A note on generalized equivariant homotopy groups

TL;DR

This paper extends the concept of equivariant homotopy groups, introduces generalized Rhodes and Fox homotopy groups, and explores their algebraic relationships and properties within a broader mathematical framework.

Contribution

It generalizes equivariant homotopy groups, establishes exact sequences linking these groups, and introduces $\Gamma$-Rhodes groups with co-grouplike structures.

Findings

Established a short exact sequence relating generalized Rhodes and Fox homotopy groups.

Introduced $\Gamma$-Rhodes groups with co-grouplike structures.

Discussed evaluation subgroups of $\Gamma$-Rhodes groups.

Abstract

In this paper, we generalize the equivariant homotopy groups or equivalently the Rhodes groups. We establish a short exact sequence relating the generalized Rhodes groups and the generalized Fox homotopy groups and we introduce $\Gamma$-Rhodes groups, where $\Gamma$ admits a certain co-grouplike structure. Evaluation subgroups of $\Gamma$-Rhodes groups are discussed.