On the Level Set of a Function with Degenerate Minimum Point

Abstract EN

For n≥2, let M be an n-dimensional smooth closed manifold and f:M→R a smooth function.

We set minf(M)=m and assume that m is attained by unique point p∈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 Sn-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.