# Homology groups of cubical sets with connections

## TL;DR

This paper investigates the homology groups of cubical sets with connections, demonstrating that connections do not affect the homology, thus simplifying the understanding of their algebraic topology.

## Contribution

It proves that connections generate an acyclic subcomplex, showing homology groups are unaffected by normalization with connections in cubical sets.

## Key findings

- Connections generate an acyclic subcomplex
- Homology groups are independent of normalization
- Connections do not contribute to nontrivial cycles

## Abstract

Toward defining commutative cubes in all dimensions, Brown and Spencer introduced the notion of "connection" as a new kind of degeneracy. In this paper, for a cubical set with connections, we show that the connections generate an acyclic subcomplex of the chain complex of the cubical set. In particular, our results show that the homology groups of a cubical set with connections are independent of whether we normalize by the connections or we do not, that is, connections do not contribute to any nontrivial cycle in the homology groups of the cubical set.