Transversal Homotopy Monoids of Complex Projective Space

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 $X_{i-1}$.