The Lawrence-Krammer-Bigelow cover

Let $C$ be the two-point configuration space of the punctured disk with basepoint $c_0$, and let $\mathrm{\Phi }:{\pi }_1(C,c_0)\to {\mathbb{Z}}^2=⟨q⟩\oplus ⟨t⟩$ be the two-variable covering homomorphism of The two-variable covering homomorphism. Let $\tilde{C}⟶C$ be the connected covering space of Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings and Connected covering spaces are classified by conjugacy classes of fundamental-group subgroups classified by the subgroup $\ker \mathrm{\Phi }\le {\pi }_1(C,c_0)$, and let ${\tilde{c}}_0\in \tilde{C}$ be a fixed point of the fibre over $c_0$. The space $\tilde{C}$ is the Lawrence-Krammer-Bigelow cover (the LKB cover).

The hypotheses of Connected covering spaces are classified by conjugacy classes of fundamental-group subgroups hold: $C$ is nonempty and path-connected, and a configuration has a neighborhood homeomorphic to the product of two disjoint small convex disk or half-disk neighborhoods missing $P$. These neighborhoods are contractible, so $C$ is locally path-connected and semilocally simply connected. Thus the specified based connected cover exists and is unique up to based isomorphism.

Regularity and deck group. Since $\ker \mathrm{\Phi }$ is a normal subgroup of ${\pi }_1(C,c_0)$, the cover is regular (Galois): the deck group $\mathrm{Deck}(\tilde{C}\mathrm{/}C)$ is isomorphic to ${\pi }_1(C,c_0)\mathrm{/}\ker \mathrm{\Phi }\cong \mathrm{im}\mathrm{\Phi }={\mathbb{Z}}^2$, so it is free abelian of rank two. Write $q$ and $t$ also for the two deck transformations corresponding to the generators; every deck transformation maps ${\tilde{c}}_0$ to a point of the fibre over $c_0$ and acts on $\tilde{C}$ by homeomorphisms commuting with the projection.

The coefficient ring and the module. Put $\mathrm{\Lambda }:=\mathbb{Z}[q^{\pm 1},t^{\pm 1}],$ the Laurent polynomial ring in two commuting variables. The absolute singular homology $H_2(\tilde{C};\mathbb{Z})$ carries a $\mathrm{\Lambda }$-module structure: define $q\cdot x=q_{\ast }x$ and $t\cdot x=t_{\ast }x$ for the induced automorphisms of $H_2(\tilde{C};\mathbb{Z})$ and extend $\mathbb{Z}$-linearly and multiplicatively; the deck transformations commute, so this is well defined and $\mathrm{\Lambda }$ acts through a ring homomorphism $\mathrm{\Lambda }\to {\mathrm{End}}_{\mathbb{Z}}(H_2(\tilde{C};\mathbb{Z}))$. All homology groups below are ordinary absolute singular homology unless another coefficient module is displayed.

The conventions fixed here are: $C$ is unordered, $c_0$ and ${\tilde{c}}_0$ are the basepoints, deck translations act on the left, and $H_2(\tilde{C})$ is always the integral absolute second homology of the covering space, with the $\mathrm{\Lambda }$-module structure just defined.