Cubical Acyclic Homotopy Excision

TL;DR

This paper investigates the relationship between homotopy pushout cubes and homotopy pullback cubes in algebraic topology, providing explicit measures of their differences using homotopy fibres and colimits.

Contribution

It introduces a method to quantify how a homotopy pushout cube deviates from being a homotopy pullback cube using homotopy fibres and colimits.

Findings

Explicit formula for the difference in terms of homotopy fibres

Connection between pushout and pullback cubes in homotopy theory

Tools for measuring homotopy colimit deviations

Abstract

Given a strong homotopy pushout cube of spaces A, we measure how far it is from also being a homotopy pullback cube. Explicitly, letting P be the homotopy colimit of the diagram obtained from A by forgetting the initial vertex $A_\varnothing$, we study the homotopy fibre of the double suspension of the comparison map $A_\varnothing \to P$. This difference is expressible in terms of the homotopy fibres of the original maps in A.