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Chern classes in equivariant bordism

Stefan Schwede

TL;DR

The paper develops equivariant Chern classes $c_k^{(m)}\in \mathbf{MU}^{2k}_{U(m)}$ that refine Conner–Floyd Chern classes and probes the structure of equivariant bordism rings for unitary groups and their products. It proves that these Chern classes form regular sequences generating the augmentation ideal, yielding a power-series completion $\mathbf{MU}_{U(m)}^*\,\widehat{}\!{}_I$ isomorphic to $\mathbf{MU}^*(B U(m))$, and extends to products of unitary groups. Consequently, the Greenlees–May local homology spectral sequence collapses for such groups, and the MU-completion theorem is recovered in a streamlined, self-contained fashion via $K(G,V)$ and the Chern-class formalism. Collectively, these results clarify the algebraic structure of $\mathbf{MU}^*_{G}$ for nonabelian compact groups and connect equivariant and nonequivariant bordism through explicit Chern-class machinery.

Abstract

We introduce Chern classes in $U(m)$-equivariant homotopical bordism that refine the Conner-Floyd-Chern classes in the $MU$-cohomology of $B U(m)$. For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees-May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the $MU$-completion theorem of Greenlees-May and La Vecchia.

Chern classes in equivariant bordism

TL;DR

The paper develops equivariant Chern classes that refine Conner–Floyd Chern classes and probes the structure of equivariant bordism rings for unitary groups and their products. It proves that these Chern classes form regular sequences generating the augmentation ideal, yielding a power-series completion isomorphic to , and extends to products of unitary groups. Consequently, the Greenlees–May local homology spectral sequence collapses for such groups, and the MU-completion theorem is recovered in a streamlined, self-contained fashion via and the Chern-class formalism. Collectively, these results clarify the algebraic structure of for nonabelian compact groups and connect equivariant and nonequivariant bordism through explicit Chern-class machinery.

Abstract

We introduce Chern classes in -equivariant homotopical bordism that refine the Conner-Floyd-Chern classes in the -cohomology of . For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees-May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the -completion theorem of Greenlees-May and La Vecchia.
Paper Structure (3 sections, 6 theorems, 37 equations)

This paper contains 3 sections, 6 theorems, 37 equations.

Table of Contents

  1. Equivariant MU-Chern classes
  2. Regularity results
  3. The MU-completion theorem via Chern classes

Key Result

Proposition 1.3

Let $i,j,k,l$ be positive natural numbers such that $i+j=k+l$. Then where $\gamma_d\in U(i+j)$ is the permutation matrix of the shuffle permutation $\chi_d\in\Sigma_{i+j}$ given by

Theorems & Definitions (17)

  • Definition 1.1
  • Remark 1.2
  • Proposition 1.3: Double coset formula
  • proof
  • Theorem 1.4
  • proof
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 7 more