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The Homology of Complex Equivariant Bordism

Julius Groenjes

TL;DR

The paper computes the $E$-homology of tom Dieck's equivariant complex bordism spectrum $MU_A$ for abelian compact Lie groups $A$, and of the geometric bordism spectrum $mU_A$, correcting a prior error in CGK02. It achieves two colimit presentations of $MU_A$ and related spectra, and derives explicit formulas for $E_*^A(\mathrm{MU}_A)$, $E_*^A(\mathrm{mU}_A)$, $E_*^A(\mathrm{MUP}_A)$, and $E_*^A(\mathrm{mUP}_A)$, including coordinatized forms via flags of the universe. The universality of $MU_A$ is established through a universal orientation $x^{\mathrm{uni}}(\epsilon)$, yielding a natural bijection between ring maps to any oriented $A$-spectrum $E$ and orientations of $E$, with careful handling of the canonical homotopy presentation. The work clarifies the role of invertible coaugmentations $\vartheta_\alpha$ and $\vartheta_\epsilon$ in localizing the $E$-homology rings and connects unitary and genuine $G$-spectra via comparison functors, contributing foundational tools for equivariant bordism and equivariant formal group laws.

Abstract

Let $A$ be an abelian compact Lie group and let $E$ be an oriented $A$-spectrum. We compute the $E$-homology of tom Dieck's homotopical $A$-equivariant complex bordism spectrum $MU_A$ in two ways, correcting an error in Cole-Greenlees-Kriz (2002). Additionally, we calculate the $E$-homology of the geometric $A$-equivariant complex bordism spectrum $mU_A$.

The Homology of Complex Equivariant Bordism

TL;DR

The paper computes the -homology of tom Dieck's equivariant complex bordism spectrum for abelian compact Lie groups , and of the geometric bordism spectrum , correcting a prior error in CGK02. It achieves two colimit presentations of and related spectra, and derives explicit formulas for , , , and , including coordinatized forms via flags of the universe. The universality of is established through a universal orientation , yielding a natural bijection between ring maps to any oriented -spectrum and orientations of , with careful handling of the canonical homotopy presentation. The work clarifies the role of invertible coaugmentations and in localizing the -homology rings and connects unitary and genuine -spectra via comparison functors, contributing foundational tools for equivariant bordism and equivariant formal group laws.

Abstract

Let be an abelian compact Lie group and let be an oriented -spectrum. We compute the -homology of tom Dieck's homotopical -equivariant complex bordism spectrum in two ways, correcting an error in Cole-Greenlees-Kriz (2002). Additionally, we calculate the -homology of the geometric -equivariant complex bordism spectrum .
Paper Structure (21 sections, 49 theorems, 147 equations)

This paper contains 21 sections, 49 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Orientations of $A$-Spectra
  3. Calculation of Homology
  4. Coordinatized Statements
  5. Universality of $\mathrm{MU}_A$
  6. Equivariant orientations
  7. Equivariant Homotopy Theory
  8. Orientations of Equivariant Spectra
  9. Grassmannians and the Thom Isomorphism
  10. The Homology of Projective Space
  11. Coaugmentations
  12. Grassmannians
  13. The Thom Isomorphism
  14. Bordism Spectra
  15. Unitary Bordism Spectra
  16. ...and 6 more sections

Key Result

Theorem 1.1

There is an $E_*^A$-algebra isomorphism

Theorems & Definitions (119)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 109 more