orbit types, namely fixed points and free orbits* *, are allowed. Our work is of further interest in two different ways. To make the computation of geomet* *ric semi-free bordism, in Theorem 3.11 we prove the semi-free case of what we call the geometric realizat* *ion conjecture (Conjec- ture 2.7), which if true in general would determine the ring structure of geome* *tric S1-bordism from the ring structure of homotopical S1-bordism given in [11]. Additionally, an applic* *ation of our results to semi- free actions with isolated fixed points which we state now gives results parall* *el to results from symplectic geometry [10]. Let P(C æ) denote the space of complex lines in C æ where æ * *is the standard complex representation of S1 (in other words, the Riemann sphere with S1 action given b* *y the action of the unit complex numbers.) Theorem 1.1. Let S1 act semi-freely with isolated fixed points on M, compatible* * with a stable complex structure on M. Then M is equivariantly cobordant to a disjoint union of produc* *ts of P(C æ). This result should be compared with the second main result of [10], which sta* *tes that when M is con- nected a semi-free Hamiltonian S1 action on M implies that M has the same Borel* * equivariant cohomology and equivariant Chern classes as a product of such P1's. Based on their results* * and ours, we make the following. Conjecture 1.2. A semi-free Hamiltonian S1 action with isolated fixed points on* * a connected manifold is equivariantly diffeomorphic to a product of P(C æ)'s. As Theorem 1.1 lead us to the more general computation of Theorem 3.12, it wo* *uld also be interesting to see if there is an analog of Theorem 3.12 for Hamiltonian S1-actions. In ge* *neral, the symplectic and cobordism approaches to transformation groups have remarkable overlaps in l* *anguage (for example, localization by inverting Euler classes of representations plays a key role in * *each theory), though the same words sometimes have different precise meanings. A synthesis of these technique* *s could perhaps address Conjecture 1.2 or other interesting questions within transformation groups. The author would like to thank Jonathan Weitsman for stimulating conversation* *s. 2. Review of complex equivariant bordism The foundational results of this section are taken from [7], and the computat* *ional results are taken from [11]. If V is a representation of G equipped with a G-invariant inner product, * *let SV denote its one-point 1 2 DEV SINHA compactification, let D(V ) be the unit disk in V , and let S(V ) be the bounda* *ry of D(V ), namely the unit sphere in V . Let U,G*denote geometric complex equivariant bordism (see for example chapte* *r 28 of [7]), the bordism ring whose representatives are stably complex G-manifolds up to cobordisms in w* *hich the bounding man- ifolds have stably complex G-actions which extend the actions on their boundari* *es. By equipping these manifolds with maps to a space X we define an equivariant homology theory U,G** *(X), which extends to a homology theory on pairs by using manifolds with boundary. Note that this equiv* *ariant homology theory has suspension isomorphisms only when G acts trivially on the suspension coordi* *nate. Let MUG*denote stable complex equivariant bordism, represented by a direct li* *mit of homotopy classes of maps from spheres SV to Thom spaces (see for example section 3 of [11]). Alt* *ernately, by a result of Bröcker and Hook [2], MUG*(X) is isomorphic to a direct limit which has non-tri* *vial G-suspensions built in, namely limV U,G*(X x D(V ), X x S(V )), where V ranges over the real repres* *entations of G. Thus, MUG*is also represented by stably complex G-manifolds M with boundary equipped * *with a map to D(V ) for some representation V such that the boundary of M maps to S(V ). Through ei* *ther the Pontryagin- Thom construction or through the Bröcker-Hook result, there is a canonical a ma* *p OE: U,G*! MUG*. The following result allows us to use homotopy theory to effectively study G-manifo* *lds. Theorem 2.1 (Comeza~na, Theorem 28.5.4 of [7]). If A is abelian, the map OE: U* *,A*! MUA*is injective. What makes MUA*at all manageable is that it in turn maps injectively to a loc* *alization which is computable. To set this stage, we recall some of the main results from [11]. Euler classes play a central role in equivariant bordism. Vector spaces with * *G-action, viewed as vector bundles over a point, may have interesting characteristic classes in the equiva* *riant cohomology of a point. For a complex representation V let eV 2 MUnG(pt.) = MUG-n, where n is the dimen* *sion of V over the reals, denote the Euler class of V . Under the Bröcker-Hook isomorphism, eV is * *represented by a point (as a zero-manifold) mapping to zero in D(V ). It is intuitively clear, and shown i* *n chapter 15 of [7], that eV is non-trivial if V G= 0. Thus, MUG*contains many classes in arbitrarily negati* *ve degrees. Euler classes multiply by the rule eVeW = eV W. Let S be the multiplicative set of non-trivi* *al Euler classes. Let Zn,V be the class in MUG*represented by P(Cn V ), the space of lines in Cn V wit* *h induced G-action. Theorem 2.2 (Theorem 1.2.5 of [11]). There are inclusions of MUT*-algebras. MU*[eV, Zn,V] ! MUT*~!S-1MUT*~=MU*[eV1, Zn,V], where V ranges over non-trivial irreducible representations of T, n 2, and ~ * *is the canonical localization map. Thus, understanding of MUT*follows from understanding of divisibility by Eule* *r classes. Crucial insight is provided by the following theorem, which is proved by applying MU*Tto the co* *fiber sequence S(V ) ! * ! SV . Let K(V ) denote the kernel of the representation V . Theorem 2.3 (Theorem 1.2 of [11]). There is an isomorphism of MUT*=(eV) with MU* *K(V*), where the quotient map from MUT*to MUT*=(eV) coincides with the restriction map rV from M* *UT*to MUK(V*). By carefully splitting the restriction map rV and composing that splitting wi* *th rV itself, in [11] we define idempotent operations fiV :MUT*! MUT*such that x - fiV(x) is divisible b* *y eV. By Theorem 2.2, the quotient of x - fiV(x) by eV is unique, and we call that quotient V(x). We* * say V is a Conner-Floyd operation. If I = V1, . .,.Vk is a finite sequence of non-trivial irreducible * *representations let I(x) = Vk Vk-1. . .V1x. Fix an ordering on the non-trivial irreducible representation* *s of T which includes the relation V < W if K(V ) K(W). Call a finite sequence of representations admis* *sible if it respects this ordering. The following is the main theorem of [11]. BORDISM OF SEMI-FREE S1-ACTIONS * * 3 Theorem 2.4. MUT*is generated as an MU*-algebra by the classes I(eV) and I(Zn* *,V), where V ranges over non-trivial irreducible representations, I ranges over all admissis* *ible sequences of non-trivial irreducible representations, and n ranges over natural numbers. Relations include the following: (1)eV V(x) = x - fiV(x) (2) V(fiV(x)) = 0 (3) V(eV) = 1 (4) V(x)y = (x - fiV(x)) V(y) + V(x)fiV(y) (5) V Wx = WV x - WV fiW (x) - WV (eW fiV( W x)) where V and W range over non-trivial irreducible representations of T and x and* * y are any classes in MUT*. For T = S1, these relations are complete. 1 Note that while MUS1*is known as a ring, U,S*is not known because it is1not * *known which classes I(x), and in particular which I(eV) can be realized by classes in U,S*. As w* *ith much of the theory, this question is best understood by applying1the localization map ~. By Proposi* *tions 4.13 and 4.5 of [11], one may compute the image of a class in U,S* MUS1*under ~ by investigating fi* *xed sets and normal bundle data of that manifold. Indeed, this is one of the main results of the pa* *per [4]. Definition 2.5. Let P* denote the sub-algebra MU*[e-1V, Zn,V] of MU*[eV1, Zn,V]. In other words P* is the sub-algebra of S-1MUT*generated by those generators * *which are in positive degrees. Note that this algebra is called F* in [11]. 1 Proposition 2.6 (See Proposition 4.13 of [11]). The image of U,S*under ~ lies * *in P*. We make the following1conjecture, a converse to Proposition 2.6, which would * *determine the ring structure of U,S*from that of MUS1*and its image under ~. 1 Conjecture 2.7 (The realization conjecture). If ~(x) 2 P* then x 2 U,S*. As evidence for this conjecture, we will prove it in the semi-free setting. 3.Semi-free bordism We now focus on the case where T = S1 and the S1 action on manifolds in quest* *ion is semi-free, so that points are either fixed or freely acted upon by S1. 1 Definition 3.1. Let SF*denote the subring of U,S*of bordism classes represent* *ed by semi-free actions. For a semi-free action, the normal bundle of a fixed set will have representa* *tion type on the fiber of æ or æ*, where æ is the standard one-dimensional representation of S1 and æ*is it* *s conjugate. Note here that bordisms between1our semi-free manifolds are allowed to have g* *eneral S1-action, so we are looking at the image in U,S of the theory which we may call SF!*in which * *all manifolds in question have semi-free action. But in fact, by looking at the families exact sequence f* *or the family consisting of all of S1 and the identity subgroup (see chapter 15 of [7], or [12]) both of th* *ese theories fit in long exact sequences 0 _ ! 21 G @ . .!.MU*-1(BS1) i! SF*~!MU*@ BU(n) A ! MU*-2(BS1) ! . .,. n>0 which map to each other and thus can be shown to be isomorphisms by the five-le* *mma. Here, if we have M mapping to BS1 we may pull back the canonical S1-bundle to get a free S1 manifo* *ld, so that i is inclusion of free S1-manifolds into the semi-free theory. And ~ in this setting sends a s* *emi-free bordism class to the 4 DEV SINHA bordism class of the normal bundle (split according to appearance of æ and æ*in* * the fiber). Finally, @ sends a manifold with a direct sum two bundles over it (classified by a map som* *e BU(i) x BU(j)) to the free S1 manifold given by the sphere bundle of that bundle, where S1 acts as æ * *on the first factor and æ* on the second. i F j 1 In fact, by identifiying MU* n>0BU(n) 2 as a sub-ring of P* S-1MUS* (aga* *in, see Proposi- tion 4.13 of [11]) we have the following. Lemma 3.2. The image under ~ of SF*lies in the sub-ring PSF*= MU*[e-1æ, e-1æ*,* * Zn,æ, Zn,æ*] of P*. We are lead to the following. Definition 3.3. Let MUSF*be the subring of MUS1*which maps under ~ to Z[eæ1, eæ* *1*, Zn,æ, Zn,æ*]. Our main results are computations of MUSF*and then, remarkably, SF*as MU*-al* *gebras. First, we pause to consider semi-free manifolds with isolated fixed points. In fact, at f* *irst we analyzed this case because of its independent interest [10] and then realized it could be used as * *a base case in a filtration to compute SF*. Later, we found that we could compute Sf*more directly from Theo* *rem 2.4 and deduce Theorem 1.1 from that computation. Now we choose to present the isolated fixed * *point case independently from the more general semi-free case (Theorem 3.12, as the two approaches are c* *omplementary. Under the identification of Lemma 3.2, semi-free actions with isolated fixed * *points have image under ~ which sit in the subring Z[e-1æ, e-1æ*]. In particular ~(P(C æ) = e-1æ+ e-1æ*. Theorem 3.4. The intersection of ~(MUS1*) with the subring Z[e-1æ, e-1æ*] is th* *e subring Z[e-1æ+ e-1æ*]. This theorem, along with Theorem 2.1 and Theorem 2.2, implies Theorem 1.1 and* * thus characterizes semi-free actions with isolated fixed points up to cobordism. Our main tool in * *this direct proof is application of Theorem 2.3, which for V = æ or æ*says that reduction modulo eæor eæ*coincid* *es with the augmentation map from MUS1*to MU*. Proof of Theorem 3.4.Let R denote the subring Z[e-1æ, e-1æ*] of S-1MUS1*, and l* *et Q denote the subring Z[e-1æ+ e-1æ*] of R, so Q = ~(Z[P(1 æ)]). Since R is graded and lies in positive degrees, we may induct by degree, focu* *sing on homogeneous elements. Suppose that a0e-næ+ a1e-(n-1)æeæ*+ . .+.ane-næ*is equal to ~(x). C* *onsider y = eæ*(x - a0[P(C æ)]n). The image ~(y) is in R and is in degree 2(n - 1), thus by induc* *tion hypothesis we may deduce that y is in Z[P(C æ)], and thus must be equal to an integral multiple* * of P(C æ)n-1. But this is not possible since by Theorem 2.3 the image of eæ*under augmentation is zero* *, thus so is the image of y, whereas it is well-known that (P1)n-1 is non-zero in MU* for any n > 0. Finally, we must estabish the base case, which is for the degree two part of * *R. Here we want to establish that if ae-1æ+ be-1æ*is ~(x) for some x, then a = b. By subtracting bP(C æ) f* *rom x, it suffices to show that no non-zero integral multiple of e-1æis in the image of ~. But if ~(z) = c* *e-1æ, then ~(eæz) = c, so that eæz = c by Theorem 2.2, which implies that 0 = c once we apply the augmenation * *map to the equality. Now we proceed with the computation of MUSF*, which follows from Theorem 2.4 * *by noting that any class in MUS1*is in MUSF*if and only if the only representations which appear i* *n its definition are æ and æ*. For these representations, we have K(æ) and K(æ*) are the trivial subg* *roup of S1 and thus the idempotents fiæ and fiæ* project onto the split image of MU* in MUT*. In t* *he "stable manifolds" interpretation of Bröcker and Hook, fiæ and fiæ*take a class M ! D(V ) and impo* *se a trivial S1-action on both M and D(V ). Let _xdenote fiæ(x). Definition 3.5. Let B be the set of MUT*elements {eæ, eæ*, Zn,æ, and Zn,æ*} whe* *re n 2. Order B by the degree of the classes, with the additional needed convention that Zn,æ< Zn,* *æ*and eæ < eæ*. BORDISM OF SEMI-FREE S1-ACTIONS * * 5 Theorem 3.6. MUSF*is generated as a ring by classes iæ jæ*(x) where x 2 B and * *if x = eæ*, j = 0. Relations are (1)eæ æ(x) = x - ~x= eæ*_æ*(x),_ (2) æ*æ(x) = ææ*(x) + æ(x) ææ*(eæ), (3) V(x)(y - ~y) = (x - ~x) V(y), where V is æ or æ*, (4) ææ*(eæ) = æ*æ(eæ*). Additionally, we require the calculuations ~eV= 0 and V(eV) = 1, where V is æ * *or æ*. An additive basis over MU* is given by monomials iæ jæ*(x)m where x 2 B and * *m is a monomial in the y x in B. Proof.The computation of ring structure is an immediate application of Theorem * *2.4, using the fact that any class in MUT*is in MUSF*if and only if the only representations which appea* *r in its definition are æ and æ*. * * Q The identification of the additive basis follows from the fact that one may t* *ake any product ikæ jkæ*(xk) where xk 2 B and use relation 3 repeatedly to reduce to a sum of monomials such* * that only the minimal element of B appearing in each monomial is operated on by any æ or æ*. Then, * *one uses relation 2 to reorder these operations so that the æ*are applied before the æ. We now turn to a computation of SF*by proving the version of Conjecture 2.7 * *for semi-free actions. We start by making geometric constructions of æ and æ*on classes represented * *by honest G- manifolds. These constructions follow ones made by Conner and Floyd (hence the name given * *to the general opeations V). Lemma 3.7. ~( æ(x)) = e-1æ(~(x) - ~x) and similarly ~( æ*(x)) = e-1æ*(~(x) - ~x* *). Definition 3.8. Define fl(M) for any stably complex S1-manifold to be the stabl* *y complex S1-manifold __ fl(M) = M xS1S3 t (-M ) x P(C æ), where S3 has the standard Hopf S1-action and the S1-action on M xS1S3 is given * *by (1) i . [m, z1,=z2][i . m, z1,.iz2] Define ~fl(M) similarly with the quotient of M x S3 now being by the S1 action * *in which ø sends m, (z1, z2) to øm, (øz1, ø-1z2) and with induced S1 action on the quotient given by -1 (2) i . [m, z1,=z2]i . m, z1, i. z2 Proposition 3.9. Let M be a stably complex S1-manifold. Then æ[M] = [fl(M)] an* *d æ*[M] = [~fl(M)] in MUS1*. Proof.By Lemma 3.7 and the injectivity of ~, it suffices to check the fixed set* *s of fl(M) and ~fl(M). One set of fixed points of fl(M) are points [m, z1, z2]such that m is fixed in * *M and z2 = 0. This fixed set is diffeomorphic to MG , and its normal bundle is the normal bundle of MG i* *n M crossed with the representation æ. In the localization, crossing with æ coincides with multiplyi* *ng by e-1æ. The second set of fixed points are [m, z1, z2]such that z1 = 0. This set of fixed points is diffe* *omorphic to M, and its normal bundle is the trivial bundle æ-1. __ * * __ Hence, if x = ~([M]), then_the image of [fl(M)] is xe-1æ+M e-1æ-1. By subtrac* *ting the image of M xP(C æ) we are left with xe-1æ- M e-1æ, which by Lemma 3.7 is ~( æ([M])). The analysis is similar for ~fl(M). Thus, the classes I(P(Cn æ)) and I(P(Cn æ*)) can be realized geometrica* *lly. Along similar lines we have the following. Lemma 3.10. ææ*(eæ) = P(C æ). 6 DEV SINHA Proof.The equality of these classes follows from computation of their image und* *er ~. Following the methods of tom Dieck [4] as applied in Proposition 4.14 of [11], the isolated f* *ixed point of P(C æ)with normal bundle æ contributes a term of e-1æto its image under ~, and similarly t* *he other fixed point contributes an e-1æ*. Thus, ~(P(C æ)) = e-1æ+_e-1æ*._To show that this is als* *o ~( æ æ*(eæ)), by applying Lemma 3.7 twice it suffices to compute that æ*(eæ)= -1. This equality in turn * *follows from giving a Bröcker-Hook model for æ*(eæ) as D(æ*) mapping to D(æ) through complex conjuga* *tion, which forgetting S1-action has degree -1. We are now ready to prove the semi-free case of the geometric realization con* *jecture. Theorem 3.11. The following square is a pull-back square SF* ----! MUSF* ?? ? y ~?y PSF*= MU*[e-1æ, e-1æ*, Zn,æ,-Zn,æ*]---!MU*[eæ1, eæ1*, Zn,æ, Zn,æ** *]. Proof.We first go through the list of generators of MUSF*, determine which map * *to P*, and show that those which do have geometric representatives. By Proposition 3.9, the iæ jæ(x) where x = P(Cn æ) or P(Cn æ*) are in S* *F*. Next, by Lemma 3.7, X ______ (3) ~( iæ(eæ*)) = eæ*e-iæ+ iæ(eæ*)e-m-iæ, which is not in PSF*as eæ*appears with a positive power. This leaves iæ jæ*(eæ), which if i = 0 has image under ~ which is not in P* * *by a computation as in Equation 3. For i > 0, note that relation 2 of Theorem 3.6 says that æ æ*(x) * *= æ* æ(x) modulo MU*[P(C æ)], which is of course in SF*. Thus, modulo SF*, we have iæ jæ*(eæ) = i-1æ j-1æ*( æ æ*(eæ)), which by Lemma 3.10 is i-1æ j-1æ*(P(C æ)). Again applying Proposition 3.9, t* *his class is in SF*. Theorem 3.6 gave an additive basis for MUSF*as given by monomials M = iæ jæ** *(x)m where x 2 B = {eæ, eæ*, P(Cn æ), P(Cn æ*)} and m is a monomial in the y x in B. If x = * *eæ and i = 0 or if x = eæ* and j = 0, then by a computation using Lemma 3.7 as above, ~(M) =2PSF*, regardl* *ess of what m is, as eæ (respectively eæ*) will appear with a positive power in the leading term of * *~(M). Otherwise, M is a product of generators which we have shown are in SF*. From the proof of Theorem 3.11, an explicit computation of SF*including geom* *etric representatives is immediate, since SF*is just the sub-ring of MUSF*generated by the iæ jæ*P(Cn * *æ) and iæ jæ*P(Cn æ*) for n 1. Given the general complexities of equivariant bordism, in particular* * for the geometric theories, SF*has a remarkably simple form. Theorem 3.12. SF*is generated as an algebra over MU* by the classes fliflj*P(C* *n æ) and fliflj*P(Cn æ*) where n 1. Relations are (1)fl(x)(y - ~y) = (x - ~x)fl(y),_and similarly for fl*, (2)fl*fl(x) = flfl*(x) + fl(x)P(C æ). BORDISM OF SEMI-FREE S1-ACTIONS * * 7 References [1]M. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology * *23 (1984) 1-28. [2]T. Bröcker and E. Hook, Stable equivariant bordism. Math. Z. 129 (1972), 26* *9-277. [3]P.E. Conner and E.E. Floyd, Differentiable Periodic Maps. Springer, Berlin-* *Heidelberg-New York, 1964. 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