# Polytope invariant of a manifold

## 1 Problem

The motivation for this question is given by [Friedl&Lück&Tillmann2016]. The authors associate to any $L^2$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}L^2$-acyclic closed oriented manifold $M$$M$ a certain invariant which can be understood as a formal difference of polytopes in $H_1(M; \mathbb{R})$$H_1(M; \mathbb{R})$, $\displaystyle \left\{ L^2\text{-acyclic closed oriented manifold } M \right\} \to \text{formal difference of polytopes in } H_1(M; \mathbb{R}).$

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.