Suspension Splitting and Cohomotopy Sets of Simply Connected $7$-manifolds

TL;DR

This paper investigates the homotopy decomposition of suspended simply connected 7-manifolds and applies these results to analyze their cohomotopy sets, providing new insights into their topological structure.

Contribution

It establishes prime-localized homotopy decompositions of suspension spaces of 7-manifolds and applies these to study their cohomotopy sets, advancing understanding of their topological properties.

Findings

Homotopy decompositions of $oldsymbol{\Sigma M}$ into simpler spaces at primes.

Descriptions of cohomotopy sets $oldsymbol{\pi^k(M)}$ and $oldsymbol{\pi^4(M;oldsymbol{\mathbb{Z}_{(p)}})}$.

New methods for analyzing the topology of simply connected 7-manifolds.

Abstract

Let $M$ be a closed simply connected $7$-manifold. In this paper we establish homotopy decompositions of the reduced suspension space $\Sigma M$ into a wedge sum of simpler spaces when localized at a set of primes. These decompositions are applied to study the cohomotopy sets $\pi^k(M)$ and the $p$-local cohomotopy sets $\pi^4(M;\mathbb{Z}_{(p)})$.