Combinatorialization of spaces of nondegenerate spherical curves

Abstract

A parametric curve $\gamma$ of class $C^n$ on the $n$-sphere is said to be nondegenerate (or locally convex) when $\det\left(\gamma(t),\gamma'(t),\cdots,\gamma^{(n)}(t)\right)>0$ for all values of the parameter $t$. We orthogonalize this ordered basis to obtain the Frenet frame $\mathfrak{F}_{\gamma}$ of $\gamma$ assuming values in the orthogonal group $\operatorname{SO}_{n+1}$ (or its universal double cover, $\operatorname{Spin}_{n+1}$), which we decompose into Schubert or Bruhat cells. To each nondegenerate curve $\gamma$ we assign its itinerary: a word $w$ in the alphabet $S_{n+1}\smallsetminus\{e\}$ that encodes the succession of non open Schubert cells pierced by the complete flag of $\mathbb{R}^{n+1}$ spanned by the columns of $\mathfrak{F}_{\gamma}$. 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 Poincar\'e 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.