On the homology of simplicial and cubical sets with symmetries

TL;DR

This paper investigates the homology of symmetric simplicial and cubical sets, showing that certain sub-complexes are acyclic over fields of characteristic zero, which simplifies computations and provides structural insights.

Contribution

It demonstrates the acyclicity of specific sub-complexes in symmetric simplicial and cubical sets over characteristic zero fields, enabling more efficient homology calculations.

Findings

Sub-complexes generated by degeneracies or connections are acyclic over characteristic zero fields.

Quotient complexes have isomorphic homology to original complexes, simplifying calculations.

Acyclicity does not hold for general coefficient rings R.

Abstract

We study the homology of simplicial and cubical sets with symmetries. These are simplicial and cubical sets with additional maps expressing the symmetries of simplices and cubes. We consider the chain complex computing the homology groups in either case. We show for coefficients in fields of characteristic $0$ that the sub-complex generated by degeneracies (simplicial case) or connections (cubical case) together with all $x - sgn(t)tx$ for symmetries $t$ and chains $x$ is acyclic. In particular, it follows that quotienting by this sub-complex yields a chain complex with isomorphic homology. The latter leads to structural insight and a speedup in explicit computations. We also exhibit examples which show that acyclicity does not hold for general coefficient rings $R$.