A remarkable DG-module model for configuration spaces

TL;DR

This paper introduces a new DG-module model for the rational homotopy type of configuration spaces in simply-connected closed manifolds, providing a potential tool for understanding their algebraic topology.

Contribution

It constructs a Sigma_k-equivariant commutative differential graded algebra model for configuration spaces, advancing the algebraic modeling of these topological objects.

Findings

Proposed a candidate DG-model for F(M,k)

Proved the model is Sigma_k-equivariant

Explored Lefschetz duality at the cochain level

Abstract

Let M be a simply-connected closed manifold and consider the (ordered) configuration space of $k$ points in M, F(M,k). In this paper we construct a commutative differential graded algebra which is a potential candidate for a model of the rational homotopy type of F(M,k). We prove that our model it is at least a Sigma_k-equivariant differential graded model. We also study Lefschetz duality at the level of cochains and describe equivariant models of the complement of a union of polyhedra in a closed manifold.