Transversal homotopy theory

TL;DR

This paper introduces new invariants for Whitney stratified manifolds using transversal homotopy theory, generalizing classical homotopy groups and revealing rich categorical structures.

Contribution

It defines transversal homotopy monoids and categories for stratified manifolds, extending classical invariants and establishing their functorial properties and categorical structures.

Findings

Transversal homotopy monoids generalize classical homotopy groups.

Transversal homotopy categories are rigid monoidal and ribbon categories.

Examples include categories of framed tangles related to stratified spheres.

Abstract

Implementing an idea due to John Baez and James Dolan we define new invariants of Whitney stratified manifolds by considering the homotopy theory of smooth transversal maps. To each Whitney stratified manifold we assign transversal homotopy monoids, one for each natural number. The assignment is functorial for a natural class of maps which we call stratified normal submersions. When the stratification is trivial the transversal homotopy monoids are isomorphic to the usual homotopy groups. We compute some simple examples and explore the elementary properties of these invariants. We also assign `higher invariants', the transversal homotopy categories, to each Whitney stratified manifold. These have a rich structure; they are rigid monoidal categories for n>1 and ribbon categories for n>2. As an example we show that the transversal homotopy categories of a sphere, stratified by a point and its complement, are equivalent to categories of framed tangles.