Short loops, the uniform-plus-length topology, short-loop homotopies, and nonshrinkability

Fix the following definitions and conventions; they are used throughout this page. Let $X$ be a compact locally CAT(1) metric space (Comparison triangles, the CAT(0) and CAT(1) inequalities, local CAT, local geodesics and round circles, Open cover, subcover, compact metric space, and compact subset of a metric space).

(1) Rectifiable loops and normalization. A loop in $X$ is a continuous map $\gamma \text{\hspace{0.17em}⁣}:[0,1]\to X$ with $\gamma (0)=\gamma (1)$; its length $L(\gamma )\in [0,\mathrm{\infty }]$ is the supremum of its polygonal sums and $\gamma$ is rectifiable if $L(\gamma )<\mathrm{\infty }$ (Length in a metric target: lower semicontinuity and arc-length reparametrization, Upper bound, least upper bound, and strict upper bound). A rectifiable loop is normalized if it is parametrized proportionally to arclength, i.e. $L(\gamma {\mathrm{\mid }}_{[s,t]})=(t-s)L(\gamma )$ for all $0\le s\le t\le 1$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length); by the arc-length parametrization clause of Length in a metric target: lower semicontinuity and arc-length reparametrization every rectifiable loop of positive length has a normalized reparametrization, and a loop of length $0$ is normalized exactly when it is constant. Henceforth every loop on this page is normalized.

(2) Short loops. A loop $\gamma$ is short if $L(\gamma )<2\pi$ (Pi is the first positive zero of sine). The constant loop at any point is short.

(3) The uniform-plus-length topology. For loops $\gamma ,{\gamma }^{\mathrm{\prime }}$ write ${\gamma }_k\to \gamma$ in the uniform-plus-length topology when ${\sup }_{t\in [0,1]}d({\gamma }_k(t),\gamma (t))\to 0$ and $L({\gamma }_k)\to L(\gamma )$ (Limits and Cauchy sequences of reals). Uniform convergence of the maps alone does not bound rectifiable lengths, so the length term belongs to the topology by definition; the resulting uniform length bound $b<2\pi$ and the common fine mesh of a compact short family are proved in Polygon transfer, the basin as the shrinkable class, and the short-loop criterion ↗.

(4) Short-loop homotopies. A short-loop homotopy from ${\gamma }_0$ to ${\gamma }_1$ is a family $({\gamma }_s{)}_{s\in [0,1]}$ of short loops with ${\gamma }_0,{\gamma }_1$ the given loops such that $s\mapsto {\gamma }_s$ is continuous for the uniform-plus-length topology (Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form). A short loop is shrinkable if it is short-loop homotopic to some constant loop, and nonshrinkable otherwise. Short-loop homotopy is an equivalence relation on short loops (reflexivity, symmetry and transitivity hold by reparametrizing the parameter interval).

(5) Comparison with ordinary null-homotopy. A null-homotopy of a short loop allows intermediate loops of arbitrary length, whereas a short-loop homotopy demands that every intermediate loop be short; the comparison of the two notions, and the fact that a closed local geodesic is never shrinkable, are theorems of this page, not part of the definition.

(6) Polygons are defined separately. The cyclic tuples, midpoint operation and zero-limit basin used below are defined without reference to short-loop homotopy; their identification with the classes of (4) is a theorem proved after the basin's topological properties are established.