TITLE: Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex
QUESTION [13 upvotes]: Let $M$ be a compact manifold (possibly non-smooth) manifold with boundary $\partial M$.
Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a CW-complex, i.e. is there a CW-complex $X$ with a subcomplex $Y$ and homotopy equivalences $g\colon X\rightarrow M$ and $h\colon Y\rightarrow \partial M$, such that the obvious diagram commutes up to homotopy?
I think this must be true, but I can't find a reference. I would be also interested, if someone has one for the smooth case.

REPLY [4 votes]: Kirby and Siebenmann proved that any compact topological manifold has the homotopy type of a finite complex (announced in Kirby, R. C.; Siebenmann, L. C. On the triangulation of manifolds and the Hauptvermutung. Bull. Amer. Math. Soc. 75 1969 742–749, with proofs in their book "Foundational essays on Topological Manifolds").