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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."

Given

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.

Given a diagram $B \leftarrow A \to C$ of spaces homotopy equivalent to CW complexes 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 follow from be a consequence of Wall's results on finiteness conditions for CW complexes. The point I guess is , or maybe one could argue directly: it seems to me that the singular chains on if $f: X \to Y$ is a map of spaces, each having the double mapping cylinder homotopy type of a finite CW complex, then there is quasi-isomorphic a factorization $X \to chain level pushout Y' \to Y$ in which $C(B) Y'$ 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 $B' \leftarrow C(A) A' \to C(C)$ of singular chain complexes, and C'$which maps by homotopy equivalences to$B \leftarrow A \to C$such that each of these chain complexes space in the new diagram is chain homotopy a finite , so complex and each map is a cofibration. Then the chain level pushout$B' \cup_{A'} C'$has the homotopy type of the original double mapping cylinder and it is tooalso a finite CW complex. 1 In Corollary 5.12 of Whitehead's book it is shown that that the cobase change of a homotopy equivalence along a cofibration (NDR pair) is again a homotopy equivalence. This ought to imply homotopy invariance of the double mapping cylinder construction (with respect to homotopy equivalences of the spaces used to form the double mapping cylinder). This is sometimes called the "gluing lemma." Given a diagram$B \leftarrow A \to C$of spaces homotopy equivalent to CW complexes 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 follow from Wall's results on finiteness conditions for CW complexes. The point I guess is that the singular chains on the double mapping cylinder is quasi-isomorphic to chain level pushout$C(B) \leftarrow C(A) \to C(C)\$ of singular chain complexes, and each of these chain complexes is chain homotopy finite, so the chain level pushout is too.