Splitting of homotopy idempotents revisited

TL;DR

This paper revisits fundamental results on homotopy idempotents, providing clearer proofs that certain homotopy idempotents split in specific categories, and addresses gaps in previous literature.

Contribution

It offers simplified, more accessible proofs of splitting results for homotopy idempotents and rectifies gaps in existing literature.

Findings

Homotopy idempotents in pointed connected CW complexes split.

Unpointed homotopy idempotents in finite-dimensional CW complexes split.

Proofs clarify and strengthen previous results.

Abstract

We are presenting proofs of fundamental results related to homotopy idempotents, proofs that are sufficiently simple so that even the author can understand them. The first one is that homotopy idempotents in the category of pointed connected CW complexes split and the second one is that unpointed homotopy idempotents in the category of finite-dimensional CW complexes split. Some of our proofs rectify gaps in the existing literature.