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.



Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex
I have no idea at the moment where to find a reference for the specific result you seek. However, it can be deduced from the following fact: topological manifolds (paracompact and Hausdorff) are absolute neighbourhood retracts, and thus have the homotopy type of CW-complexes. This is briefly stated in corollary 1 of Milnor's article [*On spaces having the homotopy type of a CW-complex*](http://www.ams.org/journals/tran/1959-090-02/S0002-9947-1959-0100267-4/) (published in Trans. Amer. Math. Soc. 90, 1959, pages 272-280). An extensive discussion of these matters is given in the thesis [*A topological manifold is homotopy equivalent to some CW-complex*](http://image.diku.dk/aasa/oldpage/aasa.pdf) by Aasa Feragen. For the specific case of compact manifolds, an elementary proof is given in the appendix of Hatcher's book *Algebraic topology* (see corollary A.12 there). 


We can now prove the result you state. Let $Y$ be a CW-complex admitting a homotopy equivalence $f : Y \to \partial M$ to the topological manifold $\partial M$. Denote by $j : \partial M \to M$ the inclusion of the boundary of $M$, and factor the composition
$$ Y \overset{f}{\longrightarrow} \partial M \overset{j}{\longrightarrow} M $$
as the inclusion of a sub-CW-complex $i : Y \to X$ followed by a weak equivalence $h : X \to M$. The weak equivalence $h$ is a homotopy equivalence since both its domain and its target have the homotopy type of CW-complexes.


Finally, let me briefly justify why we can take $i : Y \to X$ to be the inclusion of a subcomplex of a CW-complex. The usual construction of the factorization above produces $(X,Y)$ as a *relative* CW-complex, i.e. $X$ is obtained from $Y$ by adding cells in increasing order of their dimension. Now observe that this construction can be modified to inductively build $X$ as a CW-complex itself: at each stage of the construction of $X$, we can replace the attaching maps of the cells with homotopic cellular maps using the [cellular approximation theorem](https://en.wikipedia.org/wiki/Cellular_approximation_theorem).


