# Algebra Seminar

### 19/04/2018, 11:30 — 12:30 — Room P3.10, Mathematics Building

Julien Ducoulombier, *ETH Zurich*

### Swiss Cheese operad and applications to embedding spaces

During this talk, I would like to give an overview of the (relative) delooping theorems as well as applications to spaces of long embeddings. In particular, we show that the space of long embeddings and the space of ($k$)-immersions from $\mathbb{R}^d$ to $\mathbb{R}^m$ are weakly equivalent to an explicit ($d+1$)-iterated loop space and an explicit ($d+1$)-iterated relative loop space, respectively. Both of them can be expressed in term of derived mapping spaces of coloured operads. Such a pair is a typical example of Swiss-Cheese algebra.