The suspension of a 4-manifold and its applications

TL;DR

This paper provides a homotopy decomposition of the suspension of certain 4-manifolds, enabling calculations of generalized cohomology theories and understanding the homotopy types of related gauge and current groups.

Contribution

It introduces a new homotopy decomposition of the suspension of 4-manifolds with specific homology conditions, facilitating cohomology and homotopy group computations.

Findings

Homotopy decomposition of the suspension in terms of spheres, Moore spaces, and $\Sigma ext{C}P^{2}$

Calculation of generalized cohomology groups of the manifold

Determination of homotopy types of certain current and gauge groups

Abstract

Let $M$ be a smooth, orientable, closed, connected $4$-manifold and suppose that $H_1(M;\mathbb{Z})$ is finitely generated and has no $2$-torsion. We give a homotopy decomposition of the suspension of $M$ in terms of spheres, Moore spaces and $\Sigma\mathbb{C}P^{2}$. This is used to calculate any reduced generalized cohomology theory of $M$ as a group and to determine the homotopy types of certain current groups and gauge groups.