On the topology of no $k$-equal spaces

TL;DR

This paper explores the topology of no $k$-equal spaces using cellular spanning trees, establishing a connection between homology ranks and combinatorial structures in hypercubes.

Contribution

It introduces a novel approach linking homology of no $k$-equal spaces to spanning trees in hypercube skeletons, providing new topological insights.

Findings

Homology rank equals the number of facets in a spanning tree.

Establishes a combinatorial-topological correspondence.

Advances understanding of no $k$-equal space topology.

Abstract

We consider the topology of real no $k$-equal spaces via the theory of cellular spanning trees. Our main theorem proves that the rank of the $(k-2)$-dimensional homology of the no $k$-equal subspace of $\mathbb{R}$ is equal to the number of facets in a $k$-dimensional spanning tree of the $k$-skeleton of the $n$-dimensional hypercube.