Doug Ravenel (URochester) asks: %Date: Fri, 18 Oct 91 18:01:01 EDT %From: Doug Ravenel <drav@uhura.cc.rochester.edu> %Subject: Query on homotopy commutative diagrams %Dear experts, % I recently reproved the following lemma about homotopy commutative % diagrams after Peter May found an error in the proof I published 10 years ago. %(The following is written in LateX.) \documentstyle{article} \begin{document} Suppose are given a collection of pointed CW--complexes (or CW--spectra) $X_{i, j}$ for $i, j \in \ints$, maps $f:X_{i, j} \to X_{i + 1, j}$ and $g:X_{i, j} \to X_{i, j + 1}$ (we can safely omit the subscripts on $f$ and $g$) so that the diagram $$ \begin{array}{ccccc} X_{i, j} &\stackrel{f}{\longrightarrow} &X_{i + 1, j} &\stackrel{f}{\longrightarrow} & \cdots\\ {}\sp{g}\downarrow & &\downarrow {}\sp{g}\\ X_{i, j + 1} &\stackrel{f}{\longrightarrow} &X_{i + 1, j + 1} &\stackrel{f}{\longrightarrow} & \cdots\\ {}\sp{g}\downarrow & &\downarrow {}\sp{g}\\ \vdots & &\vdots \end{array} $$ commutes up to homotopy, and for each $(i, j)$ we are given a homotopy $$ X_{i, j} \times I \stackrel{h}{\longrightarrow} X_{i + 1, j + 1} $$ with $h(x, 0) = fg(x)$ and $h(x, 1) = gf(x)$. Then this diagram is homotopy equivalent to a {\em strictly} commutative diagram with spaces $X'_{i, j}$ and maps $f'$ and $g'$, each of which is a cofibration (i.e., an inclusion). The construction of this new diagram depends on the given homotopies $h$. For each $(i, j)$ there are homotopy equivalences $$ \alpha:X_{i, j} \to X'_{i, j} \qquad\mbox{and}\qquad \beta:X'_{i, j} \to X_{i, j} $$ with $f'\alpha$ homotopic to $\alpha f$, $g'\alpha$ homotopic to $\alpha g$, $\beta f'$ homotopic to $f\beta$, and $\beta g'$ homotopic to $g\beta$. \end{document} My first question is, has anyone heard of this before? The proof is elementary and should have been done thirty years ago. I would like to generalize this result to $n$--dimensional diagrams equipped with suitable higher homotopies. The proof I have in mind is still elementary in spirit but could involve some tricky combinatorics that I would like to avoid. In particular it reqires a tessellation of $n$--space with certain properties. I suspect that all of the relevant higher homotopies have been written down somewhere but I do not know where. I also suspect that someone has developed some machinery that will enable one to prove this quickly. Any suggestions? Thanks for your help. Doug Ravenel \end{document} Seems to me it is a case of rectification but I defer to the experts for the precise details/reference, especially with regard to the topology involved. jim ======================