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A geometric computation of cohomotopy groups in co-degree one

Michael Jung, Thomas O. Rot

TL;DR

This work solves the geometric computation of the cohomotopy set $\pi^n(X)$ for closed $(n+1)$-manifolds with $n\ge3$ by translating it into a problem on normally framed 1-dimensional submanifolds via the Pontryagin–Thom construction. It distinguishes Type I and Type II manifolds to describe the exact extension data: Type I yields an isomorphism $\mathbb{F}_1(X)\to H_1(X; o_X)$, while Type II yields a nontrivial extension classified by $\mathrm{Ext}(H_1(X; o_X), \mathbb{Z}_2)$ that maps to $w_1^2+w_2$; the extension splits precisely when $X$ admits a $\mathrm{Pin}^-$-structure, linking splitting maps to Spin/Pin$^-$-structures. The paper further develops a refined obstruction to nonvanishing sections of oriented spin vector bundles through a degree invariant $\kappa(E)$ in Type IIa and shows how these invariants generalize Konstantis’ counting invariant in the orientable case. Overall, the results extend Kirby–Melvin–Teichner and Konstantis to non-orientable and non-spin settings, providing explicit geometric obstructions for vector-bundle sections and a complete cohomotopy description in co-degree one.

Abstract

Using geometric arguments, we compute the group of homotopy classes of maps from a closed $(n+1)$-dimensional manifold to the $n$-sphere for $n \geq 3$. Our work extends results from Kirby, Melvin and Teichner for closed oriented 4-manifolds and from Konstantis for closed $(n+1)$-dimensional spin manifolds, considering possibly non-orientable and non-spinnable manifolds. In the process, we introduce two types of manifolds that generalize the notion of odd and even 4-manifolds. Furthermore, for the case that $n \geq 4$, we discuss applications for rank $n$ spin vector bundles and obtain a refinement of the Euler class in the cohomotopy group that fully obstructs the existence of a non-vanishing section.

A geometric computation of cohomotopy groups in co-degree one

TL;DR

This work solves the geometric computation of the cohomotopy set for closed -manifolds with by translating it into a problem on normally framed 1-dimensional submanifolds via the Pontryagin–Thom construction. It distinguishes Type I and Type II manifolds to describe the exact extension data: Type I yields an isomorphism , while Type II yields a nontrivial extension classified by that maps to ; the extension splits precisely when admits a -structure, linking splitting maps to Spin/Pin-structures. The paper further develops a refined obstruction to nonvanishing sections of oriented spin vector bundles through a degree invariant in Type IIa and shows how these invariants generalize Konstantis’ counting invariant in the orientable case. Overall, the results extend Kirby–Melvin–Teichner and Konstantis to non-orientable and non-spin settings, providing explicit geometric obstructions for vector-bundle sections and a complete cohomotopy description in co-degree one.

Abstract

Using geometric arguments, we compute the group of homotopy classes of maps from a closed -dimensional manifold to the -sphere for . Our work extends results from Kirby, Melvin and Teichner for closed oriented 4-manifolds and from Konstantis for closed -dimensional spin manifolds, considering possibly non-orientable and non-spinnable manifolds. In the process, we introduce two types of manifolds that generalize the notion of odd and even 4-manifolds. Furthermore, for the case that , we discuss applications for rank spin vector bundles and obtain a refinement of the Euler class in the cohomotopy group that fully obstructs the existence of a non-vanishing section.
Paper Structure (7 sections, 35 theorems, 35 equations, 3 figures, 1 table)

This paper contains 7 sections, 35 theorems, 35 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The kernel of the forgetful map
  4. Manifolds of type I
  5. Manifolds of type II
  6. Manifolds of type IIa
  7. Application to vector bundles

Key Result

Theorem 1.1

If $X$ is of type I, the forgetful map $h \colon \mathbb{F}_1(X) \to H_1(X; o_X)$ is an isomorphism. If $X$ is of type II, then there is a short exact sequence of abelian groups \begin{tikzcd}[sep=small] 0 \arrow[r] & \ZZ_2 \arrow[r] & \Fr_1(X) \arrow[r, "h"] & H_1(X; o_X) \arrow[r]

Figures (3)

  • Figure 1: The surface $\Sigma = D \cup \Sigma_0$ in $X \times [0,1]$ where $\nu_\Sigma$ is oriented but not trivializable with many suppressed directions.
  • Figure 2: The construction of the framed null-bordism in \ref{['prop:framing_inward_stretch']}.
  • Figure 3: The construction of framed bordisms in \ref{['lem:rank1_subbundle_bordism']}. Left the case when $\lambda$ is trivial, and right when $\lambda$ is the non-trivial bundle.

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 59 more