Multisimplicial chains and configuration spaces

TL;DR

This paper generalizes $E_ abla$-coalgebra structures to multisimplicial sets, connecting chains of configuration spaces with algebraic models via quasi-isomorphisms, advancing the understanding of algebraic topology.

Contribution

It introduces a multisimplicial extension of $E_ abla$-coalgebra structures and constructs explicit quasi-isomorphisms linking configuration space chains to algebraic models.

Findings

Established a multisimplicial $E_ abla$-coalgebra framework.

Constructed quasi-isomorphisms connecting configuration space chains to algebraic models.

Unified different chain complexes through complexity-preserving quasi-isomorphisms.

Abstract

This paper presents a generalization to multisimplicial sets of previously defined $E_\infty$-coalgebra structures on the chains of simplicial and cubical sets. We focus on the surjection chain complexes of McClure--Smith as a main example and construct a zig-zag of complexity preserving quasi-isomorphisms of $E_\infty$-coalgebras relating these to both the singular chains on configuration spaces and the Barratt--Eccles chain complexes.