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A geometric braid slices to an interior configuration loop

Statement

For a geometric braid $β = (z_{1}, \dots, z_{n})$ based at $Q$ , let $S (β) (t) : = [(z_{1} (t), \dots, z_{n} (t))],$ where the real-disc coordinates are identified with complex coordinates as in Based motions of an unordered point configuration. Then $S (β)$ is a continuous based loop in $C_{n} (int D^{2})$ . This slicing uses the given level parameter and the given labelled point motions; at height $t$ its value is exactly the unordered configuration cut out by the braid at that height.

Facts & Assumptions

Given: $n \in N$ and a geometric braid $β = (z_{1}, \dots, z_{n})$ based at the fixed tuple $Q$ .

[L1]

The ordered configuration space $F_{n} (X)$ has the subspace topology from the product $X^{n}$ with its product topology; for $n = 0$ it is the one-point empty tuple (Ordered configuration spaces $F_{n} (X)$ ).

[L2]

Each $z_{j} : I \to D^{\circ}$ is continuous (Geometric braids in the disc with setwise endpoints).

[L3]

At every height the coordinates are pairwise distinct (Geometric braids in the disc with setwise endpoints).

[L4]

Each point motion starts at the corresponding $q_{j}$ (Geometric braids in the disc with setwise endpoints).

[L5]

The set of the terminal points is $Q$ (Geometric braids in the disc with setwise endpoints).

[L6]

The fixed real-complex identification takes $D^{\circ}$ to $int D^{2}$ (Based motions of an unordered point configuration).

[L7]

The canonical map $p_{n} : F_{n} (X) \to C_{n} (X)$ , $x \mapsto [x]$ , is continuous and sends a tuple to its coordinate-permutation orbit (Unordered configuration spaces $C_{n} (X)$ ).

[L8]

The product topology on a product is the initial topology of its coordinate projections, so continuity into that product is checked coordinatewise (The product set $\prod_{i \in I} X_{i}$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).

[L9]

The $j$ th strand is the graph $t \mapsto (z_{j} (t), t)$ and its second coordinate is its height (Geometric braids in the disc with setwise endpoints).

The coordinate functions determine one tuple-valued map at each height; no choice of ordering or lift is made.

Proof

technique · direct

1.1L1L2L3L4L6L8

The ordered tuple varies continuously. Apply the fixed real-complex identification from [L6] to each coordinate $z_{j}$ . By [L2], each resulting coordinate map $I \to int D^{2}$ is continuous. The product topology on $(int D^{2})^{n}$ is the topology for which continuity into the product is checked coordinatewise [L8], so $z : I \to (int D^{2})^{n}, z (t) = (z_{1} (t), \dots, z_{n} (t))$ is continuous. Pairwise distinctness in [L3] places its image in $F_{n} (int D^{2})$ ; because that space has the subspace topology, the same map, with this restricted codomain, is continuous ([L1]). For $n = 0$ this is the constant map to the one-point empty tuple.

1.2

Pass to the unordered quotient. By [L7], $p_{n}$ is continuous, so $S (β) = p_{n} \circ z : I \to C_{n} (int D^{2})$ is continuous. At the bottom, $z (0) = Q$ by [L4], so $S (β) (0) = [Q]$ . At the top, the set of the coordinates of $z (1)$ is $Q$ by [L5], hence $z (1)$ is a coordinate permutation of $Q$ and $p_{n} (z (1)) = [Q]$ by the orbit description in [L7]. Thus $S (β) (1) = [Q]$ , as required for a based loop ([L4, L5, L7]).

1.3L3L5L7L9

Identify each height slice. The graph of the $j$ th coordinate is $t \mapsto (z_{j} (t), t)$ by [L9], so its intersection with height $t$ returns the point $z_{j} (t)$ . Taking all labels, their unordered orbit is exactly $p_{n} (z (t)) = S (β) (t)$ . The labels are retained in $z$ and then forgotten precisely by the quotient, with no reparametrization of height. For $n = 0$ the slice and loop are the unique empty configuration.

2.1step 1.1step 1.2step 1.3∎

The sliced path is continuous, has both endpoint values $[Q]$ , and at each height equals the unordered slice of the given level-preserving braid.

Depends on

Used by

Pure geometric braids and ordered configuration loops Corollary
An embedded height-folded arc has no configuration-loop slices Counterexample
An exchange closes only after forgetting labels Counterexample
Raw slicing reverses geometric stacking products Lemma
Tracing and slicing are inverse on relative classes Lemma
The geometric endpoint permutation matches covering monodromy Proposition

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Juan Gonzalez-Meneses, Basic results on braid groups, §§1.1–1.3, printed pp. 3–6 (standard reference, not scraped)
Joan S. Birman and Tara E. Brendle, Braids: A Survey, §1.1, author manuscript pp. 3–5 (standard reference, not scraped)