Homotopy type of spaces of curves with constrained curvature on flat surfaces

TL;DR

The paper determines the homotopy type of spaces of planar curves on a flat surface with fixed endpoints and tangents, whose curvature stays in a prescribed open interval. The authors develop a h-principle–style framework built around quasicritical curves, a stretching (grafting) procedure, and a good cover of a Banach-manifold of curves, to translate geometric constraints into combinatorial incidence data. They prove that each connected component of the constrained-curve space is either contractible or homotopy equivalent to a sphere $\mathbb{S}^n$ for some $n$, and they realize every $n\ge 0$ explicitly by constructing homotopy equivalences; when top sign-strings exist, the space is modeled by $\mathbf{E}\times\mathbb{S}^{n-1}$, with $\mathbf{E}$ the separable Hilbert space. The results extend and sharpen prior work on constrained-curvature spaces, providing concrete generators for homotopy groups and a precise regional description of the homeomorphism types in terms of end data $(q,z)$ on the unit tangent bundle of the plane.

Abstract

Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a given open interval, in terms of all parameters involved. Any connected component of such a space is either contractible or homotopy equivalent to an $n$-sphere, and every $n\geq 1$ is realizable. Explicit homotopy equivalences between the components and the corresponding spheres are constructed.

Figures (13)

• Figure 1: This drawing to scale indicates the homeomorphism class of $\mathscr{M}(Q)$ in terms of $q$, for $Q=(q,z)\in \C\times \Ss^1$ and a fixed $z\neq-1$ (here $z\approx \exp(\frac{i\pi}{7})$). If $q$ lies in the unshaded region, then $\mathscr{M}(Q)\approx \mathbf{E}$, the separable Hilbert space. The line segments are only auxiliary elements and do not bound any regions. The line through 0 and $1+z$ (not drawn) contains $\pm (i-iz)$ and is an axis of symmetry of the figure. The radii of the circles are indicated inside parentheses near their centers.
• Figure 2: The homeomorphism class of $\mathscr{M}(Q_x)$ as a function of $x\in \R$.
• Figure 3: Constructing a generator of $\pi_1\mathscr{M}(Q)$ when $Q=(x,1)\in \R\times \Ss^1$ and $4<x\leq 4\sqrt{2}$.
• Figure 4: The decomposition of $\R^3$ into the 13 cells $W_{\ast}$ and into the sets $M$, $S$ and $L_\sigma$, for $\left\vert\sigma\right\vert\geq 2$. More precisely, what is depicted here is the orthogonal projection of these sets onto the plane $\{(x_1,x_2,x_3)\in \R^3\text{\ \large$:$\ }x_1+x_2+x_3=0\}$.
• Figure 5: Split, level and mixed points in $\R^{10}$, respectively, represented by beads (black for odd-indexed coordinates and white for even-indexed coordinates).

Theorems & Definitions (48)

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• proof : Proof of P:decomposition
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• proof : Proof of P:nested
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