Such a manifold is homeomorphic to a sphere

Question

I recently read that if a compact differentiable manifold admits a real function with only two critical points, then it is homeomorphic to a sphere. If the function is Morse, this follows from standard Morse Theory. How does one show the general case, in which the critical points may be degenerate?

Answer 1

$\newcommand{\Reals}{\mathbf{R}}$Non-degeneracy of critical points in Morse theory is needed only to control the cell attachments at critical points with indefinite Hessian. At a relative minimum or maximum, where the attached cell has dimension or codimension $0$, you can get by with knowing the critical point is isolated.

Assume $f:M \to \Reals$ has an absolute minimum at $p$ and an absolute maximum at $q$. Remove small balls $B_{p}$ and $B_{q}$ about $p$ and $q$. Use the fact $f$ has no critical points outside these balls to deduce $M \setminus(B_{p} \cup B_{q})$ is diffeomorphic to $S^{n - 1} \times \Reals$, so that $M$ itself is obtained by capping off the boundary spheres with balls, and hence is a sphere.