On the level set of a function with degenerate minimum point (Q1751374)

Summary: For \(n \geq 2\), let \(M\) be an \(n\)-dimensional smooth closed manifold and \(f : M \rightarrow \mathbb{R}\) a smooth function. We set \(\min f(M) = m\) and assume that \(m\) is attained by unique point \(p \in M\) such that \(p\) is a nondegenerate critical point. Then the Morse lemma tells us that if \(a\) is slightly bigger than \(m\), \(f^{- 1}(a)\) is diffeomorphic to \(S^{n - 1}\). In this paper, we relax the condition on \(p\) from being nondegenerate to being an isolated critical point and obtain the same consequence. Some application to the topology of polygon spaces is also included.