Homotopy Types of Suspended $4$-manifolds

TL;DR

This paper determines the homotopy types of double suspensions of certain 4-manifolds, revealing their decomposition into elementary complexes and conditions for desuspension, advancing understanding of 4-manifold topology.

Contribution

It provides a detailed homotopy decomposition of the double suspension of 4-manifolds with 2-torsion, using Postnikov and Pontryagin squares to identify desuspension conditions.

Findings

Homotopy decomposition of $oldsymbol{ ext{Σ}^2 M}$ into elementary complexes.

Conditions for $oldsymbol{ ext{Σ}^2 M}$ to desuspend to $oldsymbol{ ext{Σ} M}$.

Application of Postnikov and Pontryagin squares in homotopy analysis.

Abstract

Given a closed, smooth, connected, orientable $4$-manifold $M$, whose integral homology groups can have $2$-torsion, we determine the homotopy decomposition of the double suspension $\Sigma^2M$ as wedge sums of some elementary $\mathbf{A}_3^3$-complexes, which are $2$-connected finite complexes of dimension at most $6$. Furthermore, we utilize the Postnikov square (or equivalently Pontryagin square) to find sufficient conditions for the homotopy decompositions of $\Sigma^2M$ to desuspend to that of $\Sigma M$.