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

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 $Spi{n}_{n+1}$. 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 $z_0,z_1\in Spi{n}_4$. 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 $z_1\in Qua{t}_4$ where $Qua{t}_4\subset Spi{n}_4$ is a finite group of order $16$. A more recent paper shows that for $z_1\in Qua{t}_4\setminus Z(Qua{t}_4)$ we have a homotopy equivalence $L_3(1;z_1)\approx \Omega Spi{n}_4$. In this paper we compute the homotopy type of $L_3(1;z_1)$ for $z_1\in Z(Qua{t}_4)$: it is equivalent to the wedge of $\Omega Spi{n}_4$ 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 $Y\subset \Omega Spi{n}_{n+1}(z_0;z_1)$ 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 $S_{n+1}\setminus \{e\}$ of nontrivial permutations.

42 pages, 5 figures; references updated