Matthew. I had a look at Wilson's paper; he is of course rigourous; he says that
- $V(c)$ is homotopy-equivalent to $S^{n-1}$ for every $n$;
- $V(c)$ is diffeomorphic to $S^{n-1}$ for every $n\neq 4, 5$
which corresponds to what was known in 1967. Nowadays, one can say a little more:
- $V(c)$ is also diffeomorphic to $S^{n-1}$ for $n=4$ (thanks to Perelman's proof of the Poincare conjecture);
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- $V(c)$ is homeomorphic to $S^{n-1}$ for every $n$ (thanks to Friedman's topological h-cobordism theorem).
Good reading!