Riemannian Exponential Map is a Homeomorphism outside the Cut Locus Let $M$ be a connected and complete Riemannian manifold.  The Hopf-Rinow Theorem guarantees that $Exp_p$, for any $p \in M$, is defined on all of $T_p(M)$.  Further, this map is a diffeomorphism on a neighbourhood of the origin.  
However, under the assumption of completeness and connectedness of $M$ is it a homeomorphism, is it a homeomorphism from $M-C_p$ to $T_p(M)$; where $C_p$ is the cut-locus of $p$?
 A: Here I'll suppose $M$ is connected and complete.
Careful there is two notions of cut locus: the cut-locus at $p$ in $T_pM$ of a riemannian manifold is a subset of $T_pM$, not $M$ : it is the set of vectors $v$ in $T_pM$ for which $\exp_p(tv)$ is minimizing for $t \in [0,1]$ but is not minimizing on $[0,1+\varepsilon]$ for every $\varepsilon >0$. The cut locus at $p$ in $M$ is its image by $\exp_p$.
Consider the cut locus in $T_pM$, denoted by $C_p$. Let $U = \{tv \mid t\in [0,1[, v \in C_p\}$. Then $U$ is open and star-shaped. You can show that if $tv \in U$, then $\mathrm{d}{\exp_p}_{tv} (w)$ is the value of a certain Jacobi field at $t$, and thus is non-zero by asumptions if $w$ is non-zero. So $\mathrm{d}\exp_{tv}$ is invertible.
Okay right now we know that $\exp : U \to \exp_p(U)$ is smooth, has everywhere invertible differential, and by the very definition of $U$, is injective. Then it is a diffeomorphism as it is a injective local diffomorphism.
In fact, we have the disjoint decomposition $M = \exp_p(U) \cup \exp_p(C_p)$.
Remark there is no result here on all of $T_pM$ or on all of $M$ : we have shown that $M$ minus a closed subset (that can look really hideous) is diffeomorphic to a star shaped open subset of $T_pM$.
Edit : I think completeness is a strong hypothesis. We can do all this if $M$ has a pole $p$, that is a point $p$ for which the exponential map $\exp_p$ is surjective. Completeness says that everypoint is a pole.
