Computations of complex equivariant bordism rings

Abstract

let [M ] denote the corresponding class in M U G* . Complex projective spaces give a rich collection of examples of G-manifolds. Given a complex representation W of G let P(W ) denote the space of complex one-dimensional subspaces of W with inherited G-action.The starting point in our work is that after inverting Euler classes, M U G * becomes computable by non-equivariant means. That we rely heavily on localization is not surprising because localization techniques have pervaded equivariant topology. For any compact Lie group G let R 0 denote the sub-algebra of M U G * generated by the e V and [P(n⊕ V )] as V ranges over non-trivial irreducible representations. Let S be the multiplicative set in R 0 of non-trivial Euler classes. By abuse, denote the same multiplicative set in M U G * by S. Then the key first result, which we emphasize is true for a large class of groups including p-groups, is the following.Theorem 1.1. Let G be a group such that any proper subgroup is contained in a proper normal subgroup. The inclusion of R 0 into M U G * becomes an isomorphism after inverting S.In other words, we may multiply any class in M U G * by some Euler class to get a class in R 0 modulo the kernel of the localization map S. We are lead to study divisibility by Euler classes as well as the kernel of this localization map. We can do so successfully in the case when the group in question is a torus.Let T be a torus, and let V be a non-trivial irreducible representation of T . Let K(V ) denote the subgroup of T which acts trivially on V . There is a restriction homorphism (of algebras) res T H : M U T * → M U H * for any subgroup H. The restriction of e V to M U K(V ) * is zero, as can be seen using an explicit homotopy. Remarkably, we have the following.

Bordism of semi-free S1-actions

“…Outline of proof. The proof of this theorem parallels the main results of [7] and section four of [21]. The sequence in question is the M U SF * long exact sequence associated to the cofiber sequence ES…”

“…In [21], M U S 1 * was computed, and it has the following prominent features, much as we have seen for semi-free bordism:…”

“…Let Ω V X denote the space of based maps from S V to X, where S 1 acts by conjugation. Let The foundational results of this section are based on [16], and the computational results parallel those of [21]. Theorem 1.1 follows from little more than the computation of Conner-Floyd and tom Dieck exact sequences adapted for semi-free bordism.…”

“…Our work is of further interest in two different ways. To make the computation of geometric semi-free bordism, in Corollary 2.12 we prove the semi-free case of what we call the geometric realization conjecture, which if true in general would determine the ring structure of geometric S 1 -bordism from the ring structure of homotopical S 1 -bordism given in [21]. Additionally, we investigate semi-free actions with isolated fixed points as a first case, and that result is parallel to results from symplectic geometry.…”

The universal equivariant genus and Krichever's formula

“…Our Gj(M 2n ) are clearly complex bordant to the non-connected manifolds γ j (M 2n ) of Theorem 6.3 in [6]. Therefore, we deduce that the completion map of the homotopical bordism ring M U S 1 * at its augmentation ideal (see Ch.…”