Combinatorial groupoids, cubical complexes, and the Lovasz conjecture

Details Combinatorial groupoids, cubical complexes, and the Lovasz conjecture Combinatorial groupoids, cubical complexes, and the Lovasz conjecture

2005-10-10

arxiv.org/abs/math/0510204v2

Mathematics

Combinatorics

Scientific paper

A foundation is laid for a theory of combinatorial groupoids, allowing 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. A new, holonomy-type invariant for cubical complexes is introduced, leading to a combinatorial ``Theorema Egregium'' for cubical complexes non-embeddable into cubical lattices. Parallel transport of Hom-complexes and maps is used as a tool for extending Babson-Kozlov-Lovasz graph coloring results to more general statements about non-degenerate maps (colorings) of simplicial complexes and graphs.

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