On Modular Cohomotopy Groups

TL;DR

This paper determines modular cohomotopy groups for CW-complexes using classical cohomology operations and unstable homotopy theory, providing explicit calculations and extension conditions, especially for the case when p=2 and dimension up to 6.

Contribution

It explicitly computes modular cohomotopy groups and analyzes their extension properties, advancing understanding of homotopy classes into Moore spaces.

Findings

Determined modular cohomotopy groups up to extensions.

Provided conditions for splitness of extensions.

Calculated specific groups like c3^3(X;) for dim(X)  6.

Abstract

Let $p$ be a prime and let $\pi^n(X;\mathbb{Z}/p^r)=[X,M_n(\mathbb{Z}/p^r)]$ be the set of homotopy classes of based maps from CW-complexes $X$ into the mod $p^r$ Moore spaces $M_n(\mathbb{Z}/p^r)$ of degree $n$, where $\mathbb{Z}/p^r$ denotes the integers mod $p^r$. In this paper we firstly determine the modular cohomotopy groups $\pi^n(X;\mathbb{Z}/p^r)$ up to extensions by classical methods of primary cohomology operations and give conditions for the splitness of the extensions. Secondly we utilize some unstable homotopy theory of Moore spaces to study the modular cohomotopy groups; especially, the group $\pi^3(X;\mathbb{Z}_{(2)})$ with $\dim(X)\leq 6$ is determined.