The cone on a manifold I believe that I have run across the statement that if $X$ is a compact smooth manifold and $CX$ is the cone on $X$, i.e. $[0,1] \times X$ modulo $(0,x)\sim(0,y)$ for all $x,y \in X$, then $CX$ admits the structure of a smooth manifold with boundary, with $\left\{1\right\} \times X$ smoothly immersed in it as the boundary, just when $X$ is diffeomorphic to a sphere in Euclidean space with its standard smooth structure. I would like a reference, or a reference to a similar statement.
 A: There is no need for smoothness here : if $X$ is a compact topological $n$-manifold and $CX$ is the cone on $X$, then $CX$ is a topological manifold if and only if $X$ is homeomorphic to a sphere.  This is trivial for $n=1$, so assume that $n \geq 2$.  The backward implication is trivial, so assume that $CX$ is a topological manifold.  Let $p_0 \in CX$ be the cone point.  The local homology groups $H_{k}(CX,CX-p_0)$ are then $\mathbb{Z}$ for $k=0,n+1$ and $0$ otherwise.  Looking at the long exact sequence for the pair $(CX,CX-p_0)$, we then get that $H_{k}(X)$ is $\mathbb{Z}$ for $k=0,n$ and $0$ otherwise.  In other words, $X$ is a homology $n$-sphere.  Next, since $n+1 \geq 3$ and $CX$ is an $(n+1)$-manifold, the space $CX$ must satisfy the following condition : for all point $q \in CX$ and all neighborhoods $U$ of $q$, there must be a neighborhood $V$ of $q$ such that $V \subset U$ and $V-q$ is simply connected.  Around the cone point $p_0$, it is easy to see that this condition implies that $X$ must be simply-connected.  The Poincare conjecture thus implies that $X$ is homeomorphic to an $n$-sphere.
