[Paper Review] Diagrams of classifying spaces and $k$-fold Boolean algebras

Eric Babson, Dmitry N. Kozlov|arXiv (Cornell University)|Apr 29, 1997

Homotopy and Cohomology in Algebraic Topology14 references3 citations

TL;DR

This paper develops a homotopy-theoretic method to compute the homology of quotients of topological spaces under group actions by representing the space as a homotopy limit of a diagram and analyzing the group action on the diagram. The key result is a complete computation of the $2$-homology of the symmetric deleted join of a simplex using classifying spaces, spectral sequences, and braid stratification, with an alternative poset-based description for finite simplices.

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.

Motivation & Objective

• To develop a general method for computing the homology of quotient spaces under group actions.
• To analyze the symmetric deleted join of a simplicial complex as a diagram of symmetric deleted products.
• To compute the $2$-homology of the symmetric deleted join of a simplex using classifying spaces and spectral sequences.
• To provide a poset-theoretic description of the problem for finite simplices.
• To reduce the homology computation to a combinatorial problem via braid stratification of the sphere.

Proposed method

• Represent the original topological space as a homotopy limit of a diagram of spaces.
• Act with a group on the diagram, tracking its action on both objects and morphisms.
• Use Nakaoka's description of the $2$-homology of symmetric groups to analyze the homology of classifying spaces in the quotient diagram.
• Apply a spectral sequence to reduce the computation to a combinatorial problem.
• Employ braid stratification of the sphere to solve the resulting combinatorial problem.
• Reformulate the problem using posets and $k$-fold Boolean algebras for finite simplices.

Experimental results

Research questions

• RQ1 How can the homology of a quotient space under a group action be computed using diagrammatic homotopy limits?
• RQ2 What is the $2$-homology of the symmetric deleted join of a simplex?
• RQ3 How does the action of the symmetric group on a simplex translate into actions on classifying spaces in a diagram?
• RQ4 Can the homology computation be reduced to a combinatorial problem via geometric stratification?
• RQ5 What is the poset-theoretic description of the symmetric deleted join for finite simplices?

Key findings

• The $2$-homology of the symmetric deleted join of an infinite simplex is computed via classifying spaces of direct products of symmetric groups.
• The spectral sequence reduction leads to a combinatorial problem solvable using braid stratification of the sphere.
• For the infinite simplex, the spaces in the quotient diagram are classifying spaces of symmetric group products, with morphisms induced by group homomorphisms.
• The computation for the infinite simplex yields a complete description of the $2$-homology in terms of the homology of symmetric groups and diagram structure.
• For a finite simplex, the problem is re-expressed in terms of posets and $k$-fold Boolean algebras, enabling a complete homology computation.
• The method successfully reduces a topological homology problem to a combinatorial one through geometric and algebraic techniques.