On the difficulty of finding spines

We prove that the set of symplectic lattices in the Siegel space ${\mathfrak{h}}_g$ whose systoles generate a subspace of dimension at least 3 in ${\mathbb{R}}^{2g}$ does not contain any $\mathrm{S}\mathrm{p}(2g,\mathbb{Z})$-equivariant deformation retract of ${\mathfrak{h}}_g$.