Diagrams of classifying spaces and $k$-fold Boolean algebras

TL;DR

This paper develops a diagrammatic homotopy-theoretic method to compute the homology groups of quotient spaces under group actions, with explicit calculations for symmetric deleted joins of simplicial complexes.

Contribution

It introduces a novel approach using homotopy limits of diagrams and group actions to compute homology of quotient spaces, including explicit computations for specific cases.

Findings

Computed homology groups for symmetric deleted joins of simplices.

Reduced the problem to a combinatorial one using spectral sequences and braid stratification.

Provided a new diagrammatic framework for analyzing classifying spaces and Boolean algebras.

Abstract

In this paper we study the problem of determining the homology groups of a quotient of a topological space by an action of a group. The method is to represent the original topological space as a homotopy limit of a diagram, and then act with the group on that diagram. Once it is possible to understand what the action of the group on every space in the diagram is, and what it does to the morphisms, we can compute the homology groups of the homotopy limit of this quotient diagram. Our motivating example is the symmetric deleted join of a simplicial complex. It can be represented as a diagram of symmetric deleted products. In the case where the simplicial complex in question is a simplex, we perform the complete computation of the homology groups with $\mathbb Z_p$ coefficients. For the infinite simplex the spaces in the quotient diagram are classifying spaces of various direct products of symmetric groups and diagram morphisms are induced by group homomorphisms. Combining Nakaoka's description of the $\mathbb Z_p$-homology of the symmetric group with a spectral sequence, we reduce the computation to an essentially combinatorial problem, which we then solve using the braid stratification of a sphere. Finally, we give another description of the problem in terms of posets and complete the computation for the case of a finite simplex.