# Curvature and asphericity of cube complexes

Let $$K$$ be a connected cube complex (one may assume that its a cellulation of a smooth, closed manifold). Such a $$K$$ comes equipped with a length metric (one assumes that each edge is of unit length). It is a well known result of Gromov that if $$K$$ is non-positively curved (in the sense of Aleksandrov) then its universal cover is contractible; equivalently $$K$$ is aspherical. Gromov also gave the following criterion to check the curvature: If the link of every vertex in $$K$$ is flag then $$K$$ is non-positively curved.

Recently I discovered a cube complex structure on some smooth, closed manifolds such that there are vertices whose link is the boundary of a simplex of appropriate dimension. However, some of these manifolds are aspherical (since tori, positive genus surfaces are on that list).
So I was wondering whether there are any other (combinatorial) criterion to decide the asphericity of a cube complex?

I would also like to know about the papers that prove a certain manifold is aspherical using the combinatorial properties of a regular CW structure on that manifold.

• I guess that asphericity of CAT(0) spaces was known long before Gromov? Gromov is known for giving a simple combinatorial characterization of being non-positively curved among cube complexes. – YCor Jul 2 '19 at 23:35
• "positively curved": what do you mean by this? – YCor Jul 2 '19 at 23:36
• I suspect that "positively curved" in the question means "not nonpositively curved". If you want an example, glue three squares together as in the boundary of an orthant in $R^3$. – Misha Jul 3 '19 at 5:31
• Yes, I meant not non-positively curved. My questions are not correctly worded; I am changing them now. – Priyavrat Deshpande Jul 3 '19 at 8:07
• Boundary of a simplex is not a flag complex. Topologically all links are codimension 1 spheres, however, in case of cube complexes links are triangulated. The triangulation plays an important role. – Priyavrat Deshpande Jul 3 '19 at 13:08

## 1 Answer

Regarding your second question, I'm aware of (at least) two combinatorial conditions (beyond Gromov's) for 3-manifolds to be aspherical. Cannon-Floyd-Parry have a combinatorial construction of 3-manifolds by "twisted face pairings". They showed that the 3-manifolds arising from "ample twisted face pairings" are aspherical and have word-hyperbolic fundamental group. However this paper was never published.

Elder-McCammond-Meier show that 3-manifolds with a triangulation in which every edge has degree 5 or 6, and every triangle has at most one degree 5 edge, admits a CAT(0) metric, answering a question of Thurston.

Regarding your first question about cube complexes, I'm not immediately aware of some combinatorial criterion in the literature. However, there ought to be more general criteria based on Gromov's condition. Suppose one has a vertex in a cube complex with link an $$n$$-simplex. Then one can glue an $$n+1$$-cube to this corner to get a simple-homotopy equivalent complex. If this complex satisfies Gromov's condition (links of the vertices are flag), then it is CAT(0) and the original complex is aspherical. For example, this works for a 2-complex which is a union of 3 squares glued cyclically around a vertex. One can try to continue in this way, gluing on higher-dimensional cubes to get rid of "positive" curvature at vertices. If this succeeds, then it gives a combinatorial condition for asphericity. The motivation for this condition comes from Sageev's theory, but I won't delve into this here.

Another attempt at a description: suppose one has a cube complex with flag links of vertices (locally CAT(0)), and there are cubes with free faces. Then one can collapse along these faces to obtain a simple-homotopy equivalent cube complex which might no longer be locally CAT(0). One would expect a construction of Sageev to construct cube complexes with higher dimensional cubes which collapse to a manifold associated to e.g. a surface with filling curves, or to a hyperbolic 3-manifold with filling quasi-convex surfaces. Reversing this process gives a sequence of expansions of the cube complex to a CAT(0) complex that one might be able to locate combinatorially, and hence give a combinatorial criterion for asphericity. I haven't thought about whether this might be functorial though - in some situations it might be.