Mapping spaces of Swiss cheese operads

TL;DR

This paper proves a homotopy equivalence between the derived mapping spaces of Swiss cheese operads and little discs operads, providing insights into disc embedding spaces.

Contribution

It establishes a weak homotopy equivalence between the mapping spaces of Swiss cheese and little discs operads, advancing understanding of operad relationships.

Findings

The color restriction map is a weak homotopy equivalence.

This result aids in studying disc concordance embedding spaces.

Provides a new tool for analyzing operad mapping spaces.

Abstract

We show that the color restriction map $\mathrm{Op}^h({\mathcal{SC}}_m,{\mathcal{SC}}_n)\to \mathrm{Op}^h({\mathcal E}_{m-1},{\mathcal E}_{n-1})$ from the derived mapping space of Swiss cheese operads to that of little discs operads, is a weak homotopy equivalence. We explain how this can help in the study of disc concordance embedding spaces.