The unreduced Burau relative homology module

Given: The Burau cover $p:\tilde{X}\to X$ with deck group $\{T_{t^k}:k\in \mathbb{Z}\}$, the discrete fibre $p^{-1}d$, and the abelian group $U=H_1(\tilde{X},p^{-1}d;\mathbb{Z})$.

Every deck transformation is a homeomorphism of $\tilde{X}$ over $X$; it maps the fibre $p^{-1}d$ to itself, hence is a homeomorphism of the pair $(\tilde{X},p^{-1}d)$; the deck group is a group under composition with $T_{t^k}\circ T_{t^l}=T_{t^{k+l}}$ and $T_{t^k}^{-1}=T_{t^{-k}}$ (Deck transformations and the deck-transformation group of a covering, The Burau infinite cyclic cover).

A continuous map of pairs $f:(X,A)\to (Y,B)$ induces $f_{\ast }:H_n(X,A;G)\to H_n(Y,B;G)$ with identity and composite laws (Relative singular homology, Functoriality of relative homology).

The cover is regular, so its deck group acts transitively on the fibre, and deck transformations act freely on the connected total space; hence $p^{-1}d=\{T_{t^k}\tilde{d}:k\in \mathbb{Z}\}$ is a $\mathbb{Z}$-torsor. A fibre of a covering is discrete (Regular coverings, On a connected covering space, a deck transformation is determined by one point and the deck action is free, For a nonempty path-connected total space, a covering fibre is in bijection with the right cosets of the induced fundamental-group subgroup).

${\mathrm{\Lambda }}_1$ has the universal property that each unit $u$ of a unital ring $A$ determines a unique unital ring homomorphism ${\mathrm{\Lambda }}_1\to A$ with $t\mapsto u$ (The Laurent polynomial ring as the principal localisation of Z[t] at t).

There is a long exact sequence $\cdots \to H_1(p^{-1}d)\to H_1(\tilde{X})\to H_1(\tilde{X},p^{-1}d)\stackrel{\mathrm{\partial }}{\to }{H}_0(p^{-1}d)\to H_0(\tilde{X})\to \cdots$ (Long exact sequence of a pair).

The relative lifted-edge basis $({ϵ}_1,\dots ,{ϵ}_n)$ of $H_1(\mathrm{\Sigma },{\mathrm{\Sigma }}^0)$ and the ${\mathrm{\Lambda }}_1$-module isomorphisms $H_1(\tilde{X},p^{-1}d)\cong H_1(\mathrm{\Sigma },{\mathrm{\Sigma }}^0)$ and $H_1(\tilde{X})\cong H_1(\mathrm{\Sigma })$ are fixed in The cyclic cover retracts onto the lifted flower and has a deck-equivariant spine model.