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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 $\Gamma$ in the group $Spin_{n+1}$; recall that $\Pi: Spin_{n+1} \to Flag_{n+1}$ is the universal cover of the space of flags. Let $L(z_0;z_1)$ 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 $z_0, z_1 \in Spin_4$. Recall that $Spin_4 = S^3 \times S^3 \subset H \times H$, where $H$ is the ring of quaternions. This paper relies heavily on previous publications. The earliest such 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 $16$ with center $Z(Quat_4) = \{(\pm 1,\pm 1)\}$. 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 infinite wedge of $\Omega Spin_4$ 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 $Y \subset L_n(z_0;z_1)$ for which the inclusion $Y \subset \Omega Spin_{n+1}(z_0;z_1)$ is a homotopy equivalence. For $n = 3$, 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 $S_{n+1} \smallsetminus \{e\}$ of nontrivial permutations., Comment: 34 pages, 4 figures

Locally convex curves in the sphere $S^n$ have been studied for several reasons, including the study of linear ordinary differential equations. Taking Frenet frames obtains corresponding curves $\Gamma$ in the group $Spin_{n+1}$; $\Pi: Spin_{n+1} \to Flag_{n+1}$ is the universal cover of the space of flags. Determining the homotopy type of spaces of such curves $\Gamma$ with prescribed initial and final points appears to be a hard problem. We may focus on $L_n$, the space of locally convex curves $\Gamma: [0,1] \to Spin_{n+1}$ with $\Gamma(0) = 1$, $\Pi(\Gamma(1)) = \Pi(1)$. Convex curves form a contractible connected component of $L_n$; there are $2^{n+1}$ other components, one for each endpoint. The homotopy type of $L_n$ has so far been determined only for $n=2$. This paper is a step towards solving the problem for larger values of $n$. The itinerary of $\Gamma$ belongs to $W_n$, the set of finite words in the alphabet $S_{n+1} \setminus \{e\}$. The itinerary of a curve lists the non open Bruhat cells crossed. Itineraries stratify the space $L_n$. We construct a CW complex $D_n$ which is a kind of dual of $L_n$ under this stratification: the construction is similar to Poincar\'e duality. The CW complex $D_n$ is homotopy equivalent to $L_n$. The cells of $D_n$ are naturally labeled by words in $W_n$; $D_n$ is locally finite. Explicit glueing instructions are described for lower dimensions. We describe an open subset $Y_n \subset L_n$, a union of strata of $L_n$. In each non convex component of $L_n$, the intersection with $Y_n$ is connected and dense. Most connected components of $L_n$ are contained in $Y_n$. For $n > 3$, in the other components the complement of $Y_n$ has codimension at least $2$. The set $Y_n$ is homotopy equivalent to the disjoint union of $2^{n+1}$ copies of $\Omega Spin_{n+1}$. For all $n \ge 2$, all connected components of $L_n$ are simply connected., Comment: 56 pages, 15 figures. Updated references and new results. Minor corrections. Text overlap with arXiv:1810.08632