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Endpoint monodromy of an unordered configuration loop as a permutation of the labels

Definition

Fix $n \in N$ and the base configuration $q = (q_{1}, \dots, q_{n})$ of pairwise distinct points of $int D^{2}$ used in The pure braid group $P B_{n}$ as the fundamental group of an ordered configuration space and The configuration braid group $B_{n}^{c o n f}$ as the fundamental group of an unordered configuration space, and let $p : F_{n} (D^{2}) ⟶ C_{n} (D^{2})$ be the quotient map, which is an $n!$ -sheeted covering with deck group $S_{n}$ acting by coordinate permutations (Ordered configuration spaces cover the unordered ones regularly with deck group $S_{n}$ , Disjoint coordinate neighbourhoods evenly cover the unordered configuration space); the theorem applies to $M = D^{2}$ because the closed disk is a nonempty connected Hausdorff topological $2$ -manifold with boundary (The closed disk $D^{2}$ is a connected Hausdorff topological $2$ -manifold with boundary).

Now let $α : I \to C_{n} (D^{2})$ be a based loop at the orbit $[q]$ , that is $α (0) = [q] = α (1)$ (Based loops and the fundamental group). By the path-lifting property of a covering (Existence and uniqueness of path lifts through a covering map) there is a unique path $\tilde{α} : I ⟶ F_{n} (D^{2}), \tilde{α} (0) = q, p \circ \tilde{α} = α .$ Its endpoint $\tilde{α} (1)$ lies in the fibre $p^{- 1} ([q])$ , which is exactly the orbit $S_{n} \cdot q = {σ \cdot q : σ \in S_{n}}$ (Unordered configuration spaces $C_{n} (X)$ ); since the action is free, there is a unique permutation $σ_{α} \in S_{n}$ with $\tilde{α} (1) = σ_{α} \cdot q .$ The endpoint monodromy of $α$ is $π ([α]) : = σ_{α} \in S_{n} .$

The label form and the inverse. Equivalently, reading the endpoint tuple position by position, define $e_{α} \in S_{n}$ by $\tilde{α} (1)_{i} = q_{e_{α} (i)} (1 \leq i \leq n),$ so that the point standing at position $i$ at the end of the lifted motion is the one that carried label $e_{α} (i)$ at the start. Comparing with the coordinate formula $(σ \cdot q)_{i} = q_{σ^{- 1} (i - 1) + 1}$ of The symmetric group acts continuously and freely on $F_{n} (X)$ by permuting labels gives $e_{α} = σ_{α}^{- 1}, equivalently π ([α]) = e_{α}^{- 1} .$ The naive endpoint record $e$ is an antihomomorphism for the library's first-then-second loop product, $e_{α β} = e_{β} \circ e_{α}$ , as verified in step 3.1 below; the inversion in the definition of $π$ is exactly what turns it into the group homomorphism that the next results need.

Relation to the published monodromy action. For the right action $e \cdot [α]$ of the fundamental group on the fibre recorded in The monodromy right action on a covering fibre and its equivalent left-action convention one has $q \cdot [α] = \tilde{α} (1) = σ_{α} \cdot q = π ([α]) \cdot q$ . Thus $π ([α])$ is the unique permutation $σ$ satisfying $q \cdot [α] = σ \cdot q$ : the endpoint monodromy is the published covering monodromy, translated into the coordinate-permutation labels of $F_{n} (D^{2})$ . The corresponding left-action element of The monodromy right action on a covering fibre and its equivalent left-action convention is $[α] \cdot q = π ([α])^{- 1} \cdot q$ .

Scope and trivial cases. The map $π$ is the homomorphism $π : B_{n}^{c o n f} = π_{1} (C_{n} (D^{2}), [q]) ⟶ S_{n}$ whose image records the permutation of the labels effected by a loop; it is the last arrow of the configuration braid short exact sequence proved in The configuration braid short exact sequence $1 \to P B_{n} \to B_{n}^{c o n f} \to S_{n} \to 1$ . For $n \leq 1$ the group $S_{n}$ is trivial, so $π$ is the trivial homomorphism; the case $n = 0$ concerns the one-point space $C_{0} (D^{2})$ .

Facts & Assumptions

Given: A natural number $n$ , the base configuration $q \in F_{n} (int D^{2})$ , the covering $p : F_{n} (D^{2}) \to C_{n} (D^{2})$ , and a based loop $α : I \to C_{n} (D^{2})$ at $[q]$ .

[F1]

Points of $F_{n} (X)$ are tuples $(x_{1}, \dots, x_{n})$ of pairwise distinct points, with $F_{0} (X)$ a one-point space and labels identified with $n = {0, \dots, n - 1}$ by $κ (i) = i - 1$ (Ordered configuration spaces $F_{n} (X)$ ).

[L2]

$C_{n} (X) = F_{n} (X) / S_{n}$ with the quotient topology of $p$ , which is a covering map here; two tuples have the same image exactly when they differ by a permutation of coordinates, and the fibre $p^{- 1} ([q])$ is the orbit $S_{n} \cdot q$ (Unordered configuration spaces $C_{n} (X)$ , Ordered configuration spaces cover the unordered ones regularly with deck group $S_{n}$ ). The closed disk $D^{2} \subseteq C$ is nonempty, connected, Hausdorff and a topological $2$ -manifold with boundary, so that theorem applies with $M = D^{2}$ and $d = 2 \geq 2$ , and $p : F_{n} (D^{2}) \to C_{n} (D^{2})$ is an $n!$ -sheeted covering whose deck group $S_{n}$ acts by coordinate permutations (The closed disk $D^{2}$ is a connected Hausdorff topological $2$ -manifold with boundary).

[L3]

The action $(σ \cdot x)_{i} = x_{σ^{- 1} (i - 1) + 1}$ is a continuous, free left action of $S_{n}$ on $F_{n} (X)$ , so $\tilde{α} (1) = σ \cdot q$ determines $σ$ uniquely and the action law $(σ τ) \cdot x = σ \cdot (τ \cdot x)$ holds (The symmetric group acts continuously and freely on $F_{n} (X)$ by permuting labels, Group and abelian group).

[L4]

For a covering $p$ and a path $α$ in the base there is a unique lift with a prescribed starting point, and the endpoints of lifts of path-homotopic paths with the same initial point coincide (Existence and uniqueness of path lifts through a covering map, The endpoint of a lifted path depends only on its endpoint-fixed homotopy class).

[L5]

The monodromy $e \cdot [α]$ is the endpoint of the unique lift of $α$ beginning at $e$ , and with the first-then-second product $[α] [β] = [α * β]$ it is a right action, so $e \cdot ([α] [β]) = (e \cdot [α]) \cdot [β]$ (The monodromy right action on a covering fibre and its equivalent left-action convention, Based loops and the fundamental group).

[L6]

Loop classes at a point form a group under $[α] [β] = [α * β]$ (Loop classes form the group $π_{1} (X, x_{0})$ under concatenation), and a homomorphism of groups is a map preserving products (Monoid homomorphism and group homomorphism).

Proof

technique · direct

1.1

The lift and its endpoint permutation. By [L4] the lift $\tilde{α}$ with $\tilde{α} (0) = q$ exists and is unique. Its endpoint satisfies $p (\tilde{α} (1)) = α (1) = [q]$ , so $\tilde{α} (1) \in p^{- 1} ([q]) = S_{n} \cdot q$ by [L2]; thus there is $σ_{α} \in S_{n}$ with $\tilde{α} (1) = σ_{α} \cdot q$ , and it is unique by freeness in [L3]. So $e_{α}$ and $σ_{α}$ are related by $\tilde{α} (1)_{i} = q_{σ_{α}^{- 1} (i - 1) + 1}$ by the coordinate formula of [L3], that is $e_{α} (i) = σ_{α}^{- 1} (i - 1) + 1$ , which is $e_{α} = σ_{α}^{- 1}$ under the label identification of [F1].

F1L2L3L4

2.1

Independence of the representative. Let $α^{'} ≃ α$ rel endpoints be another based loop at $[q]$ . A path homotopy rel endpoints from $α$ to $α^{'}$ lifts, by [L4], to a homotopy of paths from $\tilde{α}$ to the lift of $α^{'}$ starting at $q$ , keeping the starting point fixed; in particular the two lifts have the same endpoint, so $σ_{α^{'}} = σ_{α}$ by uniqueness in step 1.1. Hence $π ([α]) : = σ_{α}$ is well defined on classes.

step 1.1L4

2.2

The endpoint permutation is multiplicative. Let $α, β$ be based loops at $[q]$ and let $\tilde{α}, \tilde{β}$ be their lifts starting at $q$ . The path $s \mapsto σ_{α} \cdot \tilde{β} (s)$ is a path in $F_{n} (D^{2})$ starting at $σ_{α} \cdot q = \tilde{α} (1)$ and covering $β$ , because $p (σ_{α} \cdot x) = p (x)$ for every $x$ by [L2]; by uniqueness of lifts in [L4] it is the lift of $β$ beginning at $\tilde{α} (1)$ . Therefore the concatenation $s \mapsto \tilde{α} (2 s)$ for $s \leq \frac{1}{2}$ and $s \mapsto σ_{α} \cdot \tilde{β} (2 s - 1)$ for $s \geq \frac{1}{2}$ is a path in $F_{n} (D^{2})$ starting at $q$ and covering $α * β$ — the two pieces agree at $s = \frac{1}{2}$ at the point $\tilde{α} (1)$ — so by uniqueness it is the lift of $α * β$ starting at $q$ . Its endpoint is $σ_{α} \cdot \tilde{β} (1) = σ_{α} \cdot (σ_{β} \cdot q) = (σ_{α} σ_{β}) \cdot q$ by the action law of [L3]. Hence $σ_{α β} = σ_{α} σ_{β}$ , that is $π ([α] [β]) = π ([α]) π ([β])$ by [L5] and [L6].

step 1.1L2L3L4L5L6

2.3

Relation to the published monodromy. By [L5] and step 1.1, $q \cdot [α]$ is the endpoint of the lift of $α$ beginning at $q$ , namely $σ_{α} \cdot q = π ([α]) \cdot q$ ; since the action is free by [L3], $π ([α])$ is the unique $σ$ with $q \cdot [α] = σ \cdot q$ .

step 1.1L3L5

3.1

The label form is an antihomomorphism. For based loops $α, β$ at $[q]$ , step 1.1 gives $e_{α β} = σ_{α β}^{- 1}$ and $e_{α} = σ_{α}^{- 1}$ , $e_{β} = σ_{β}^{- 1}$ ; by step 2.2 and the group law $(σ_{α} σ_{β})^{- 1} = σ_{β}^{- 1} σ_{α}^{- 1}$ of [L3], $e_{α β} = σ_{β}^{- 1} σ_{α}^{- 1} = e_{β} \circ e_{α}$ in the composition convention of $S_{n}$ . Thus the endpoint record $e$ reverses the order of the product, while $π = e^{- 1}$ does not.

step 1.1step 2.2L3

4.1

Conclusion. Steps 1.1, 2.1 and 2.2 show that $π ([α]) = σ_{α} = e_{α}^{- 1}$ is a well-defined group homomorphism $B_{n}^{c o n f} \to S_{n}$ , step 3.1 records that the label form $e$ itself is an antihomomorphism, and step 2.3 identifies $π$ with the published covering monodromy at the element $q$ . For $n \leq 1$ , $S_{n}$ is trivial and $π$ is trivially a homomorphism.

step 1.1step 2.1step 2.2step 3.1step 2.3L6∎

Depends on

Used by

The configuration braid short exact sequence 1→ PBₙ→ Bₙᶜᵒⁿᶠ→ Sₙ→ 1 Theorem

Dependency tree · two levels

77 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, sections 1.1-1.3 and 2.1, printed pp. 3-6, 11-13 (standard reference, not scraped)