Stratification of spaces of locally convex curves by itineraries

Abstract

Locally convex (or nondegenerate) curves in the sphere (or projective space) have been studied for several reasons, including the study of linear ordinary differential equations. Taking Frenet frames allows us to translate such curves into corresponding curves in the flag space, the orthogonal group or its cover $Spin_{n+1}$. Determining the homotopy type of the space of such closed curves or, more generally, of spaces of such curves with prescribed initial and final jets appears to be a hard problem, which has been solved for $n=2$ but otherwise remains open. This paper is a step towards solving the problem for larger values of $n$. In the process, we prove a related conjecture of B. Shapiro and M. Shapiro. We define the itinerary of a locally convex curve $\Gamma:[0,1]\to Spin_{n+1}$ as a word $w$ in the alphabet of non-trivial permutations. This word encodes the succession of non-open Bruhat cells of $Spin_{n+1}$ pierced by $\Gamma$. We prove that, for each word $w$, the subspace of curves of itinerary $w$ is an embedded contractible (topological) submanifold of finite codimension, thus defining a stratification of the space of curves. We show how to obtain explicit (topologically) transversal sections for each of these submanifolds. We study both a space of curves with minimum regularity hypotheses, where only topological transversality applies, and spaces of sufficiently regular curves. In both cases we study the neighboring relation between strata. This is an important step in the construction of CW cell complexes mapped into the original space of curves by weak homotopy equivalences. Our stratification is not as nice as might be desired, lacking for instance the Whitney property. The differentiability class of the curves affects some properties of the stratification. The necessary ingredients for the construction of a dual CW complex are proved.