Combinatorial Groupoids, Cubical Complexes, and the Lovász Conjecture

Abstract

This paper lays the foundation for a theory of combinatorial groupoids that allows us to use concepts like "holonomy", "parallel transport", "bundles", "combinatorial curvature", etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. We introduce a new, holonomy-type invariant for cubical complexes, leading to a combinatorial "Theorema Egregium" for cubical complexes that are non-embeddable into cubical lattices. Parallel transport of Hom-complexes and maps is used as a tool to extend BabsonKozlov-Lovász graph coloring results to more general statements about nondegenerate maps (colorings) of simplicial complexes and graphs.Keywords Combinatorial groupoids · Lovász conjecture · Cubical complexes · Discrete differential geometry IntroductionThere has been new and encouraging evidence that the language and methods of the theory of groupoids, after being successfully applied in other major mathematical fields, offer new insights and perspectives for applications in combinatorics and discrete and computational geometry.The groupoids (groups of projectivities) have recently appeared in geometric combinatorics in the work of M. Joswig [28]; see also a related paper with Izmestiev [27] and the references to these papers, where they have been applied to toric manifolds, branched coverings over S 3 , colorings of simple polytopes etc.

Any smooth knot $\mathbb{S}^{n}\hookrightarrow\mathbb{R}^{n+2}$ is isotopic to a cubic knot contained in the canonical scaffolding of ℝ n+2

“…Cubic complexes may play a role in extending these invariants to higher dimensional knots. Recent work on the subject can be found in the work of Louis Funar [8] and Rade T. Živaljević [15]. The results of the present paper were used in an essential way to construct higher dimensional wild knots as limit sets of geoemtrically-finite conformal groups in our article [1].…”

Buchstaber invariant theory of simplicial complexes and convex polytopes

“…Note that even a more general definition of such invariants was given independently byŽivaljević in [59], where he expressed hope that new original invariants of simplicial complexes may arise in this way. Thus, the Buchstaber numbers give an example of such invariants.…”

Set Partition Complexes

“…Hence one realizes that the structure of the independent sets and not the maximal degree implies the connectivity bounds for Hom complexes, and this inspired the study of set partition complexes. Other ways to generalize Hom complexes have been pursued, for example, by Žival-jević [19].…”