Cohomotopy sets of 4-manifolds

TL;DR

This paper uses geometric methods to compute cohomotopy sets of 4-manifolds, completing a long-standing project and extending recent homotopy theory results, with applications to Morse 2-functions.

Contribution

It provides elementary geometric proofs for cohomotopy sets of 4-manifolds, completing previous work and extending recent homotopy theoretic results.

Findings

Computed homotopy classes of maps from 4-manifolds to S^3

Enumerated homotopy classes of maps to S^2

Established a framework for Morse 2-functions on 4-manifolds

Abstract

Elementary geometric arguments are used to compute the group of homotopy classes of maps from a 4-manifold X to the 3-sphere, and to enumerate the homotopy classes of maps from X to the 2-sphere. The former completes a project initiated by Steenrod in the 1940's, and the latter provides geometric arguments for and extensions of recent homotopy theoretic results of Larry Taylor. These two results complete the computation of all the cohomotopy sets of closed oriented 4-manifolds and provide a framework for the study of Morse 2-functions on 4-manifolds, a subject that has garnered considerable recent attention.