[Paper Review] Topology of the space of locally convex curves on the 3-sphere

Emília Alves|arXiv (Cornell University)|Aug 16, 2016

Geometric Analysis and Curvature Flows17 references3 citations

TL;DR

This paper investigates the topology of spaces of locally convex curves on the 3-sphere, focusing on four non-generic homotopy types in the case $ n = 3 $. By decomposing generic curves in $ \mathbb{S}^3 $ into pairs of curves in $ \mathbb{S}^2 $, the author proves that two of these spaces have nontrivial cohomology beyond that of the space of generic curves, establishing they are topologically distinct and providing a criterion for global convexity using the 2-sphere projection.

ABSTRACT

A (positive) locally convex curve in the 2-sphere is a curve with positive geodesic curvature (i.e., which always turns left). In the 3-sphere, it is a curve with positive torsion. In this work we discussed the topology of spaces of such curves with prescribed initial and final jets. The case of the 2-sphere is understood (Saldanha-2013); the case of n=3 is not yet thoroughly clarified. In order to obtain partial information about the homotopy type of such spaces in the case n=3, we represented a positive locally convex curve as a pair of curves on the 2-sphere with some restrictions.

Motivation & Objective

• To determine the topological structure of non-generic components in the space of locally convex curves on $ \mathbb{S}^3 $, particularly those not homeomorphic to the space of generic curves.
• To extend the classification of locally convex curve spaces from $ n = 2 $ to $ n = 3 $, where Saldanha previously resolved the $ n = 2 $ case.
• To develop a decomposition method for generic curves in $ \mathbb{S}^3 $ into pairs of curves in $ \mathbb{S}^2 $, preserving local convexity in one component.
• To establish a criterion for global convexity of a curve in $ \mathbb{S}^3 $ based on the local convexity of its projected curve in $ \mathbb{S}^2 $.
• To prove that certain spaces of locally convex curves in $ \mathbb{S}^3 $ have nontrivial cohomology in even degrees beyond the generic curve space, implying topological distinction.

Proposed method

• Use of the Frenet frame lift to define $ \mathcal{L}\mathbb{S}^n(z) $, the space of locally convex curves with fixed initial and final lifted frames in $ \mathrm{Spin}_{n+1} $.
• Application of Bruhat cell decomposition and homotopy-theoretic techniques to analyze the homotopy type of curve spaces.
• Construction of maps $ \hat{h}_{2k-2} : \mathbb{S}^{2k-2} \to \mathcal{L}\mathbb{S}^3(z_l, z_r) $ to detect nontrivial homotopy groups.
• Definition of cohomology class $ \hat{m}_{2k-2} $ on $ \mathcal{L}\mathbb{S}^3(z_l, z_r) $, with $ \hat{m}_{2k-2}(\hat{h}_{2k-2}) = \pm 1 $, indicating nontriviality.
• Use of the pullback map $ h_{2k-2}^* $ and its lift $ \widehat{h_{2k-2}^*} $ to relate curve spaces in $ \mathbb{S}^2 $ and $ \mathbb{S}^3 $.
• Leveraging the fact that $ h_{2k-2} $ is homotopic to a constant in $ \mathcal{G}\mathbb{S}^2(\mathbf{1}) $ to deduce homotopy triviality in $ \mathcal{G}\mathbb{S}^3(\mathbf{1}, -\mathbf{k}) $, while detecting extra topology in $ \mathcal{L}\mathbb{S}^3 $.

Experimental results

Research questions

• RQ1 Are the non-generic components of the space of locally convex curves on $ \mathbb{S}^3 $ topologically distinct from the space of generic curves?
• RQ2 Can the topology of $ \mathcal{L}\mathbb{S}^3(z) $ be determined for $ z \in \mathrm{Spin}_4 $ beyond the known generic case?
• RQ3 Does a locally convex curve in $ \mathbb{S}^3 $ decompose into a pair of curves in $ \mathbb{S}^2 $, with one component preserving local convexity?
• RQ4 Can global convexity of a curve in $ \mathbb{S}^3 $ be determined from the local convexity of its projection to $ \mathbb{S}^2 $?
• RQ5 What is the cohomological structure of $ \mathcal{L}\mathbb{S}^3(z) $, and how does it differ from $ \mathcal{G}\mathbb{S}^3(z) $?

Key findings

• For $ n = 3 $, the space $ \mathcal{L}\mathbb{S}^3(-\mathbf{1}, \mathbf{1}) $ has cohomology dimension at least $ l+2 $ in degree $ j = 2l $ when $ l $ is odd, exceeding the dimension of the generic curve space.
• Similarly, $ \mathcal{L}\mathbb{S}^3(\mathbf{1}, -\mathbf{1}) $ has cohomology dimension at least $ l+2 $ in degree $ j = 2l $ when $ l $ is even, indicating topological distinction from the generic case.
• The maps $ \hat{h}_{2k-2} : \mathbb{S}^{2k-2} \to \mathcal{L}\mathbb{S}^3(z_l, z_r) $ are homotopic to constant maps in $ \mathcal{G}\mathbb{S}^3(z_l, z_r) $, but detect nontrivial cohomology classes $ \hat{m}_{2k-2} $, proving extra topology in $ \mathcal{L}\mathbb{S}^3(z_l, z_r) $.
• Any generic curve in $ \mathbb{S}^3 $ can be decomposed into a pair of curves in $ \mathbb{S}^2 $, and if the original curve is locally convex, one of the projected curves is also locally convex.
• A locally convex curve in $ \mathbb{S}^3 $ is globally convex if and only if its associated curve in $ \mathbb{S}^2 $ is globally convex, providing a criterion based on the 2-sphere projection.
• The spaces $ \mathcal{L}\mathbb{S}^3(z) $ for $ z = (-\mathbf{1})^k $ and $ z = (-\mathbf{1})^{k-1}\mathbf{k} $, with $ k \geq 2 $, have nontrivial cohomology beyond the generic curve space, confirming they are not homotopy equivalent to it.