The relative pairing modules as stabilized direct limits

Let $D$ be the closed unit disk, $P=\{p_1,\dots ,p_n\}\subset \mathrm{int}D$, let $C$ be the two-point configuration space of the punctured disk with the distance function $f(\{x,y\})=\min (\mathrm{\mid }x-y\mathrm{\mid },\mathrm{dist}(x,P),\mathrm{dist}(y,P)),$ and let $\tilde{C}\to C$ be the LKB cover of The Lawrence-Krammer-Bigelow cover.

For $\epsilon >0$ put ${\nu }_{\epsilon }=\{\{x,y\}\in C:f(\{x,y\})<\epsilon \}=\{\{x,y\}:\mathrm{\mid }x-y\mathrm{\mid }<\epsilon \text{ or }\mathrm{dist}(x,P)<\epsilon \text{ or }\mathrm{dist}(y,P)<\epsilon \},$ and let ${\tilde{\nu }}_{\epsilon }$ be the preimage of ${\nu }_{\epsilon }$ in $\tilde{C}$. Thus ${\nu }_{\delta }\subseteq {\nu }_{\epsilon }$ and ${\tilde{\nu }}_{\delta }\subseteq {\tilde{\nu }}_{\epsilon }$ whenever $\delta <\epsilon$, and the identity inclusions of pairs $(C,{\nu }_{\delta })\to (C,{\nu }_{\epsilon }),(C,\mathrm{\partial }C\cup {\nu }_{\delta })\to (C,\mathrm{\partial }C\cup {\nu }_{\epsilon })$ induce the natural relative-homology maps from small radii to large radii.

The stabilized limits. By Equivariant stabilization of the LKB end neighbourhoods there is ${\epsilon }_0>0$ such that for all $0<\delta <\epsilon <{\epsilon }_0$ these inclusions are homotopy equivalences of pairs, with explicit inverse homotopy equivalences of pairs, and so induce isomorphisms in relative homology in every degree. The relative pairing modules are the direct limits $H_2(\tilde{C},\tilde{\nu }):={\lim }_{\epsilon \to 0}{H}_2(\tilde{C},{\tilde{\nu }}_{\epsilon }),H_2(\tilde{C},\mathrm{\partial }\tilde{C}\cup \tilde{\nu }):={\lim }_{\epsilon \to 0}{H}_2(\tilde{C},\mathrm{\partial }\tilde{C}\cup {\tilde{\nu }}_{\epsilon }),$ taken in the category of abelian groups along the inverse transition maps $H_2(\tilde{C},{\tilde{\nu }}_{\epsilon })⟶H_2(\tilde{C},{\tilde{\nu }}_{\delta }),H_2(\tilde{C},\mathrm{\partial }\tilde{C}\cup {\tilde{\nu }}_{\epsilon })⟶H_2(\tilde{C},\mathrm{\partial }\tilde{C}\cup {\tilde{\nu }}_{\delta })(\delta <\epsilon <{\epsilon }_0),$ which are the unique inverses of the natural maps. The stabilization lemma supplies these inverses explicitly and proves that they compose compatibly for $\gamma <\delta <\epsilon$; consequently the inverse system is constant up to canonical isomorphism on $(0,{\epsilon }_0)$ and each of the two direct limits is canonically isomorphic, as an abelian group, to every group $H_2(\tilde{C},{\tilde{\nu }}_{\epsilon })$ with $0<\epsilon <{\epsilon }_0$ (respectively to every boundary-union group). This is the stabilized direct limit convention: the natural relative maps themselves run from small to large radii, and the transition maps toward zero are their canonical inverses, not the natural maps.

The $\mathrm{\Lambda }$-module structure. A deck transformation of $\tilde{C}$ commutes with the projection, hence carries ${\tilde{\nu }}_{\epsilon }$ onto ${\tilde{\nu }}_{\epsilon }$ and preserves both pairs; the induced automorphisms commute with the inclusions and with the inverse transition maps, so they induce automorphisms of both direct limits. Extending multiplicatively gives both limits the structure of $\mathrm{\Lambda }$-modules, where $\mathrm{\Lambda }=\mathbb{Z}[q^{\pm 1},t^{\pm 1}]$ acts through the deck translations $q$ and $t$. These two $\mathrm{\Lambda }$-modules are the targets of the LKB pairing defined below; no element of either limit is claimed to be an absolute class.