A purely homotopy theoretic proof of the Blakers-Massey Theorem for $n$-cubes

TL;DR

This paper provides a new, purely homotopy-theoretic proof of the Blakers-Massey Theorem for n-cubes, avoiding transversality and using elementary methods based on classical homotopy arguments.

Contribution

It introduces a homotopy-theoretic proof of a key lemma in the Blakers-Massey Theorem, simplifying the proof and removing the need for transversality arguments.

Findings

Proof is elementary and purely homotopy-theoretic.

Generalizes Puppe's argument for squares to n-cubes.

Simplifies the understanding of the Blakers-Massey Theorem.

Abstract

Goodwillie's proof of the Blakers-Massey Theorem for $n$-cubes relies on a lemma whose proof invokes transversality. The rest of his proof follows from general facts about cubes of spaces and connectivities of maps. We present a purely homotopy-theoretic proof of this lemma. The methods are elementary, using a generalization and modification of an argument originally due to Puppe used to prove the Blakers-Massey Theorem for squares.