Mapping spaces of Swiss cheese operads

TL;DR

The paper proves that the color restriction maps between derived mapping spaces of Swiss cheese operads and little discs operads are weak equivalences, bridging boundary-interacting operadic structures with classical disc operads within the Goodwillie–Weiss framework. It employs the Fulton–MacPherson model ${\mathcal{FS}}_m$ for ${\mathcal{SC}}_m$ and analyzes truncated mapping spaces ${\mathrm{T}}_k{\mathrm{Op}}^h(-,-)$ using Reedy model structures, cofibrancy/fibrancy, and a fiberwise contractibility argument. The main result provides a robust tool for translating problems about disc concordance embeddings into operadic maps, with extensions to rational models and potential applications to ranges where $n-m\ge 3$. The work also details a technical proposition ensuring certain inclusions are trivial cofibrations, a key step in the inductive argument. Overall, the results advance the understanding of how boundary and interior configurations interact operadically and offer a concrete pathway to study embedding spaces via operadic methods.

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.

Theorems & Definitions (11)

• Theorem A
• Definition 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Lemma 3.1
• proof
• Lemma 3.2
• Example 3.3
• proof : Proof of Lemma \ref{['l:triv_cof']}