Stiefel Manifolds and Coloring the Pentagon

TL;DR

This paper proves a conjecture linking the Lovász complex of graph multimorphisms from a 5-cycle to a complete graph with a classical geometric space, the Stiefel manifold, using equivariant topology.

Contribution

It establishes a Z/2Z-equivariant homeomorphism between the Lovász complex Hom(C_5,K_n) and the Stiefel manifold V(n-1,2), advancing understanding of graph complexes and topology.

Findings

Confirmed Csorba's conjecture.

Developed equivariant piecewise-linear topology methods.

Connected graph complexes with classical geometric spaces.

Abstract

We prove Csorba's conjecture that the Lov\'asz complex Hom(C_5,K_n) of graph multimorphisms from the 5-cycle C_5 to the complete graph K_n is Z/2Z-equivariantly homeomorphic to the Stiefel manifold, V(n-1,2), the space of (ordered) orthonormal 2-frames in R^{n-1}. The equivariant piecewise-linear topology that we need is developed along the way.