A proof of the homotopy push-out and pull-back lemma (Q2719023)

This paper provides an explicit, detailed proof of the homotopy pushout and pullback lemma. In particular, the author proves that for NDR-pairs \((X,A)\) and \((Y,B)\), with inclusions \(i:A\rightarrow X\) and \(j:B\rightarrow Y\), and maps \(f:Z\rightarrow X\) and \(g:Z\rightarrow Y\) from a space \(Z\), the homotopy pushout of \(\Omega _{f,i}\leftarrow \Omega _{(f,g),i\times j}\rightarrow \Omega _{g,j}\) is naturally of the same homotopy type as the homotopy pullback of \(Z\rightarrow X\times Y\leftarrow X\times B\cup A\times Y\). Here \(\Omega _{f,i}\), \(\Omega _{(f,g),i\times j}\), and \(\Omega _{g,j}\) are the homotopy pullbacks of \((Z\buildrel f\over \rightarrow X\buildrel i\over \leftarrow A)\), \((Z\buildrel (f,g)\circ \Delta\over \longrightarrow X\times Y\buildrel i\times j\over \longleftarrow A\times B)\), and \((Z\buildrel g\over\rightarrow Y\buildrel j\over\leftarrow B)\), respectively. \textit{N. Iwase} established this result for connected CW pairs \((X,A)\) and \((Y,B)\) [Bull. Lond. Math. Soc. 30, No. 6, 623-634 (1998; Zbl 0947.55006)]. Both theorems are generalizations of a result of \textit{T. Ganea} [Comment. Math. Helv. 39, 295-322 (1965; Zbl 0142.40702)]. The author concludes by stating a version of the theorem for the category of quasi-fibrations.