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A Guide for Computing Stable Homotopy Groups

Agnes Beaudry, Jonathan A. Campbell

TL;DR

The paper provides a concise, practitioner-oriented tour of stable homotopy techniques relevant to Freed–Hopkins’ invertible extended TQFT classification, focusing on low-dimensional computations via the Adams spectral sequence. It surveys the stable homotopy category, the Steenrod algebra and its finite subalgebras, and the construction and use of Thom and Madsen–Tillmann spectra, all framed to compute $\pi_*MTH$ and related cobordism theories. By reducing Ext computations to $\mathcal{A}_1$-modules and illustrating with explicit cases ($MTG^{+}$, $MTG^{-}$, $MTG^{0}$, and complex analogues), it demonstrates how to extract concrete homotopy groups in small degrees and to connect these results to the FH tables. The work emphasizes practical algebraic topology tools—Anderson/Brown–Peterson dualities, Adams resolutions, and change-of-rings—yielding explicit, usable data for classifying low-dimensional topological field theories in physics. Overall, the text bridges deep homotopy-theoretic machinery with physically motivated cobordism problems, enabling concrete, readable computations in a critical range of dimensions.

Abstract

This paper contains an overview of background from stable homotopy theory used by Freed--Hopkins in their work on invertible extended topological field theories. We provide a working guide to the stable homotopy category, to the Steenrod algebra and to computations using the Adams spectral sequence. Many examples are worked out in detail to illustrate the techniques.

A Guide for Computing Stable Homotopy Groups

TL;DR

The paper provides a concise, practitioner-oriented tour of stable homotopy techniques relevant to Freed–Hopkins’ invertible extended TQFT classification, focusing on low-dimensional computations via the Adams spectral sequence. It surveys the stable homotopy category, the Steenrod algebra and its finite subalgebras, and the construction and use of Thom and Madsen–Tillmann spectra, all framed to compute and related cobordism theories. By reducing Ext computations to -modules and illustrating with explicit cases (, , , and complex analogues), it demonstrates how to extract concrete homotopy groups in small degrees and to connect these results to the FH tables. The work emphasizes practical algebraic topology tools—Anderson/Brown–Peterson dualities, Adams resolutions, and change-of-rings—yielding explicit, usable data for classifying low-dimensional topological field theories in physics. Overall, the text bridges deep homotopy-theoretic machinery with physically motivated cobordism problems, enabling concrete, readable computations in a critical range of dimensions.

Abstract

This paper contains an overview of background from stable homotopy theory used by Freed--Hopkins in their work on invertible extended topological field theories. We provide a working guide to the stable homotopy category, to the Steenrod algebra and to computations using the Adams spectral sequence. Many examples are worked out in detail to illustrate the techniques.

Paper Structure

This paper contains 34 sections, 11 theorems, 163 equations, 37 figures.

Table of Contents

  1. Introduction and organization
  2. Introduction
  3. Organization
  4. A working guide to spectra
  5. The categories
  6. Homotopies and homotopy groups
  7. The homotopy category of spectra and its triangulation
  8. Cohomology and Homology Theories.
  9. Connective spectra
  10. Multiplicative Homology Theories.
  11. Spanier--Whitehead duality
  12. Brown-Comenetz and Anderson duality
  13. Thom Spectra
  14. The Steenrod algebra
  15. Cohomology operations and the Steenrod algebra
  16. ...and 19 more sections

Key Result

Theorem 3.1.1

For each $k \geq 0$, there exists a stable cohomology operation of degree $k$ called the $k$th Steenrod square. For $X$ a topological space, the Steenrod squares satisfy the following properties:

Figures (37)

  • Figure 1: A cell diagram, used to depict the Steenrod operations on the cohomology of a space or spectrum. Each $\bullet$ denotes a generator of ${{\mathbb{Z}}}/2$. The difference in cohomological degree of the generators is represented vertically. Straight lines denote the action of $Sq^1$ and curved lines denote the action of $Sq^2$.
  • Figure 2: The structure of $H^*(\mathbb{R} P^2)$ (left), $H^*(\mathbb{C} P^2)$ (right) as modules over $\mathcal{A}$. The class $w_1$ is in $H^1(\mathbb{R} P^{2})$. The class $w_2$ is in $H^2(\mathbb{C} P^2)$.
  • Figure 3: $\mathcal{A}_1$ (left) and its subalgebra $\mathcal{E}_1$ (right). The dashed lines represent the action of $Q_1 = Sq^1Sq^2 + Sq^2 Sq^1$.
  • Figure 4: From the left, the structures of $H^*(BO_1)$, $H^*(MO_1)$, $H^*(BU_1)$ and $H^*(MU_1)$ as $\mathcal{A}_1$-modules.
  • Figure 5: The $\mathcal{A}_1$-submodule of $H^*(MU_1 \wedge MO_1)$ generated by $U \otimes U$, $U\otimes w_1^2U + w_2U \otimes U$, $w_2^2U \otimes U$ and $w_2^2U \otimes w_1^2U + w_2^3U \otimes U$. The class $U\otimes U \in H^3( MU_1 \wedge MO_1)$. All classes of of degree $* \leq 5$ in $H^*(\Sigma^{-3} MU_1 \wedge MO_1)$ are contained in this submodule.
  • ...and 32 more figures

Theorems & Definitions (82)

  • Definition 2.1.1: Prespectra
  • Remark 2.1.1
  • Definition 2.1.2: Spectra
  • Definition 2.1.3: CW-prespectra
  • Example 2.1.1
  • Example 2.1.2
  • Example 2.1.3
  • Example 2.1.4
  • Definition 2.1.4
  • Remark 2.1.2
  • ...and 72 more