Surgery on manifold operads

TL;DR

This paper explores cobordisms and surgery on manifold operads, generalizing the Fulton-MacPherson operad, and constructs new manifold operads with specific cobordism properties.

Contribution

It introduces a surgery theory for manifold operads and demonstrates the existence of infinitely many operads cobordant but not homotopy equivalent to the Fulton-MacPherson operad.

Findings

Constructed infinitely many manifold operads with specific cobordism properties.

Developed combinatorial results for trees associated to operadic bimodules.

Extended the theory of cobordisms to a class of topological operads.

Abstract

We study cobordisms of a class of topological operads called ``manifold operads''. These operads are generalizations of the Fulton-MacPherson operad: an operad built from configurations of points in Euclidean space. Cobordism of manifold operads, along with the associated theory of surgery, depends crucially on delicate combinatorial results for trees associated to operadic bimodules. As an application of surgery, we produce infinitely many manifold operads which are left or right ``bimodule cobordant'' to, but not homotopy equivalent to the Fulton-MacPherson operad.