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

TL;DR

This paper determines the homotopy type of spaces of locally convex curves in the 3-sphere, extending previous results for lower dimensions and providing explicit constructions for the homotopy equivalences.

Contribution

It computes the homotopy type of spaces of locally convex curves in S^3 for all endpoint conditions, introducing new algebraic and combinatorial methods for the analysis.

Findings

Homotopy type of L_3(z_0;z_1) for all endpoints determined.

Homotopy type of closed locally convex curves in S^3 and P^3 characterized.

Explicit subsets Y are constructed to establish homotopy equivalences.

Abstract

Locally convex (or nondegenerate) curves in the sphere $S^n$ 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 $\Gamma$ in the group $Spin_{n+1}$. Let $L_n(z_0;z_1)$ be the space of such curves $\Gamma$ with prescribed endpoints $\Gamma(0) = z_0$, $\Gamma(1) = z_1$. The aim of this paper is to determine the homotopy type of the spaces $L_3(z_0;z_1)$ for all $z_0, z_1 \in Spin_4$. As a corollary, we obtain the homotopy type of the space of closed locally convex curves in either $S^3$ or $P^3$. There are many previous papers addressing related questions. An early paper solves the corresponding problem for curves in $S^2$. Another previous result (with B. Shapiro) reduces the problem to $z_0 = 1$ and $z_1 \in Quat_4$ where $Quat_4 \subset Spin_4$ is a finite group of order $16$. A more recent paper shows that for $z_1 \in Quat_4 \smallsetminus Z(Quat_4)$ we have a homotopy equivalence $L_3(1;z_1) \approx \Omega Spin_4$. In this paper we compute the homotopy type of $L_3(1;z_1)$ for $z_1 \in Z(Quat_4)$: it is equivalent to the wedge of $\Omega Spin_4$ with an infinite countable family of spheres (as for the case $n = 2$). The structure of the proof can be compared to that of the case $n = 2$ but some of the steps require the creation of new theories, involving algebra and combinatorics. We construct explicit subsets $Y \subset L_n(z_0;z_1)$ for which the inclusion $Y \subset \Omega Spin_{n+1}(z_0;z_1)$ is a homotopy equivalence. For $n = 2$, there is a simple geometric description of $Y$; for $n = 3$, the far less natural construction is based on the theory of itineraries of such curves. The itinerary of a curve in $L_n(1;z_1)$ is a finite word in the alphabet $S_{n+1} \smallsetminus \{e\}$ of nontrivial permutations.