Transversal Homotopy Monoids of Complex Projective Space

TL;DR

This paper provides a geometric description of the nth transversal homotopy monoid of complex projective space, relating it to isotopy classes of certain filtrations of spheres with specific bundle properties.

Contribution

It introduces a new geometric characterization of transversal homotopy monoids for complex projective spaces using filtrations with manifold differences and orientable bundle conditions.

Findings

Isomorphism between transversal homotopy monoids and sphere filtrations

Characterization of filtrations via nested subspaces with bundle conditions

Explicit description of the monoid structure in geometric terms

Abstract

We will give a geometric description of the nth transversal homotopy monoid of k-dimensional complex projective space, where we stratify by lower dimensional complex projective spaces in the usual way. Transversal homotopy monoids are defined as classes of based transversal maps into Whitney stratified spaces up to equivalence through such maps. We will show the nth transversal homotopy monoid of k-dimensional complex projective space is isomorphic to isotopy classes of certain filtrations of the n-sphere. The required filtrations are by nested closed subspaces $X_0 \subset ... \subset X_k=S^n$, such that the difference between any two is a manifold and the normal bundle of $X_i$ in $X_{i+1}$ is an orientable real 2-dimensional vector bundle with Euler class represented by the Poincar\'e dual of $X_{i-1}$.