The homotopy type of spaces of locally convex curves with fixed endpoints in Spin n+1 , the universal covering the orthogonal group SO n+1 for n ≥ 2, has been determined for n = 2 but is in general not known. The results in this paper have as one important aim trying to make progress in this problem. In the process, we prove a related conjecture of B. Shapiro and M. Shapiro regarding the behavior of fundamental systems of solutions to linear ordinary differential equations. We define the itinerary of a locally convex curve Γ : [0, 1] → Spin n+1 as a (finite) word w in the alphabet S n+1 {e} of non-trivial permutations. This word encodes the succession of non-open Bruhat cells of Spin n+1 pierced by Γ(t) as t ranges from 0 to 1. We prove that, for each word w, the subspace of curves of itinerary w is an embedded contractible (globally collared 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 also study the neighboring relation between strata. This is an important step in the construction of abstract cell complexes mapped into the original space of curves by weak homotopy equivalences, which we cover in a follow-up paper.

A parametric curve γ of class C n on the n-sphere is said to be nondegenerate (or locally convex) when det γ(t), γ (t), • • • , γ (n) (t) > 0 for all values of the parameter t. We orthogonalize this ordered basis to obtain the Frenet frame Fγ of γ assuming values in the orthogonal group SO n+1 (or its universal double cover, Spin n+1 ), which we decompose into Schubert or Bruhat cells. To each nondegenerate curve γ we assign its itinerary: a word w in the alphabet S n+1 {e} that encodes the succession of non open Schubert cells pierced by the complete flag of R n+1 spanned by the columns of Fγ . Without loss of generality, we can focus on nondegenerate curves with initial and final flags both fixed at the (non oriented) standard complete flag. For such curves, given a word w, the subspace of curves following the itinerary w is a contractible globally collared topological submanifold of finite codimension. By a construction reminiscent of Poincaré duality, we define abstract cell complexes mapped into the original space of curves by weak homotopy equivalences. The gluing instructions come from a partial order in the set of words. The main aim of this construction is to attempt to determine the homotopy type of spaces of nondegenerate curves for n > 2. The reader may want to contrast the present paper's combinatorial approach with the geometry-flavoured methods of previous works.

Locally convex (or nondegenerate) curves in the sphere S n (or the projective space) have been studied for several reasons, including the study of linear ordinary differential equations of order n + 1. Taking Frenet frames allows us to obtain corresponding curves Γ in the group Spin n+1 ; recall that Π : Spin n+1 → Flag n+1 is the universal cover of the space of flags. Determining the homotopy type of spaces of such curves Γ with prescribed initial and final points appears to be a hard problem. Due to known results, we may focus on L n , the space of (sufficiently smooth) locally convex curves Γ : [0, 1] → Spin n+1 with Γ(0) = 1 and Π(Γ(1)) = Π(1). Convex curves form a contractible connected component of L n ; there are 2 n+1 other components, corresponding to non convex curves, one for each endpoint. The homotopy type of L n has so far been determined only for n = 2 (the case n = 1 is trivial). This paper is a step towards solving the problem for larger values of n.The itinerary of a locally convex curve Γ : [0, 1] → Spin n+1 belongs to W n , the set of finite words in the alphabet S n+1 {e}. The itinerary of a curve lists the non open Bruhat cells crossed by the curve. Itineraries yield a stratification of the space L n . We construct a CW complex D n which is a kind of dual of L n under this stratification: the construction is similar to Poincaré duality. The CW complex D n is homotopy equivalent to L n . The cells of D n are naturally labeled by words in W n so that D n is infinite but locally finite. Explicit glueing instructions are described for lower dimensions.As an application, we describe an open subset Y n ⊂ L n , a union of strata of L n . In each non convex component of L n , the intersection with Y n is connected and dense. Most connected components of L n are contained in Y n . For n > 3, in the other components the complement of Y n has codimension at least 2. We prove that Y n is homotopically equivalent to the disjoint union of 2 n+1 copies of Ω Spin n+1 . In particular, for all n ≥ 2, all connected components of L n are simply connected.

No abstract