On the difficulty of finding spines

TL;DR

This paper proves that the set of symplectic lattices with systoles generating a subspace of dimension at least 3 cannot be simplified via an equivariant deformation retraction, highlighting the complexity of the space.

Contribution

It establishes a non-existence result for certain equivariant deformation retracts in the space of symplectic lattices based on systole properties.

Findings

The set of lattices with systoles generating a high-dimensional subspace is topologically complex.

No $ ext{Sp}(2g, ext{Z})$-equivariant deformation retract exists for this set.

The result impacts understanding of the geometric structure of symplectic lattices.

Abstract

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 $\mathrm{Sp}(2g,\mathbb{Z})$-equivariant deformation retract of $\mathfrak{h}_g$.