Suspension Homotopy of $(n-1)$-connected $(2n+2)$-dimensional Poincar\'{e} Duality Complexes

TL;DR

This paper investigates the homotopy decompositions of suspensions of certain high-dimensional Poincaré duality complexes, providing complete classifications for specific 6-manifolds and partial results for higher dimensions after localization.

Contribution

It offers a complete homotopy type classification for suspensions of simply-connected 6-manifolds with 2-torsion, and extends decomposition results to higher dimensions after localization away from 2.

Findings

Complete homotopy types for suspensions of 6-manifolds with 2-torsion.

Homotopy decompositions for dimensions 3 to 5 after localization.

Extensions of decomposition techniques to higher dimensions.

Abstract

We study the homotopy decompositions of the suspension $\Sigma M$ of an $(n-1)$-connected $(2n+2)$ dimensional Poincar\'{e} duality complex $M$, $n\geq 2$. In particular, we completely determine the homotopy types of $\Sigma M$ of a simply-connected orientable closed (smooth) $6$-manifold $M$, whose integral homology groups can have $2$-torsion. If $3\leq n\leq 5$, we obtain homotopy decompositions of $\Sigma M$ after localization away from $2$.