The Functor $A^{\min}$ for $(p-1)$-cell Complexes and $\EHP$ Sequences

TL;DR

This paper investigates the structure of a specific functor applied to certain co-$H$-spaces with even-dimensional cells, determining their homology and establishing an $ ext{EHP}$ sequence to understand their loop space properties.

Contribution

It introduces the functor $ar A^{ ext{min}}$ for $(p-1)$-cell complexes and explicitly computes the homology of the resulting spaces, along with their $ ext{EHP}$ sequences.

Findings

Homology of $ar A^{ ext{min}}(X)$ determined.

$ ext{EHP}$ sequence constructed for $ar A^{ ext{min}}(X)$.

Retraction of loop space $ar A^{ ext{min}}(X)$ identified.

Abstract

Let $X$ be a co-$H$-space of $(p-1)$-cell complex with all cells in even dimensions. Then the loop space $\Omega X$ admits a retract $\bar A^{\min}(X)$ that is the evaluation of the functor $\bar A^{\min}$ on $X$. In this paper, we determine the homology $H_*(\bar A^{\min}(X))$ and give the $\EHP$ sequence for the spaces $\bar A^{\min}(X)$.