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The geometric and configuration braid models agree at the fixed base configuration

Statement

At the explicit geometric base tuple $Q$ , let $G_{n}$ be the geometric braid group, $B_{n}^{c o n f} = π_{1} (C_{n} (D^{2}), [Q])$ , and $P B_{n} = π_{1} (F_{n} (D^{2}), Q)$ . The inverse-loop slicing map $Φ : G_{n} ⟶ B_{n}^{c o n f}, [β] ⟼ (ι_{*}^{C} [S (β)])^{- 1}$ is the canonical group isomorphism. It intertwines the endpoint maps: $π_{c o n f} (Φ ([β])) = π_{g e o} ([β]) ([β] \in G_{n}) .$ If $G_{n}^{p u r e} : = \ker π_{g e o}$ , then its restriction, followed by the inverse of the quotient-induced map $p_{*} : P B_{n} \to \ker π_{c o n f}$ , is the isomorphism $Ψ : G_{n}^{p u r e} ⟶ P B_{n}, Ψ ([β]) = (ι_{*}^{F} [z_{β}])^{- 1},$ where $z_{β} (t) = (z_{1} (t), \dots, z_{n} (t))$ for a pure braid. Thus pure geometric braids and ordered configuration loops agree through the same fixed-basepoint construction. No canonical identification at another basepoint or Artin-presentation claim is included.

Facts & Assumptions

Given: $n$ , the explicit tuple $Q$ , the geometric and configuration braid groups based at $Q$ and $[Q]$ , endpoint monodromy, and the ordered quotient map.

[L1]

At this exact basepoint, inverse-loop slicing $Φ ([β]) = (ι_{*}^{C} [S (β)])^{- 1}$ is a group isomorphism $G_{n} \to B_{n}^{c o n f}$ (Geometric braid classes and the unordered configuration fundamental group).

[L2]

The isomorphism $Φ$ carries the pure geometric subgroup onto $\ker π_{c o n f} = im p_{*}$ and restricts to an isomorphism onto that kernel (Pure geometric braids and ordered configuration loops).

[L3]

The quotient-induced map $p_{*}$ identifies $P B_{n}$ with that same kernel, and the resulting pure-group isomorphism has the formula $Ψ ([β]) = (ι_{*}^{F} [z_{β}])^{- 1}$ (Pure geometric braids and ordered configuration loops).

[L4]

For every geometric braid class $[β]$ , $π_{c o n f} (Φ ([β])) = π_{g e o} ([β])$ (The geometric endpoint permutation matches covering monodromy).

The common tuple $Q$ and all coordinate paths are specified. The inherited isomorphisms use no arbitrary change-of-basepoint path, and no choice principle is used.

Proof

technique · direct

1.1L1

The unordered configuration model. By [L1], the inverse-loop slicing map is a group isomorphism at the shared basepoint $Q$ , with $B_{n}^{c o n f} = π_{1} (C_{n} (D^{2}), [Q])$ . No path to another basepoint enters this map.

1.2L2L3

The pure ordered model. By [L2], $Φ$ takes $G_{n}^{p u r e}$ onto $\ker π_{c o n f}$ . The quotient-induced isomorphism from [L3] identifies $P B_{n}$ with that kernel, so composing its inverse with the pure restriction of $Φ$ gives $Ψ : G_{n}^{p u r e} \to P B_{n}$ . The same fact [L3] supplies the coordinate formula and makes it representative-independent.

1.3L4

The endpoint square. Equation [L4] states exactly that the diagram with top map $Φ$ , vertical maps $π_{g e o}$ and $π_{c o n f}$ , and bottom map ${id}_{S_{n}}$ commutes. Thus the isomorphism preserves endpoint permutations.

2.1L1L2L3L4∎

Degenerate strand counts. The supplied isomorphism and both component claims [L1]–[L4] apply for every $n \geq 0$ , including the empty braid at $n = 0$ and the trivial permutation targets at $n = 1$ . Hence the packaged theorem and its pure restriction include these cases.

Remarks

The two commuting squares display the pure subgroup and the endpoint maps at the same fixed base tuple $Q$ . The left upper vertical map is subgroup inclusion; $p_{*}$ on the right has image $\ker π_{c o n f}$ . The horizontal maps $Φ$ and $Ψ$ are isomorphisms.

The upper square says $p_{*} \circ Ψ = Φ ∣_{G_{n}^{p u r e}}$ , and the lower square says $π_{c o n f} \circ Φ = π_{g e o}$ . Both use inverse-loop slicing: the strand graphs $(z_{j} (t), t)$ are read from height $0$ to $1$ , giving the raw slice $S (β) (t) = [(z_{1} (t), \dots, z_{n} (t))]$ , whereas $Φ ([β])$ is the inverse of its included loop class. For a pure braid the ordered tuple is a loop at $Q$ , and $Ψ$ uses its inverse class as well. This inversion matches the local stacking convention: $β$ lies below $γ$ in $γ ⋆ β$ , so raw slicing traverses $S (β)$ before $S (γ)$ . The figure summarizes the maps in the statement; the proof above is independent of it.

Depends on

Used by

All four classical braid models realize the Artin presentation Corollary

Dependency tree · two levels

24 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)