论文标题
球体s^3中局部凸曲线空间的同质型类型
Homotopy type of spaces of locally convex curves in the sphere S^3
论文作者
论文摘要
由于几个原因，已经研究了球体$ s^n $的本地凸（或非等级）曲线，包括研究$ n+1 $的线性普通微分方程。服用Frenet帧使我们能够在组中获得相应的曲线$γ$ $ spin_ {n+1} $。令$ l_n（Z_0; Z_1）$为此类曲线$γ$的空间，并有规定的端点$γ（0）= Z_0 $，$γ（1）= Z_1 $。本文的目的是确定所有$ z_0，z_1 \ in spin_4 $中的空间的同型类型$ l_3（z_0; z_1）$。作为推论，我们以$ s^3 $或$ p^3 $中的封闭本地凸曲线的空间的同置类型的类型。 以前有许多论文解决相关问题。早期论文解决了$ s^2 $中的曲线的相应问题。另一个先前的结果（使用B. shapiro）将问题降低到$ z_0 = 1 $和$ z_1 \ quat_4 $中的$ quat_4 \ subset spin_4 $是订单$ 16 $的有限组。最近的一篇论文表明，对于$ z_1 \，在quat_4 \ smallsetminus z（quat_4）$中，我们有一个同型等价$ l_3（1; z_1）\ ofωspin_4$。在本文中，我们计算$ l_3（1; z_1）$的同型类型$ z_1 \ in z（quat_4）$：相当于$ωspin_4$的楔形，带有无限可计数的球体（如$ n = 2 $）。 可以将证明的结构与$ n = 2 $的情况进行比较，但是某些步骤需要创建新理论，涉及代数和组合。我们构建显式子集$ y \ subset l_n（z_0; z_1）$，其中包含$ y \ y \ subsetωspin_{n+1}（z__0; z_1）$是同质的同等性。对于$ n = 2 $，有一个简单的几何描述为$ y $；对于$ n = 3 $，自然构造的基于此类曲线的行程理论。 $ l_n（1; z_1）$中的曲线的行程是字母$ s_ {n+1} \ smallSetMinus \ {e \} $中的一个有限单词。
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 $Γ$ in the group $Spin_{n+1}$. Let $L_n(z_0;z_1)$ be the space of such curves $Γ$ with prescribed endpoints $Γ(0) = z_0$, $Γ(1) = z_1$. 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 $L_3(1;z_1) \approx ΩSpin_4$. 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 $ΩSpin_4$ 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 $Y \subset ΩSpin_{n+1}(z_0;z_1)$ 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.
