From maps between coloured operads to Swiss-Cheese algebras

TL;DR

This paper establishes a connection between maps of coloured operads and Swiss-Cheese algebras, providing explicit algebraic models for certain topological pairs and exploring their applications to knot theory under a conjecture.

Contribution

It introduces a method to derive Swiss-Cheese algebra structures from coloured operad maps and applies this to knot spaces assuming a conjecture.

Findings

Pairs of topological spaces from operad maps are weakly equivalent to SC_{1}-algebras.

The space of long knots and polynomial approximations form an SC_{d+1}-algebra under the Dwyer-Hess conjecture.

Provides explicit algebraic models for topological pairs related to operads.

Abstract

In the present work, we extract pairs of topological spaces from maps between coloured operads. We prove that those pairs are weakly equivalent to explicit algebras over the one dimensional Swiss-Cheese operad SC_{1}. Thereafter, we show that the pair formed by the space of long knots and the polynomial approximation of (k)-immerions from R^{d} to R^{n} is an SC_{d+1}-algebra assuming the Dwyer-Hess'conjecture.