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The Pontryagin-Thom theorem for families of framed equivariant manifolds

Lucas Williams

TL;DR

The paper extends the Pontryagin–Thom framework to equivariant and fiberwise settings for framed manifolds under groups G that are products of a finite group with a torus. It develops a robust equivariant bordism theory using V-framings, manifolds with corners, and genuine G-spectra, and proves an isomorphism \\omega_V^G(X,A) \\cong \\pi_V^G(\\Sigma^ olinebreak \\infty X/A). It further generalizes to families parameterized over a base B via parameterized equivariant spectra and derived section functors, yielding \\omega_V^G((X,A)\\xrightarrow{\\psi} B) \\cong \\pi_V^G(\\mathbb{R}\\Gamma_B(\\Sigma_B^ olinebreak \\infty X\\cup_A B)). The results unify geometric and homotopical perspectives in equivariant and fiberwise topology, offering a canonical bridge between framed G-bordism and equivariant stable homotopy theory with fixed-point compatibilities.

Abstract

The Pontryagin-Thom theorem gives an isomorphism from the cobordism group of framed $n$-manifolds to the $n$th stable homotopy group of the sphere spectrum. In this paper, we prove the generalization of the Pontryagin-Thom theorem for families of framed equivariant manifolds parameterized over a compact base space.

The Pontryagin-Thom theorem for families of framed equivariant manifolds

TL;DR

The paper extends the Pontryagin–Thom framework to equivariant and fiberwise settings for framed manifolds under groups G that are products of a finite group with a torus. It develops a robust equivariant bordism theory using V-framings, manifolds with corners, and genuine G-spectra, and proves an isomorphism \\omega_V^G(X,A) \\cong \\pi_V^G(\\Sigma^ olinebreak \\infty X/A). It further generalizes to families parameterized over a base B via parameterized equivariant spectra and derived section functors, yielding \\omega_V^G((X,A)\\xrightarrow{\\psi} B) \\cong \\pi_V^G(\\mathbb{R}\\Gamma_B(\\Sigma_B^ olinebreak \\infty X\\cup_A B)). The results unify geometric and homotopical perspectives in equivariant and fiberwise topology, offering a canonical bridge between framed G-bordism and equivariant stable homotopy theory with fixed-point compatibilities.

Abstract

The Pontryagin-Thom theorem gives an isomorphism from the cobordism group of framed -manifolds to the th stable homotopy group of the sphere spectrum. In this paper, we prove the generalization of the Pontryagin-Thom theorem for families of framed equivariant manifolds parameterized over a compact base space.

Paper Structure

This paper contains 7 sections, 10 theorems, 43 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The equivariant version
  3. Preliminaries
  4. Equivariant framed bordism
  5. Equivariant stable homotopy theory
  6. Isotropy separation
  7. Proof of the equivariant Pontryagin-Thom theorem

Key Result

Theorem 1.1

Let $G$ be a product of a finite group and a torus. There is an isomorphism from the group of $V$-framed $G$-manifolds admitting a continuous equivariant map of pairs $(M,\partial M)\to (X,A)$ to the $V$th equivariant stable homotopy group of the suspension spectrum of $X/A$. Here a $V$-framed manifold is a manifold $M$ equipped with an equivalence class of stable bundle is

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 51 more