Umkehr Maps Patched via the Compactified Cleavage Operad

TL;DR

This paper introduces a novel approach to intersection theory and string topology using a compactified cleavage operad, providing new algebraic structures on mapping spaces into manifolds.

Contribution

It develops a compactification of the cleavage operad that enables intersection theory and higher-dimensional string topology actions on mapping spaces.

Findings

Provides a new intersection theory along the base manifold.

Establishes a homotopy theoretic framework for higher-dimensional string topology.

Introduces a compactified operad structure for advanced topological analysis.

Abstract

Via a compactification of the cleavage operad, we describe two actions on the mapping spaces from spheres into compact, oriented manifolds. The map from the compactified version of the cleavage operad provides a version of intersection theory along the underlying base manifold lifted to the mapping space. The other action map only works for the uncompactified cleavage operad, and gives a homotopy theoretic version of higher dimensional string topology.