# Polytope invariant of a manifold

## 1 Problem


$\displaystyle \left\{ L^2\text{-acyclic closed oriented manifold } M \right\} \to \text{formal difference of polytopes in } H_1(M; \mathbb{R}).$

Question: Is there a geometric definition of that invariant?

Question: What about realization results? In dimension $3$$3$ this is an old unsolved question of which Thurston polytopes are realized by aspherical 3-manifolds.

In dimensions $9$$9$ and higher, using surgery theory, we know exactly which polytopes can be realized, but in dimensions $4$$4$ to $8$$8$, it is not known. In dimension $4$$4$ this may be difficult and depend on whether we study smooth or topological manifolds.

The paper by Friedl, Lück and Tillmann mentioned above proposes several open questions on this topic.

This question was posed by Stefan Friedl at the MATRIX meeting on Interactions between high and low dimensional topology.