On homology of map spaces

TL;DR

This paper establishes an isomorphism between two models of map space chain complexes, enabling the application of convergence theorems and reducing homotopy equivalence problems to homological algebra.

Contribution

It constructs an isomorphism between Anderson's and Arone's complexes, facilitating new applications of convergence theorems and simplifying homotopy problems.

Findings

Established an isomorphism between Anderson's and Arone's complexes

Applied Shipley's convergence theorem to Arone's model

Reduced homotopy equivalence problems to homological algebra

Abstract

Following an idea of Bendersky-Gitler, we construct an isomorphism between Anderson's and Arone's complexes modelling the chain complex of a map space. This allows us to apply Shipley's convergence theorem to Arone's model. As a corollary, we reduce the problem of homotopy equivalence for certain "toy" spaces to a problem in homological algebra.