Combinatorialization of spaces of nondegenerate spherical curves

Abstract

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.

Finiteness of rank for Grassmann convexity

“…The Grassmann convexity conjecture, formulated in [8], says that, for any positive integer k satisfying 0 < k < n, the number of real zeros in I of the Wronskian of any k linearly independent solutions to a disconjugate differential equation of order n on I is bounded from above by k(n − k), which is the dimension of the Grassmannian G(k, n; R). This conjecture can be reformulated in terms of convex curves in the nilpotent lower triangular group (see Section 2 and also [3,5,4]). The number k(n − k) has already been shown to be a correct lower bound for all k and n. Moreover it gives a correct upper bound for the case k = 2 (see [7]).…”

A CW complex homotopy equivalent to spaces of locally convex curves

“…The map φ : As another example of a lower set, let I (ω) be the set of words of dimension 0, i.e., words whose letters are generators a k . (The subscript is an ordinal, a relic of notation used in [11], starting in Section 14. The notation shall not be defined or needed in its general form.)…”

“…Let J ⊂ R be an interval. A sufficiently smooth curve γ : J → S n ⊂ R n+1 is (positive) locally convex [2,22,23] or (positive) nondegenerate [11,16,18,20] if it satisfies ∀t ∈ J, det(γ(t), γ (t), . .…”

Grassmann convexity and multiplicative Sturm theory, revisited

“…In this section we follow the notation of [5,6] and use matrix realizations of flag curves obtained as osculating curves of convex projective curves. (Such realizations were frequently used in the earlier papers by the authors).…”

“…[ [1,2,3]], [ [3,4]] and [ [4,5]]. Hence, Cont(w) = 0, and rk(w) = 2; (4) for w 4 = 415234 ′ 1 ′ 5 ′ 2 ′ 3 ′ , one gets s = 3, Cont(w) = 0 and rk(w) = 2.…”