In Kosinski's Differential manifolds, he shows (in page 93) by imbedding $M\#(-M)$ (where $\#$ is the connected sum for differential manifolds and $M$ is an oriented manifold) in $M\times[-1,1]$ that $M\#(-M)$ bounds a manifold that has $M$ with the interior of a disc deleted as a deformation retract. I can get that. However, after showing that, he asserts that if $\Sigma$ is a homotopy sphere, then $\Sigma\#(-\Sigma)$ bounds a contractible manifold. If $\Sigma=S^m$, it is clear, but I don't see why this is true for a homotopy sphere.
Why is this true for a homotopy sphere? (I'm preparing an expository talk which includes this issue, but my knowledge on algebraic topology is not very good: I'm just starting to learn algebraic topology. Maybe this is a silly question).
As far as I got, from the first statement, it is sufficient to show the following: If $\Sigma$ is a homotopy sphere then the space $\Sigma'$ obtained by deleting an open ball from $\Sigma$ is contractible. I think I can show this (although by a rather cumbersome argument).
Firstly note that $\Sigma$ is in particular a homology sphere by Hurewicz theorem (i.e. has homology isomorphic to that of a sphere) .
We show by the Mayer-Vietoris sequence $H_{i}(\Sigma',\mathbb{Z}) = 0$ for $i>0$. In the notation of the Wikipedia article on Mayer-Vietoris sequence, take $X = \Sigma$, $A = \Sigma'$ and $B$ a ball in $\Sigma$ slightly bigger than the one you deleted to get $\Sigma'$, then note that $A \cap B$ will be homotopy equivalent to a sphere of dimension $\dim(\Sigma)-1$).
Let $n = \dim(\Sigma)$. The vanishing of $H_{i}(\Sigma',\mathbb{Z}) $ follows immediately from the long exact sequence for $0<i<n-1$ since there are zeros either side. For $i=n-1$ note that for $n-1$ the boundary map $H_{n}(X) \rightarrow H_{n-1}(A \cap B)$ is an isomorphism since it maps the fundamental cycle to the fundamental cycle, thus proving that $H_{n-1}(\Sigma',\mathbb{Z})= 0$ (see the description of the boundary map in the Wikipedia article). Finally, all higher homology groups vanish since $\Sigma'$ is a non-compact manifold of dimension $n$ (of finite topological type).
Next, it follows from the Hurewicz theorem (plus the fact that $\pi_{1}(\Sigma') \cong \pi_{1}(\Sigma) = \{1\} $) that all of the homotopy groups $\pi_{i}(\Sigma')$ are trivial. Finally, by Whiteheads theorem the inclusion of a point in $\Sigma'$ induces an isomorphism on all homotopy groups hence is a homotopy equivelance, i.e. $\Sigma'$ is a contractible space.