  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 $X_{i-1}$.
