On the locus of multiple maximizing geodesics on a globally hyperbolic spacetime

Extending the recent work of Cannarsa, Cheng and Fathi, we investigate topological properties of the locus $\mathcal{N}\mathcal{U}(M,g)$ of multiple maximizing geodesics on a globally hyperbolic spacetime $(M,g)$, i.e.\ the set of causally related pairs $(x,y)$ for which there exists more than one maximizing geodesic (up to reparametrization) from $x$ to $y$. We will prove that this set is locally contractible. We will also define the notion of a Lorentzian Aubry set $\mathcal{A}$ and prove that the inclusions $\mathcal{N}\mathcal{U}(M,g)\hookrightarrow {\mathrm{Cut}}_M\hookrightarrow J^{+}\mathrm{\backslash }\mathcal{A}$ are homotopy equivalences.