# Is the cut-locus of $p$ homotoy equivalent to $M\setminus \{ p\}$?

Suppose $$p$$ is a point in a connected and compact Riemannian manifold $$M$$. Does it hold that the cut locus $$\text{Cut}(p)$$ is homotopy equivalent to $$M \setminus \{p \}$$? I can show that every point in $$M$$ is either in a normal neighborhood of $$p$$ or else it is contained in its cut locus. From here my first candidate for a homotopy would be to take $$v = \exp_p ^{-1}(q)$$, find its cut time $$t(p, v/|v|)$$ and then send $$q$$ to $$\exp_p(t(p, v/|v|) v/|v|)$$. This is certainly continuous and maps the part of $$M$$ diffeomorphic to a neighborhood of $$0 \in T_pM$$ to $$\text{Cut}(p)$$. To extend this function to the cut-locus, however, I am having more trouble because I don't know what the behavior of $$\exp_p ^{-1}$$ is as we approach a cut point.

Furthermore, I still have to find the map from the cut locus to $$M$$ which can be deformed into the identity. What confuses me here is that the cut locus may consist of a single point, so that essentially any constant map is homotopic to the identity. This feels like I am either missing something or the homotopy equivalence condition is fairly weak. Maybe I am going on the totally wrong path. Is the claim even true?

This is true. It's Problem 10-22 in my Introduction to Riemannian Manifolds (2nd ed.). Here's a small hint: it's probably easiest to start by showing that $$\overline{\text{ID}(p)}\smallsetminus \{p\}$$ is homotopy equivalent to $$\partial( \text{ID}(p))$$, where $$\text{ID}(p)\subseteq T_pM$$ denotes the injectivity domain of $$p$$ (see Chapter 10 of my book for definitions).