The spine that was no spine (Q1002482)

For \(n \geq 2\) let \(\text{T}_{n}\) be the Teichmüller space of flat metrics with unit volume on the \(n\)-dimensional torus \(T^{n} = {\mathbb R}^{n}/{\mathbb Z}^{n}\) and identify \(SL_{n}{\mathbb Z}\) with the corresponding mapping class group. The systole \(syst(\rho)\) of a point \(\rho \in \text{T}_{n}\) is the length of the shortest homotopically essential geodesic in the flat torus \((T^{n}, \rho).\) Let \(S(\rho)\) be the set of homotopy classes of geodesics in \((T^{n}, \rho)\) with length \(syst(\rho);\) the elements in \(S(\rho)\) are known as the systoles of \((T^{n}, \rho).\) Theorem 1.1 (Ash). The subset \(X\) of \(T_n\) consisting of those points \(\rho\) with the property that \(S(\rho)\) generates a finite index subgroup of \(\pi_{1}(T^{n})\) is an \(SL_{n}{\mathbb Z}\)-equivariant spine of \(T_n,\) i.e. a deformation retract for \(\text{T}_{n}.\) The authors prove the following theorem: For \(n \geq 5,\) the subset \(Y\) of \(T_n\) consisting of extremely well-rounded points, i.e. those points \(\rho\) with the property that \(S(\rho)\) generates \(\pi_{1}(T^{n}),\) is not contractible and hence is not an \(SL_{n}{\mathbb Z}\)-equivariant spine.