Homotopy excision and cellularity

TL;DR

This paper investigates the relationship between homotopy push-outs and pull-backs in topological spaces, extending classical results like the Blakers-Massey Theorem by analyzing homotopy fibers and connectivity.

Contribution

It introduces a new perspective on comparing homotopy push-outs and pull-backs using homotopy fibers, generalizing classical connectivity theorems.

Findings

Established a measure of the difference between A and the homotopy pull-back via homotopy fibers.

Reproduced the classical Blakers-Massey Theorem as a special case.

Provided a framework for analyzing the connectivity of maps in homotopy diagrams.

Abstract

Consider a push-out diagram of spaces C <-- A --> B, construct the homotopy push-out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. We compare the difference between A and this homotopy pull-back. This difference is measured in terms of the homotopy fibers of the original maps. Restricting our attention to the connectivity of these maps, we recover the classical Blakers-Massey Theorem.