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The Burau infinite cyclic cover

Definition

Let $n \geq 1$ (as in the definition of the total winding homomorphism: for $n = 0$ the map $π_{1} (X, d) \to Z$ is trivial, not surjective, and there is no infinite cyclic cover). Let $X = D^{2} ∖ Q_{n}$ , let $d$ be the boundary basepoint, and let $ω : π_{1} (X, d) \to Z$ be the total winding homomorphism of The total winding homomorphism of the punctured disk. Put $K = \ker ω$ . The Burau infinite cyclic cover is the based connected covering $p : (\tilde{X}, \tilde{d}) ⟶ (X, d)$ classified by the subgroup $K$ , so that $p_{*} π_{1} (\tilde{X}, \tilde{d}) = K$ and $\tilde{d}$ is the chosen lift of $d$ . It exists and is unique up to a unique based isomorphism (Every subgroup acts on the universal cover with a connected quotient covering that realizes it, Connected covering spaces are classified by conjugacy classes of fundamental-group subgroups) because $X$ is nonempty, path-connected, locally path-connected and semilocally simply connected (The punctured disk is path-connected, locally path-connected and semilocally simply connected), and $[π_{1} (X, d) : K] = \infty$ . Because $K$ is normal, the cover is regular, its deck group is $Deck (\tilde{X} / X) ≅ π_{1} (X, d) / K ≅ Z$ (A connected covering is regular exactly when its induced subgroup is normal, exactly when deck transformations act transitively on a fibre, A regular connected covering has deck group $π_{1} (B, b_{0}) / p_{*} π_{1} (E, e_{0})$ , $Deck (E / B) ≅ N_{G} (H) / H$ for a connected covering), and we write $t$ for the deck transformation corresponding to the positive generator of $Z$ , that is, the deck transformation whose monodromy raises total winding by $1$ (The monodromy right action on a covering fibre and its equivalent left-action convention). Then $Deck (\tilde{X} / X) = {t^{k} : k \in Z}$ , deck transformations act on the left (Deck transformations and the deck-transformation group of a covering), and $t^{- 1}$ is the negative generator. The sign of $t$ is fixed once and for all by the positive orientation of the standard meridians and the identification $π_{1} / K ≅ Z$ .

Facts & Assumptions

Given: The punctured disk $X = D^{2} ∖ Q_{n}$ with basepoint $d$ , the total winding homomorphism $ω : π_{1} (X, d) \to Z$ , and $K = \ker ω$ .

[F1]

Since $X$ is nonempty, path-connected, locally path-connected and semilocally simply connected, every subgroup $H \leq π_{1} (X, d)$ is realised by a based connected covering $p_{H} : (E_{H}, e_{H}) \to (X, d)$ with $(p_{H})_{*} π_{1} (E_{H}, e_{H}) = H$ (Every subgroup acts on the universal cover with a connected quotient covering that realizes it, The punctured disk is path-connected, locally path-connected and semilocally simply connected).

[F2]

For such a base, the assignment $[p : (E, e_{0}) \to (X, d)] \mapsto p_{*} π_{1} (E, e_{0})$ is a bijection from based-isomorphism classes of based connected coverings to subgroups of $π_{1} (X, d)$ ; hence the based connected covering with image subgroup $K$ is unique up to a unique based isomorphism (Connected covering spaces are classified by conjugacy classes of fundamental-group subgroups); any two based covering isomorphisms are lifts of the same projection and agree at the basepoint, so they are equal by Two lifts from a connected space that agree at one point agree everywhere.

[F3]

A connected covering is regular exactly when its image subgroup is normal, and then its deck group acts transitively on fibres (A connected covering is regular exactly when its induced subgroup is normal, exactly when deck transformations act transitively on a fibre, Regular coverings).

[F4]

For a regular connected covering of a path-connected locally path-connected base, $Deck (E / B) ≅ G / H$ , where $G = π_{1} (B, b_{0})$ and $H = p_{*} π_{1} (E, e_{0})$ (A regular connected covering has deck group $π_{1} (B, b_{0}) / p_{*} π_{1} (E, e_{0})$ ); equivalently $Deck (E / B) ≅ N_{G} (H) / H$ ( $Deck (E / B) ≅ N_{G} (H) / H$ for a connected covering).

[F5]

The first isomorphism theorem: $ω$ factors as an isomorphism $G / \ker ω \to im ω$ (First isomorphism theorem for groups: $G / \ker f ≅ im f$ ); the total winding homomorphism is surjective with $im ω = Z$ (The total winding homomorphism of the punctured disk).

[F6]

Monodromy is the right action in which $e \cdot [α]$ is the endpoint of the lift of $[α]$ starting at $e$ , and the corresponding left action is $[α] \cdot e : = e \cdot [α]^{- 1}$ (The monodromy right action on a covering fibre and its equivalent left-action convention).

[F7]

A fibre of a covering with nonempty path-connected total space is in bijection with the set of right cosets of the image subgroup, so the index of $K$ equals the cardinality of the fibre (For a nonempty path-connected total space, a covering fibre is in bijection with the right cosets of the induced fundamental-group subgroup).

Proof

technique · direct

1.1F5

The kernel and its quotient. The kernel $K = \ker ω$ is a normal subgroup of $π_{1} (X, d)$ , and by [F5] the map $ω$ induces an isomorphism $π_{1} (X, d) / K \to im ω = Z$ . Hence $π_{1} (X, d) / K$ is infinite and the index of $K$ is infinite: a finite index would make the quotient finite.

1.2F1F2

Existence and uniqueness of the cover. The four hypotheses of [F1] hold by The punctured disk is path-connected, locally path-connected and semilocally simply connected, so the subgroup $K$ is realised by a based connected covering $p : (\tilde{X}, \tilde{d}) \to (X, d)$ with $p_{*} π_{1} (\tilde{X}, \tilde{d}) = K$ ; by the bijection [F2] any two such based coverings are uniquely based-isomorphic. This defines the Burau infinite cyclic cover.

2.1F3F4F6F7step 1.1∎

Regularity, deck group and the generator $t$ . Since $K$ is normal, [F3] makes $p$ regular, and [F4] together with step 1.1 gives $Deck (\tilde{X} / X) ≅ π_{1} (X, d) / K ≅ Z$ ; by [F7] the fibre is in bijection with the set of right cosets of $K$ , so the fibre is infinite. Fixing the isomorphism as the one induced by $ω$ , the positive generator of $Z$ corresponds to a deck transformation $t$ ; by [F6] the monodromy of $t$ raises the total winding of loops by $1$ . The group generated by $t$ is all of $Deck (\tilde{X} / X)$ , so $Deck (\tilde{X} / X) = {t^{k} : k \in Z}$ with $t^{- 1}$ the negative generator, and the left-action convention on the total space is the one recorded in Deck transformations and the deck-transformation group of a covering.

Depends on

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Sources

Vasudha Bharathram, Joan S. Birman and Tara E. Brendle, The Burau representation is faithful for n = 4, arXiv:2607.05283v1 (6 July 2026), Introduction and section 2 (printed pp. 1-5), and section 4 (Theorem 4.1) (standard reference, not scraped)
Joan S. Birman and Tara E. Brendle, Braids: A Survey (background on Burau matrices, the cyclic cover and absolute homology) (standard reference, not scraped)
Allen Hatcher, Algebraic Topology, section 1.3 (covering spaces) and section 2.2 (cellular homology and its agreement with singular homology) (standard reference, not scraped)