# Is $\partial \Gamma\hookrightarrow \Gamma$ a Serre cofibration?

Question : Let $M$ be a (say smooth, possibly non compact) manifold with boundary. Is the inclusion $\partial M\hookrightarrow M$ a cofibration in the Serre-Quillen model structure of topological spaces ?

I'm only really interested in the following special case :

Let $C\subset\Bbb R^n$ be a closed convex subset with non empty interior. Assume, if necessary, $C$ to be a (possibly unbounded) polyhedron, in that locally near any of its points, $C$ looks like the intersection of a finite number of closed half-spaces $D_i=\lbrace x\mid\varphi(x)\geq\alpha\rbrace$, $i=1,\dots,N$, with linear forms $\varphi_i$ and real numbers $\alpha_i$. Let $F\subset\partial C$ be a closed subset of the boundary; if necessary, assume $F$ to be the union of facets of the polyhedron $C$.

It seems reasonable to expect $\Gamma=C-F$ to be homeomorphic to $B-G$, where $B$ is the closed unit ball, and $G\subset S$ is a closed subset of the unit sphere $S=\partial B$, so that $\Gamma$ is homeomorphic to the smooth manifold with boundary $B-G$ (whose boundary as a manifold ought to be $S-G$).

Real Question : let $G\subset S$ be a closed subset of the sphere, is the inclusion $S-G\hookrightarrow B-G$ a Serre cofibration ?

If this fails for general closed subsets $G\subset S$, will $\partial^{\text{man}}\Gamma\hookrightarrow \Gamma$, that is $\partial C-F\hookrightarrow C-F$ for $C$ closed polyhedron, and $F$ a closed subset of $\partial C$ union of facets, be a Serre cofibration ?

I expect the inclusion of the boundary of a smooth manifold $M$ to be a Hurewicz cofibration by virtue of the existence of collarings. I was told, and have found supporting evidence online, that smooth manifolds admit smooth triangulations, so that one might expect $(M,\partial M)$ to be pair homeomorphic to $(X,A)$ for $X$ a CW complex and $A$ a subcomplex.

Are these guesses correct ? Are inclusions $\partial C-F\hookrightarrow C-F$ Serre cofibrations ? A simple Yes / No answser, and if possible, a reference would satisfy me!

(I picked what seemed like a reasonable set of tags, but feel free to change them.)

EDIT. Milnor-Stasheff provide just the theorem I need in chapter 20 of their book Characteristic Classes, along with two references, one being @ThiKu's reference to Whitehead's paper.

• If you believe that a smooth manifold and its boundary can be realised by a pair of CW-complexes, then this is just the elementary fact that inclusion of CW-complexes is a cofibration. – ThiKu May 29 '17 at 10:41
• @ThiKu This is why I included such a statement in the post. Do you have a reference for that fact ? For the pair $(M,\partial M)$ being homeomorphic to a pair $(X,A)$ ? Also, is the assertion that $(\Gamma,\partial^\text{man}\Gamma)$ is homeomorphic to $(B-G,S-G)$ correct ? Do you have a reference for this ? – Olivier Bégassat May 29 '17 at 10:42
• No, at least it is not stated as such in Whitehead's paper maths.ed.ac.uk/~aar/papers/jhcw4.pdf . One may still try whether Whitehead's proof may give this stronger result as well. – ThiKu May 29 '17 at 10:44
• In any case, you don't need a triangulation, a CW decomposition will be enough. – ThiKu May 29 '17 at 10:46
• @ThiKu Yes${}$. – Olivier Bégassat May 29 '17 at 10:49

Regarding the first question: assume $M$ is compact. According to the "fundamental theorem" of Morse theory, there is a filtration $$M_{-1} \subset M_0\subset \cdots \subset M_m = M$$ where $M_{-1} = \partial M \times [0,1]$ and $M_k$ is obtained from $M_{k-1}$ by attaching a finite collection (possibly empty) of $k$-handles (a $k$-handle is of the form $D^k \times D^{m-k}$ and is attached to $M_{k-1}$ along an embedding $S^{k-1} \times D^{m-k} \to \partial_+ M_{k-1}$, where $\partial_+ M_{k-1} := \partial M_{k-1} \setminus \partial M$).
The Serre cofibration property for $\partial M \subset M$ is then immediate.