The maximal-abelian-cover Alexander module of an oriented link

Assume the Axiom of Choice. Let $L=K_1\cup \cdots \cup K_r\subset S^3$ be a nonempty oriented smooth link with $r\ge 1$ components, and let $X_L=S^3\setminus L$ be its open complement. For each component choose a smooth fibrewise norm on its normal plane bundle $\nu (K_i)$ and a tubular map ${\mathrm{\Phi }}_i$ from an open neighborhood of the zero section, fixing the zero section, together with a radius ${ϵ}_i>0$ whose closed disk subbundle lies in that neighborhood. Require the resulting closed tubes $N(K_i)={\mathrm{\Phi }}_i(D_{{ϵ}_i}\nu (K_i))$ to be pairwise disjoint, and put $N(L)={\bigcup }_{i=1}^{r}N(K_i)$ and $E_L=S^3\setminus {\mathrm{int}}_{S^3}N(L)$, the resulting compact smooth $3$-manifold with boundary. The tube choices are part of the input. Choose a basepoint $b\in E_L\subset X_L$. Orient each normal disk so that the positive tangent to $K_i$ followed by its positive normal basis agrees with the fixed orientation of $S^3$. A positive meridian ${\mu }_i$ of $K_i$ is the oriented boundary of the radius-${ϵ}_i$ normal disk in one fibre of $\nu (K_i)$; the curve lies on $\mathrm{\partial }{E}_L$ and is disjoint from every component of $L$; its spanning normal disk meets $K_i$ transversely once with positive sign and misses every other component. Alexander duality identifies $H_1(X_L;\mathbb{Z})$ with ${\mathbb{Z}}^r$ and identifies the classes of ${\mu }_1,\dots ,{\mu }_r$ with its standard basis. Write $G={\pi }_1(X_L,b)$ and let $\varphi :G\to H_1(X_L;\mathbb{Z})$ be the first Hurewicz homomorphism, equivalently abelianization. The subgroup $\ker \varphi$ is normal. The connected covering $p:X_L^{\mathrm{a}\mathrm{b}}\to X_L$ classified by $\ker \varphi$ is the maximal abelian cover; it is regular and its deck group is identified with $G\mathrm{/}\ker \varphi \cong H_1(X_L;\mathbb{Z})$. Put ${\mathrm{\Lambda }}_r=\mathbb{Z}[H_1(X_L;\mathbb{Z})]\cong \mathbb{Z}[t_1^{\pm 1},\dots ,t_r^{\pm 1}]$, where $t_i$ is the deck transformation corresponding to $[{\mu }_i]$. The isomorphism sends each variable to its group-ring basis element; monomials $t_1^{n_1}\cdots t_r^{n_r}$ correspond bijectively to the basis elements indexed by $n_1[{\mu }_1]+\cdots +n_r[{\mu }_r]$. Define the absolute multivariable Alexander module by

$A_L=H_1(X_L^{\mathrm{a}\mathrm{b}};\mathbb{Z}),$

as a left ${\mathrm{\Lambda }}_r$-module under the deck action. “Absolute” means this is ordinary homology of the covering space, not relative homology with respect to a fibre over the basepoint.

The one-variable Alexander module of an oriented link

Oriented links in the three-sphere and ambient isotopy

Normal and conormal bundles of an embedded submanifold

Tubular neighbourhoods of embedded submanifolds

Assuming countable choice, normal and conormal bundles are smooth vector bundles

The complement of an oriented link is a connected smooth three-manifold

Alexander duality for compact locally contractible subsets of a sphere

The first Hurewicz map is abelianization

Connected covering spaces are classified by conjugacy classes of fundamental-group subgroups

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 ${\pi }_1(B,b_0)\mathrm{/}{p}_{\ast }{\pi }_1(E,e_0)$

Deck transformations and the deck-transformation group of a covering

The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements

The group ring $R[G]$ is a unital $R$-algebra with basis $G$, and each $g\in G$ is a unit of $R[G]$

Knot exterior

Meridian and preferred longitude

The Axiom of Choice