# Codimension one submanifold gives cofibrant pair

Let $$M$$ be a smooth manifold, and $$N$$ be an embedded smooth submanifold of $$M$$ with $$\partial M=\varnothing=\partial N$$. Suppose, $$\dim M-\dim N=1$$, and $$N$$ is a closed subset of $$M$$.

Does the inclusion map $$i: N\hookrightarrow M$$ a cofibration, i.e. does this pair $$(M, N)$$ have homotopy extension property?

$$\bullet$$ I am not assuming $$N$$ is compact in general. In case, $$N$$ is compact the uniform tubular neighborhood gives positive solution.

Can someone suggest me some references?

• Milnor&Stasheff's "Characteristic Classes" Theorem 11.1 claims that tubular neghbourhoods exist in general, and cite Lang's "Introduction to differentiable manifolds" for the proof. I also believe that it should be possible to triangulate the pair $(M,N)$, so that the result follows from the general fact that a subcomplex of a CW-complex is a cofibration, see my answer to mathoverflow.net/q/206212/8103 and the comments there. (Not an answer as I don't have time to check these references myself!) Commented Mar 11, 2021 at 10:44