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Let $X$ be a topological manifold with boundary.What is the idea behind the fact that emoving the boundary doesn't change the homotopy type of the manifold; i.e.,that is the manifold $X$ has the same homotopy type of $X-\partial X$?

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up vote 5 down vote accepted

In a smooth manifold, there is a neighborhood of the boundary $V$ which is diffeomorphic to $\partial M\times [0,1)$, called a collar neighborhood. (Google "Collar neighborhood theorem.") Removing the boundary would give you $\partial M\times (0,1)$. These are homotopy-equivalent in a way that fixes $\partial M\times [.5,1)$, so the homotopy equivalence extends to the interior of the manifold $M$. This homotopy equivalence can be visualized by collapsing both $\partial M\times (0,1)$ and $\partial M\times [0,1)$ down to $\partial M\times [.5,1)$.

For a topological (non-smoothable) manifold, I believe the collar neighborhood theorem also holds, but I'm not as sure about that.

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(In response to a no-longer existent comment:) I want to define two homotopy-equivalences. $f\colon \partial M\times [0,1)\to \partial M\times[.5,1)$ and $g\colon \partial M\times (0,1)\to\partial M\times [.5,1)$. Both $f$ and $g$ are defined by $(x,t)\mapsto (x,.5)$ if $t\geq .5$ and $(x,t)\mapsto(x,t)$ otherwise. The reverse maps are just inclusions. One has to check that the compositions in both directions are homotopic to the identity. – Grumpy Parsnip Oct 12 '12 at 22:26
yes i figured that out that's why i deleted the previous comment and added a new comment. I'msorry for that. – palio Oct 12 '12 at 22:31
what do you mean by "the homotopy equivalence extends to the interior of $M$? do you mean that the homotopy equivalence $h:\partial M\times [0,1)\to \partial M \times (0,1)$ extends to a homotopy equivalence $\tilde h:M\times [0,1)\to M \times (0,1)$? – palio Oct 12 '12 at 22:48
what we have showed so far is that the collar neighborhood of the boundary $N(\partial M)$ is homotopy equivalent to $N(\partial M)-\partial M$; the collar neighborhood minus the boundary. How can we generalize this to a statement $M\simeq M-\partial M$? – palio Oct 13 '12 at 8:37
You define all of your maps to be the identity away from the boundary of the manifold. That way, using the gluing lemma, you can glue together the maps relating $M$ and $M\setminus\partial M$ and the homotopies between their compositions. – Grumpy Parsnip Oct 13 '12 at 13:46

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