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The topological and matrix Burau representations agree

Statement

Assume AC (inherited from the topological definition of the reduced representation and the lift of braid mapping classes). Write $U = H_{1} (\tilde{X}, p^{- 1} d; Z)$ and $M_{r e d} = H_{1} (\tilde{X}; Z)$ over $Λ_{1} = Z [t^{\pm 1}]$ , with the relative lifted-edge basis $e_{1}, \dots, e_{n}$ of The unreduced Burau matrices and the reduced basis $g_{i} = t e_{i} - e_{i + 1}$ ( $1 \leq i \leq n - 1$ ) of $\ker σ = \ker \partial_{*}$ from The unreduced module fits an exact sequence with the reduced module and The invariant vector and the invariant covectors of the unreduced Burau. Then:

(1) the topological action of $B_{n}$ on $U$ in the basis $e_{1}, \dots, e_{n}$ is the matrix representation $ρ_{n}^{m a t}$ of The unreduced Burau matrices, i.e. $ρ_{n}^{m a t} (β)$ is the matrix of the lifted braid action for every $β \in B_{n}$ ;

(2) under the isomorphism $M_{r e d} ≅ \ker \partial_{*}$ induced by the inclusion of the pair, the reduced Burau representation ${\overset{ˉ}{ρ}}_{n}$ of The reduced Burau representation acts in the basis $(g_{1}, \dots, g_{n - 1})$ by the formulas $g_{j - 1} ⟼ g_{j - 1} + g_{j}, g_{j} ⟼ - t g_{j}, g_{j + 1} ⟼ t g_{j} + g_{j + 1},$ every other $g_{i}$ fixed; in particular, for $n = 3$ , in the basis $(g_{1}, g_{2})$ , ${\overset{ˉ}{ρ}}_{3} (σ_{1}) = (\begin{array}{cc} - t & t \\ 0 & 1 \end{array})$ , ${\overset{ˉ}{ρ}}_{3} (σ_{2}) = (\begin{array}{cc} 1 & 0 \\ 1 & - t \end{array})$ , and ${\overset{ˉ}{ρ}}_{3} (Δ^{2}) = t^{3} I_{2}$ for the full twist $Δ^{2}$ .

Caveat: the inclusion carries the fixed basis $b_{i}$ of The reduced Burau representation to $t^{i} (ϵ_{i} - ϵ_{i + 1}) = t e_{i} - e_{i + 1} = g_{i}$ , since $e_{i} = t^{i - 1} ϵ_{i}$ . Thus this identifies the two representations as matrices in the frozen bases; it does not claim that the integral extension $0 \to M_{r e d} \to U \overset{\partial_{*}}{\to} Λ_{1} \overset{ε}{\to} Z \to 0$ splits.

Facts & Assumptions

Given: AC; the unreduced module $U$ with its relative lifted-edge basis $e_{1}, \dots, e_{n}$ ; the reduced module $M_{r e d}$ with its fixed basis; the exact sequence $0 \to M_{r e d} \to U \overset{\partial_{*}}{\to} Λ_{1} \overset{ε}{\to} Z \to 0$ of The unreduced module fits an exact sequence with the reduced module; the candidate reduced basis $g_{i} = t e_{i} - e_{i + 1}$ .

[F1]

The topological action gives a homomorphism $B_{n} \to {Aut}_{Λ_{1}} (U)$ , $β \mapsto (\tilde{h})_{*}$ , where $\tilde{h}$ is the basepoint-normalised lift of a representative homeomorphism of $β$ ; isotopic representatives give the same automorphism (Braid mapping classes lift equivariantly to the Burau cover, Braid group as boundary-fixed punctured-disk mapping classes, The Artin presentation is complete for geometric braids).

[F2]

$ρ_{n}^{m a t} : B_{n} \to {GL}_{n} (Λ_{1})$ is the homomorphism with $ρ_{n}^{m a t} (σ_{i}) = B_{i}$ , and on the generator $σ_{i}$ the topological action on $U$ has matrix exactly $B_{i}$ in the basis $e_{1}, \dots, e_{n}$ (The unreduced Burau matrices satisfy the Artin relations, The geometric half twist acts on the lifted-edge basis by the Burau block).

[F3]

The inclusion of the pair induces an isomorphism $M_{r e d} \to \ker \partial_{*}$ , and $\partial_{*} = (t - 1) σ$ where $σ (x) = \sum_{i} t^{i - 1} x_{i}$ ; the maps are compatible with the braid actions in the sense that the inclusion $M_{r e d} \to U$ is natural for homeomorphisms of the pair, so the action on $M_{r e d}$ corresponds to the restriction of the action on $U$ to $\ker \partial_{*}$ (The unreduced module fits an exact sequence with the reduced module, Functoriality of relative homology, Relative singular homology).

[F4]

The matrices act on column vectors, and the matrix of a $Λ_{1}$ -linear map in a fixed basis has the images of the basis vectors as its columns; $Λ_{1}$ is a domain and $t \neq 1$ (Invertible square matrices and similarity over a commutative ring, Units, powers and the domain property of the Laurent polynomial ring, The Laurent polynomial ring as the principal localisation of Z[t] at t).

Proof

technique · direct

1.1F1F2F4

Clause (1). Both $β \mapsto (\tilde{h})_{*}$ (in the basis $e_{1}, \dots, e_{n}$ ) and $ρ_{n}^{m a t}$ are homomorphisms from the presented group $B_{n}$ to ${G L}_{n} (Λ_{1})$ by [F1], [F2]; they agree on each generator $σ_{i}$ , where the matrix of the topological action is $B_{i}$ by [F2]. Since $B_{n}$ is generated by the $σ_{i}$ , the two homomorphisms agree on all of $B_{n}$ , which is clause (1).

1.2F3algebra

The basis $(g_{1}, \dots, g_{n - 1})$ of $\ker \partial_{*}$ . By [F3], $\partial_{*} (x) = 0$ means $\sum_{i} t^{i - 1} x_{i} = 0$ for $x = \sum_{i} x_{i} e_{i}$ . Given such an $x$ , set $c_{k} : = - \sum_{i > k} t^{i - 1 - k} x_{i}$ for $1 \leq k \leq n - 1$ and $c_{0} : = c_{n} : = 0$ ; then $\sum_{k = 1}^{n - 1} c_{k} g_{k} = \sum_{j} (t c_{j} - c_{j - 1}) e_{j}$ with $t c_{j} - c_{j - 1} = x_{j}$ for every $j$ , the case $j = 1$ using $\sum_{i} t^{i - 1} x_{i} = 0$ and the case $j = n$ the definition $c_{n} = 0$ . Hence the $g_{i}$ span $\ker \partial_{*}$ . They are independent: if $\sum_{k = 1}^{n - 1} c_{k} g_{k} = 0$ , the coefficient of $e_{n}$ is $- c_{n - 1}$ , then the coefficient of $e_{n - 1}$ is $t c_{n - 1} - c_{n - 2} = - c_{n - 2}$ , and induction downwards gives $c_{k} = 0$ for all $k$ . So $(g_{1}, \dots, g_{n - 1})$ is a $Λ_{1}$ -basis of $\ker \partial_{*}$ .

2.1F2F3F4step 1.2algebra

Clause (2): the action on the reduced basis. For the generator $σ_{j}$ , the unreduced action of clause (1) is $e_{j} \mapsto (1 - t) e_{j} + e_{j + 1}$ , $e_{j + 1} \mapsto t e_{j}$ and $e_{k} \mapsto e_{k}$ otherwise [F2]. Substituting, $g_{j - 1} = t e_{j - 1} - e_{j} \mapsto t e_{j - 1} - (1 - t) e_{j} - e_{j + 1} = (t e_{j - 1} - e_{j}) + (t e_{j} - e_{j + 1}) = g_{j - 1} + g_{j}$ ; $g_{j} = t e_{j} - e_{j + 1} \mapsto t (1 - t) e_{j} + t e_{j + 1} - t e_{j} = - t (t e_{j} - e_{j + 1}) = - t g_{j}$ ; $g_{j + 1} = t e_{j + 1} - e_{j + 2} \mapsto t^{2} e_{j} - e_{j + 2} = (t^{2} e_{j} - t e_{j + 1}) + (t e_{j + 1} - e_{j + 2}) = t g_{j} + g_{j + 1}$ ; and every other $g_{i}$ involves only basis vectors outside ${e_{j}, e_{j + 1}}$ and is fixed. Boundary conventions: for $j = 1$ there is no $g_{0}$ , for $j = n - 1$ there is no $g_{n}$ . By [F3] the action on $M_{r e d}$ corresponds to this action on $\ker \partial_{*}$ , so the matrix of ${\overset{ˉ}{ρ}}_{n} (σ_{j})$ in the basis $(g_{i})$ is as displayed.

3.1step 2.1algebra

The case $n = 3$ . In the basis $(g_{1}, g_{2})$ the formulas of step 2.1 for $j = 1$ give $g_{1} \mapsto - t g_{1}$ , $g_{2} \mapsto t g_{1} + g_{2}$ , i.e. ${\overset{ˉ}{ρ}}_{3} (σ_{1}) = (\begin{array}{cc} - t & t \\ 0 & 1 \end{array})$ , and for $j = 2$ give $g_{1} \mapsto g_{1} + g_{2}$ , $g_{2} \mapsto - t g_{2}$ , i.e. ${\overset{ˉ}{ρ}}_{3} (σ_{2}) = (\begin{array}{cc} 1 & 0 \\ 1 & - t \end{array})$ . Multiplying, ${\overset{ˉ}{ρ}}_{3} (σ_{1}) {\overset{ˉ}{ρ}}_{3} (σ_{2}) {\overset{ˉ}{ρ}}_{3} (σ_{1}) = (\begin{array}{cc} 0 & - t^{2} \\ - t & 0 \end{array})$ , whose square is $t^{3} I_{2}$ ; since $Δ^{2} = (σ_{1} σ_{2} σ_{1})^{2}$ in $B_{3}$ , this is ${\overset{ˉ}{ρ}}_{3} (Δ^{2}) = t^{3} I_{2}$ .

4.1step 1.1step 2.1step 3.1∎

Conclusion. Clause (1) is step 1.1 and clause (2) is steps 2.1 and 3.1; the identification is in the frozen bases and no $Λ_{1}$ -linear splitting of the exact sequence of [F3] is constructed or claimed.

Depends on

Used by

Cited to discharge well-definedness by The unreduced Burau matrices.

Dependency tree · two levels

103 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

Joan S. Birman and Tara E. Brendle, Braids: A Survey (background on Burau matrices, the cyclic cover and absolute homology) (standard reference, not scraped)
Vasudha Bharathram, Joan S. Birman and Tara E. Brendle, The Burau representation is faithful for n = 4, arXiv:2607.05283v1 (6 July 2026), section 2 (printed pp. 1-5) (standard reference, not scraped)
Stephen J. Bigelow, The Burau representation is not faithful for n = 5, Geometry & Topology 3 (1999) 397-404, Definition 1.1 (printed pp. 397-398) (standard reference, not scraped)