# Cancellation and homotopy rigidity of classical functors

## TL;DR

This paper establishes prime decompositions for certain spaces in homotopy theory, develops a $p$-local version of Gray's correspondence, and proves homotopy rigidity of specific classical functors on $p$-local spaces.

## Contribution

It introduces a prime decomposition theorem for co-$H$-spaces and $H$-spaces, and demonstrates homotopy rigidity of $ ext{Sigma} ext{Omega}$ and $ ext{Omega}$ functors on $p$-local spaces.

## Key findings

- Prime decompositions are unique in the homotopy category for these spaces.
- A $p$-local Gray's correspondence is established.
- Homotopy rigidity of $ ext{Sigma} ext{Omega}$ and $ ext{Omega}$ on certain spaces is proved.

## Abstract

We first show that simply connected co-$H$-spaces and connected $H$-spaces can be uniquely decomposed into prime factors in the homotopy category of pointed $p$-local spaces of finite type, which is used to develop a $p$-local version of Gray's correspondence between homotopy types of prime co-$H$-spaces and homotopy types of prime $H$-spaces, and the split fibration which connects them as well. Further, we use the unique decomposition theorem to study the homotopy rigidity problem for classic functors. Among others, we prove that $\Sigma \Omega$ and $\Omega$ are homotopy rigid on simply connected $p$-local co-$H$-spaces of finite type, and $\Omega\Sigma $ and $\Sigma$ are homotopy rigid on connected $p$-local $H$-spaces of finite type.