Explicit Homotopy Equivalences Between Some Operads

TL;DR

This paper constructs explicit homotopy equivalences between important operads, such as the Fulton MacPherson and little disks operads, providing concrete links between geometric and algebraic structures in topology.

Contribution

It introduces explicit operad morphisms that are homotopy equivalences, including for associahedra and extensions to Kontsevich compactification and Swiss-cheese operad.

Findings

Explicit homotopy equivalence between Fulton MacPherson and little disks operads.

Operadic homotopy equivalence between associahedra and little intervals.

Extension of constructions to Kontsevich compactification and Swiss-cheese operad.

Abstract

In this work we present an explicit operad morphism that is also a homotopy equivalence between the operad given by the real Fulton MacPherson compactification of configuration spaces and the little $n$-disks operad. In particular, the construction gives an operadic homotopy equivalence between the associahedra and the little intervals explicitly. It can also be extended to the case of Kontsevich compactification and Voronov swiss-cheese operad.