Cohomotopy Sets of $(n-1)$-connected $(2n+2)$-manifolds for small $n$

TL;DR

This paper investigates the cohomotopy sets of certain highly connected manifolds using Postnikov towers and homotopy decompositions, providing complete characterizations for most cases when the manifolds have torsion-free homology.

Contribution

It combines advanced homotopy theoretic tools to explicitly compute cohomotopy sets of specific manifolds, extending previous knowledge in the field.

Findings

Complete characterization of $\pi^i(M)$ for $n=2,3,4$ with torsion-free homology.

Identification of the unresolved case $\pi^4(M)$ for $n=3,4$.

Application of Postnikov towers and homotopy decompositions to manifold invariants.

Abstract

Let $M$ be a closed orientable $(n-1)$-connected $(2n+2)$-manifold, $n\geq 2$. In this paper we combine the Postnikov tower of spheres and the homotopy decomposition of the reduced suspension space $\Sigma M$ to investigate the cohomotopy sets $\pi^\ast(M)$ for $n=2,3,4$, under the assumption that $M$ has $2$-torsion-free homology. All cohomotopy sets $\pi^i(M)$ of such manifolds $M$ are characterized except $\pi^4(M)$ for $n=3,4$.