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Equivariant Framed 1-Manifolds and the Pontryagin-Thom Isomorphism

Lucas Williams

TL;DR

The paper provides explicit geometric generators for the low-lying equivariant stable stems of the sphere spectrum for $C_2$, by computing the images of the equivariant framed cobordism groups $\omega_1^{C_2}$ and $\omega_\sigma^{C_2}$ under the Pontryagin–Thom isomorphism. Using the tom Dieck splitting, it identifies $\pi_1^{C_2}(\mathbb{S}) \cong \pi_1(\mathbb{S}) \oplus H_0(BC_2;\mathbb{Z}/2) \oplus H_1(BC_2;\mathbb{Z})$ and gives explicit coordinates for the images of the generators $S^1$, $C_2\times S^1$, and $S(2\sigma)$ in terms of $(\pi_1(\mathbb{S}), H_0(BC_2;\mathbb{Z}/2), H_1(BC_2;\mathbb{Z}))$, yielding maps $S^1_n\mapsto (n,0,0)$, $C_2\times S^1_n\mapsto (0,n,0)$, and $S(2\sigma)_n\mapsto (0,n+1,1)$. For the $\sigma$-framed case, the paper shows $\pi_\sigma^{C_2}(\mathbb{S})\cong \mathbb{Z}$ is generated by the equivariant Hopf map $\eta$, with $S(1+\sigma)_n$ mapping to $n \bmod 2$, and explains why the equivariant Hopf map has infinite order. The results illuminate the interplay between equivariant framing, orbifold contributions, and stable homotopy, and provide a clean geometric dataset for further study in equivariant stable homotopy theory and related areas.

Abstract

The Pontryagin-Thom theorem gives an isomorphism between the cobordism group of framed $n$-dimensional manifolds, $ω_n$, and the $n^{th}$ stable homotopy group of the sphere spectrum, $π_n(\mathbb{S})$. The equivariant analogue of this theorem, gives an isomorphism between the equivariant cobordism group of $V$-framed $G$-manifolds, $ω_V^G$, and the $V^{th}$ equivariant stable homotopy group of the $G$-sphere spectrum, $π_V^G(\mathbb{S})$, for a finite group $G$ and a $G$-representation, $V$. In this paper, we explicitly identify the images of each element of $ω_1^{C_2}$ and $ω_σ^{C_2}$ in $π_1^{C_2}(\mathbb{S})$ and $π_σ^{C_2}(\mathbb{S})$ under the equivariant Pontryagin-Thom isomorphism.

Equivariant Framed 1-Manifolds and the Pontryagin-Thom Isomorphism

TL;DR

The paper provides explicit geometric generators for the low-lying equivariant stable stems of the sphere spectrum for , by computing the images of the equivariant framed cobordism groups and under the Pontryagin–Thom isomorphism. Using the tom Dieck splitting, it identifies and gives explicit coordinates for the images of the generators , , and in terms of , yielding maps , , and . For the -framed case, the paper shows is generated by the equivariant Hopf map , with mapping to , and explains why the equivariant Hopf map has infinite order. The results illuminate the interplay between equivariant framing, orbifold contributions, and stable homotopy, and provide a clean geometric dataset for further study in equivariant stable homotopy theory and related areas.

Abstract

The Pontryagin-Thom theorem gives an isomorphism between the cobordism group of framed -dimensional manifolds, , and the stable homotopy group of the sphere spectrum, . The equivariant analogue of this theorem, gives an isomorphism between the equivariant cobordism group of -framed -manifolds, , and the equivariant stable homotopy group of the -sphere spectrum, , for a finite group and a -representation, . In this paper, we explicitly identify the images of each element of and in and under the equivariant Pontryagin-Thom isomorphism.
Paper Structure (9 sections, 14 theorems, 38 equations, 2 figures)

This paper contains 9 sections, 14 theorems, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organization
  3. Acknowledgments
  4. Preliminaries
  5. Equivariant framed cobordism
  6. The equivariant framed Pontryagin-Thom isomorphism
  7. Converting Tangent and Normal Bundle Trivializations
  8. The image of TEXT in TEXT
  9. The image of TEXT in TEXT

Key Result

Theorem 1.2

The equivariant Pontryagin-Thom isomorphism sends the elements of $\omega_1^{C_2}$ to $\pi_1^{C_2}(\mathbb{S})\cong \mathbb{Z}/2^{\oplus 3}$ as depicted in the following table:

Figures (2)

  • Figure 1: The action of $F$ on the unit $z$ vectors in $T\mathbb R^{3}|_{S^1}$
  • Figure 2: The action of $F$ on the unit $x$ and $y$ vectors in $T\mathbb R^{3}|_{S^1}$

Theorems & Definitions (36)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Remark 2.7
  • ...and 26 more