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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?

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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).

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    $\begingroup$ Hi Jack, I am a big fan of your books! I have been working through this one and I just finished writing the solution to this problem last week. I am not sure if it is okay to post it as the answer though. $\endgroup$
    – Katerina
    Commented Sep 9, 2022 at 2:18

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