The one-variable Alexander module of an oriented link

Given: AC and an oriented link $L=K_1\cup \cdots \cup K_r\subset S^3$ with $r\ge 1$ components, complement $X_L$ and fundamental group $G={\pi }_1(X_L)$.

The Axiom of Choice: every family of nonempty sets has a choice function (The Axiom of Choice). It supplies Alexander duality in [F2], cellular cohomology comparison, and the complement supplier [F1], including that supplier’s countable-choice tubular-neighbourhood hypothesis.

$L$ is nonempty and compact, and $X_L$ is connected, locally path-connected and semilocally simply connected, so the classification of connected coverings applies to $X_L$ (The complement of an oriented link is a connected smooth three-manifold).

Alexander duality gives ${\tilde{H}}_1(X_L;\mathbb{Z})\cong {\tilde{H}}^1(L;\mathbb{Z})$, and $X_L$ is connected, so $H_1(X_L;\mathbb{Z})={\tilde{H}}_1(X_L;\mathbb{Z})$; the disjoint-union structure of $L$ gives ${\tilde{H}}^1(L;\mathbb{Z})\cong {⨁}_i{H}^1(K_i;\mathbb{Z})\cong {\mathbb{Z}}^r$, each $H^1(K_i;\mathbb{Z})$ being freely generated by the class dual to the orientation class of $K_i$: the one-vertex, one-edge CW structure of the oriented circle has zero cellular differential, so its degree-one cochain group and cohomology are $\mathbb{Z}$ (cellular comparison with constant coefficients Cellular cochains compute cohomology with local coefficients) (Alexander duality for compact locally contractible subsets of a sphere, The complement of an oriented link is a connected smooth three-manifold).

For a subgroup $H\le {\pi }_1(B,b_0)$ of the fundamental group of a nonempty, path-connected, locally path-connected and semilocally simply connected base, there is a connected covering classified by $H$, obtained as the quotient of a universal cover by $H$; a normal subgroup gives a regular covering, whose deck group is $G\mathrm{/}H$ (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, A regular connected covering has deck group ${\pi }_1(B,b_0)\mathrm{/}{p}_{\ast }{\pi }_1(E,e_0)$, Regular coverings, A connected covering is regular exactly when its induced subgroup is normal, exactly when deck transformations act transitively on a fibre, Deck transformations and the deck-transformation group of a covering).

The Hurewicz map ${\pi }_1(X_L)\to H_1(X_L;\mathbb{Z})$ is surjective with kernel the commutator subgroup (The first Hurewicz map is abelianization), so $\phi$ is equivalently a surjection $G\to \mathbb{Z}$; its kernel is normal in $G$.

$\mathrm{\Lambda }=\mathbb{Z}[t^{\pm 1}]$ is the localisation of $\mathbb{Z}[t]$ at the powers of $t$; it is commutative with unit and $t$ is a unit. A $\mathrm{\Lambda }$-module structure on an abelian group $M$ is the same as a unital ring homomorphism $\mathrm{\Lambda }\to \mathrm{End}(M)$, equivalently (by the universal property of the localisation) the datum of a group homomorphism $\mathbb{Z}\to \mathrm{Aut}(M)$ (The Laurent polynomial ring as the principal localisation of Z[t] at t, Universal property of localisation: maps that invert $S$ factor uniquely through $S^{-1}R$).

A deck transformation of a covering acts by a homeomorphism of the cover and hence by a group automorphism of every homology group; the deck group acts on the left (Deck transformations and the deck-transformation group of a covering).

In the orientation-normalized Alexander duality of [F2], a positive meridian of $K_i$ maps to the cohomology class evaluating $1$ on its oriented circle and $0$ on the others. Here a positive meridian is the oriented boundary of a small normal disk intersecting $K_i$ once positively and missing the other components. This local description follows from the supplier’s construction (Proof 4.1–6.1): excision restricts to that normal disk, the pair connector is its boundary map, and capping with the ambient orientation evaluates the single oriented transverse intersection as $+1$. A disk disjoint from another component gives zero there. Thus this is a local computation of the duality map, not a choice of an arbitrary isomorphism $H_1(X_L)\cong {\mathbb{Z}}^r$.