Questions in Geometric Group Theory — Q 1.23

Definitions and mathematical statement

Let $X$ be a connected finite simplicial complex. A local coefficient system $\mathcal M$ on $X$ assigns an abelian group to each point and coherent transport isomorphisms along paths; equivalently, after a basepoint choice it is a module with an action of $\pi_1(X)$. A map $f:Y\to X$ pulls it back to a system $f^*\mathcal M$ on $Y$.

Does there always exist a connected locally CAT($-1$) polyhedral complex $Y$ and a map $f:Y\to X$ such that \[ f_*:\pi_1(Y)\twoheadrightarrow\pi_1(X),\qquad f_*:H_j(Y;f^*\mathcal M)\xrightarrow{\cong}H_j(X;\mathcal M) \] for every integer $j\geq0$ and every local coefficient system $\mathcal M$ on $X$? Locally CAT($-1$) means that each point has a neighborhood with a geodesic metric satisfying comparison with hyperbolic triangles of curvature $-1$; the polyhedral metrics agree along glued faces.

This asks for a negatively curved version of the Kan--Thurston realization theorem. Its group-theoretic variant asks for a homology realization by an aspherical space whose fundamental group is word-hyperbolic. The requested local-coefficient isomorphisms are stronger than equality of ordinary integral homology. In particular, the fundamental-group map need not be injective or an isomorphism. Merely placing the fundamental group inside a hyperbolic group does not supply the map or the homology equivalences required here.

Short English statement

Can every finite complex have all its local-coefficient homology realized by a negatively curved aspherical complex?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100123 (AMR-010-0123), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 1.23 (PDF page 6). Original source indexed by AMR-010-0123. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.