Does anyone know a reference for the fact that a homotopy pushout (double mapping cylinder) of spaces which are homotopy equivalent to finite CW complexes is also homotopy equivalent to a finite CW complex?

$\begingroup$ You might take a look at the article by Gutzwiller and Mitchell in Transform. Groups, 14(3), 541–556, 2009. They prove something similar for infinite mapping telescopes (see Section 3.2 of their paper). I would think the same ideas extend to more general homotopy colimits, but I could be wrong. $\endgroup$– Dan RamrasFeb 27, 2011 at 3:06
2 Answers
In Corollary 5.12 of Whitehead's book it is shown that that the cobase change of a homotopy equivalence along a cofibration (NDR pair) is again a homotopy equivalence. This ought to imply homotopy invariance of the double mapping cylinder construction (with respect to homotopy equivalences of the spaces used to form the double mapping cylinder). A result along these lines is sometimes called the "gluing lemma."
ADDED: Here's a better reference for the gluing lemma: Tammo tom Dieck, Partitions of unity in homotopy theory, Composito Math. 23 (1971) pp 159–167 (Numdam)
Consider a diagram $B \leftarrow A \to C$ of spaces homotopy equivalent to CW complexes to which we will take the mapping cylinder. Use the map of diagrams $$ S.B \leftarrow \quadS.A \rightarrow S.C $$
$$ \qquad \downarrow \qquad \qquad \qquad \downarrow \qquad \qquad \quad \downarrow \qquad $$
$$ B \qquad \leftarrow \qquad A \qquad \to C $$
where $S.$ in each case means geometric realization of the total singular complex. Then homotopy invariance applied to the above shows that the double mapping cylinder of the top line, which is a CW complex, has the homotopy type of the double mapping cylinder of the bottom line.
As far as finiteness goes, that should be a consequence of Wall's finiteness conditions for CW complexes, or maybe one could argue directly: it seems to me that if $f: X \to Y$ is a map of spaces, each having the homotopy type of a finite CW complex, then there is a factorization $X \to Y' \to Y$ in which $Y'$ is obtained from $X$ by attaching finitely many cells and $Y' \to Y$ is a homotopy equivalence. If we use this, then it seems to me that we can find a diagram $B' \leftarrow A' \to C'$ which maps by homotopy equivalences to $B \leftarrow A \to C$ such that each space in the new diagram is a finite complex and each map is a cofibration. Then the pushout $B' \cup_{A'} C'$ has the homotopy type of the original double mapping cylinder and it is also a finite CW complex.

$\begingroup$ Of course, the gluing lemma is a special case of a more general fact, namely the homotopy invariance property of homotopy colimits. $\endgroup$ Feb 27, 2011 at 22:58
I don't have a reference, but here is an easier argument, based, like John's, on the homotopy invariance of the homotopy pushout.
The invariance implies that you can replace the maps $i: A\to B$ and $j: A\to C$ with homotopic maps and get the same homotopy pushout, up to homotopy type. So assume they are cellular. It is easy to give the inclusions $A\hookrightarrow M_i$ and $A\hookrightarrow M_j$ finite CW structures so that $A$ is a subcomplex of each, and then the union $M_i \cup_A M_j$ inherits a finite CW structure.