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

TL;DR

The paper advances the understanding of cohomotopy sets for highly connected $(2n+2)$-manifolds with $2$-torsion-free homology by combining the Postnikov tower of spheres with a precise suspension-decomposition of $\Sigma M$. It provides complete unstable cohomotopy descriptions for $n=2,3,4$ (i.e., 6-, 8-, 10-manifolds) and analyzes the stable range, yielding explicit formulas and exact sequences governed by Steenrod operations and torsion phenomena. Key contributions include explicit structures for $\pi^5$, $\pi^6$, $\pi^3$ in low dimensions, demonstrations of abelianness for certain cases, and detailed decompositions $\Sigma M\simeq$ wedges of spheres, Moore spaces, and Chang-type complexes with attaching maps described by concrete coefficients. The results sharpen the link between cohomotopy, Steenrod operations, and manifold topology, with potential implications for geometry and gauge theory via the unstable cohomotopy 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 $ΣM$ to investigate the cohomotopy sets $π^\ast(M)$ for $n=2,3,4$, under the assumption that $M$ has $2$-torsion-free homology. All cohomotopy sets $π^i(M)$ of such manifolds $M$ are characterized except $π^4(M)$ for $n=3,4$.

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2: cf. TodaBook
• Lemma 2.3: see HL-7mfd
• Lemma 2.4