Homotopy type of spaces of locally convex curves in the sphere S^3

Abstract

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. Let L n (z 0 ; z 1 ) be the space of such curves Γ with prescribed endpoints Γ(0) = z 0 , Γ(1) = z 1 . The aim of this paper is to determine the homotopy type of the spaces L 3 (z 0 ; z 1 ) for all z 0 , z 1 ∈ Spin 4 . Recall that Spin 4 = S 3 × S 3 ⊂ H × H, where H is the ring of quaternions.This paper relies heavily on previous publications by the authors. The earliest such paper solves the corresponding problem for n = 2 (i.e., for curves in S 2 ). Another previous result (with B. Shapiro) reduces the problem to z 0 = 1 and z 1 ∈ Quat 4 where Quat 4 ⊂ Spin 4 is a finite group of order 16 with center Z(Quat 4 ) = {(±1, ±1)}. A more recent paper shows that for z 1 ∈ Quat 4 Z(Quat 4 ) we have a homotopy equivalence L 3 (1; z 1 ) ≈ Ω Spin 4 . In this paper we compute the homotopy type of L 3 (1; z 1 ) for z 1 ∈ Z(Quat 4 ): it is equivalent to the infinite wedge of Ω Spin 4 with an infinite countable family of spheres (as for the case n = 2).The structure of the proof is roughly the same as for the case n = 2 but some of the steps are much harder. In both cases we construct explicit subsets Y ⊂ L n (z 0 ; z 1 ) for which the inclusion Y ⊂ Ω Spin n+1 (z 0 ; z 1 ) is a homotopy equivalence. For n = 3, this is done by considering the itineraries of such curves. The itinerary of a curve in L n (1; z 1 ) is a finite word in the alphabet S n+1 {e} of nontrivial permutations.