The two-variable covering homomorphism

Let $C$ be the two-point configuration space of the punctured disk, with basepoint $c_0=\{d_1,d_2\}$ and boundary-fixed action, of The two-point configuration space of a punctured disk. Let ${\mathbb{Z}}^2=⟨q⟩\oplus ⟨t⟩$ denote the free abelian group written multiplicatively with basis the two letters $q,t$.

The winding form. Let $\alpha$ be a loop in $C$ at $c_0$ (Based loops and the fundamental group). It can be written $\alpha (s)=\{{\alpha }_1(s),{\alpha }_2(s)\}$ for continuous arcs ${\alpha }_1,{\alpha }_2$ in $D\setminus P$. The symmetric nonvanishing functions $W(\{x,y\})={\prod }_{j=1}^n(x-p_j)(y-p_j),V(\{x,y\})=(x-y{)}^2$ define closed loops $W\circ \alpha ,V\circ \alpha$ in ${\mathbb{C}}^{\ast }$ even when the mobile labels exchange. Define $a(\alpha )$ and $b(\alpha )$ as their integer winding numbers, using continuous argument lifts on the compact parameter interval. For piecewise smooth tracks this is equivalently $a(\alpha )=\frac{1}{2\pi i}{\sum }_{j=1}^n({\int }_{{\alpha }_1}\frac{dz}{z-p_j}+{\int }_{{\alpha }_2}\frac{dz}{z-p_j}),b(\alpha )=\frac{1}{\pi i}{\int }_{{\alpha }_1-{\alpha }_2}\frac{dz}{z}\mathrm{.}$ The winding definition applies to arbitrary continuous loops; no differentiability of $\alpha$ is presumed. The total puncture winding is $a$, and $b$ is the mutual half-twist exponent, even for returning labels and odd for exchanged labels. A loop with one mobile point going positively around a single puncture in a small disk missing the other mobile point has $(a,b)=(1,0)$. Exchanging the two mobile points by a positive half rotation in a small disk missing all punctures gives $(a,b)=(0,1)$. Join these local configurations to $c_0$ by the finite-buffer paths of the configuration-space definition; conjugating the local loops does not alter winding numbers. Hence $\mathrm{\Phi }$ is surjective, with images $q$ and $t$, including when $n=1$.

The exponent-sum form. Ignoring the punctures turns $\alpha$ into a loop in the space of unordered pairs of points of the disk, hence into a braid in $B_2$ (The braid group by Artin presentation); let $b$ be its exponent of ${\sigma }_1$. Adjoining the $n$ fixed punctures turns $\alpha$ into a loop of unordered $(n+2)$-tuples in the disk; that loop is a braid in $B_{n+2}$, and the exponent sum of that braid in the Artin generators ${\sigma }_1,\dots ,{\sigma }_{n+1}$ is written $b^{\mathrm{\prime }}$. Then $b^{\mathrm{\prime }}\equiv b(\mathrm{m}\mathrm{o}\mathrm{d}2)$ and $a=\frac{1}{2}(b^{\mathrm{\prime }}-b)\in \mathbb{Z}\mathrm{.}$ Equivalently, the parity statement is the relation $b^{\mathrm{\prime }}=2a+b$: each mobile–puncture difference occurs squared in the full discriminant, while the fixed–fixed factors are constant and the mutual half-twist of the two mobile points contributes exactly the parity of $b$.

The homomorphism. The two-variable covering homomorphism is $\mathrm{\Phi }:{\pi }_1(C,c_0)⟶{\mathbb{Z}}^2,\mathrm{\Phi }(\alpha )=q^a{t}^b,a=\frac{1}{2}(b^{\mathrm{\prime }}(\alpha )-b(\alpha ))\mathrm{.}$ It is well defined and a homomorphism of groups: path-homotopy classes have well-defined winding numbers and well-defined exponent sums, because the defining relations of the Artin presentation preserve the total exponent sum, and both $a$ and $b$ are additive under concatenation of loops, which is the product of Based loops and the fundamental group. Since the integers $a$ and $b$ are determined by the class $[\alpha ]$, the formula defines a map; additivity of winding numbers and of exponent sums gives $\mathrm{\Phi }([\alpha ][\beta ])=\mathrm{\Phi }([\alpha ])\mathrm{\Phi }([\beta ])$. The case $n=1$ is included, with $P=\{p_1\}$.

The cover classified by $\ker \mathrm{\Phi }$ and the resulting LKB module are built from this homomorphism on the same page; the variables $q$ and $t$ are the deck generators used there.