2011/04/07 by Conor Smyth, Smyth, Conor
Mathematics · #57R99 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:57R99
paper · pdf · doi:10.48550/arxiv.1104.1325
8 pages
arxiv created 2011/04/07 · openalex publication_date 2011/04/07 · arxiv updated 2011/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 X0 ⊂ ... ⊂ Xk=Sn, such that the difference between any two is a manifold and the normal bundle of Xi in Xi+1 is an orientable real 2-dimensional vector bundle with Euler class represented by the Poincaré dual of Xi-1.