Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
    
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. –  Dan Ramras Feb 27 '11 at 3:06
add comment

2 Answers 2

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: tom Dieck, Tammo Partitions of unity in homotopy theory. Composito Math. 23 (1971), 159–167

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 $$ |S.B| \leftarrow \quad|S.A| \rightarrow |S.C| $$

$$ \qquad \downarrow \qquad \qquad \qquad \downarrow \qquad \qquad \quad \downarrow \qquad $$

$$ B \qquad \leftarrow \qquad A \qquad \to C $$

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.

share|improve this answer
    
Of course, the gluing lemma is a special case of a more general fact, namely the homotopy invariance property of homotopy colimits. –  John Klein Feb 27 '11 at 22:58
add comment

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.