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);
• • $V(c)$ is homeomorphic to $S^{n-1}$ for every $n$ (thanks to Friedman's topological h-cobordism theorem).

Good reading!

Matthew. I had a look at Wilson's paper; he is of course rigourous; he says that

• $V^{-1}(c)$ is homotopy-equivalent to $S^{n-1}$ for every $n$;
• $V^{-1}(c)$ 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:

• $V^{-1}(c)$ is also diffeomorphic to $S^{n-1}$ for $n=4$ (thanks to Perelman's proof of the Poincare conjecture);
• • $V^{-1}(c)$ is homeomorphic to $S^{n-1}$ for every $n$ (thanks to Friedman's topological h-cobordism theorem).

Good reading!