The Z^2 cover of the projectivized tangent bundle and bigraded curves

The braid action is used only through the mapping class group. Assume the Axiom of Choice (The Axiom of Choice), used here only to pass between braid classes and boundary-fixed mapping classes of the punctured disk, so that the braid group acts on bigraded curves through $G={\pi }_0\mathrm{Diff}(D,\mathrm{\partial }D;\mathrm{\Delta })$ via Artin presentation completeness The Artin presentation is complete for geometric braids and the smooth configuration-space comparison of Khovanov–Seidel, Section 3b, equation (3.1), printed p. 19. The corresponding topological comparison uses Braid group as boundary-fixed punctured-disk mapping classes and AC. Everything else in this definition — the cover, its deck group and the preferred lifts — is produced by the covering-space classification and the lifting criterion and is choice-free.

Let $(D,\mathrm{\Delta })$ be the marked disk of Curves and geometric intersection numbers on the marked disk, and let $D\setminus \mathrm{\Delta }\subseteq \mathbb{C}$ carry its subspace topology. Write $P:=\mathbf{P}(T(D\setminus \mathrm{\Delta }))$ for the real projectivization of the tangent bundle, the space of tangent lines $T_{z}c$ of curves at unmarked points; the embedding $D\subset \mathbb{C}$ trivializes $TD$, so $P$ is identified with $({\mathbb{C}}^{\ast }\mathrm{/}{\mathbb{R}}^{\times })\times (D\setminus \mathrm{\Delta })$ and is a smooth manifold of real dimension $3$ with boundary. Fix once and for all a polynomial $h\in \mathbb{C}[z]$ with simple zeros exactly at the points of $\mathrm{\Delta }$ (for instance $h(z)={\prod }_{a\in \mathrm{\Delta }}(z-a)$), and define ${\delta }_P\text{\hspace{0.17em}⁣}:P⟶({\mathbb{C}}^{\ast }\mathrm{/}{\mathbb{R}}_{>0})\times ({\mathbb{C}}^{\ast }\mathrm{/}{\mathbb{R}}_{>0}),{\delta }_P(\zeta ,z):=(h(z{)}^{-2}{\zeta }^2,-h(z))\mathrm{.}$ This is well defined: $\zeta$ is a class modulo real scalars, so ${\zeta }^2$ is a class modulo positive real scalars and $h(z{)}^{-2}{\zeta }^2$ likewise, and both coordinates are nonzero because $\zeta \mathrm{\ne }0$ and $h(z)\mathrm{\ne }0$ on $D\setminus \mathrm{\Delta }$.

The cover. Let $\exp \text{\hspace{0.17em}⁣}:{\mathbb{R}}^2⟶({\mathbb{C}}^{\ast }\mathrm{/}{\mathbb{R}}_{>0}{)}^2,\exp ({\xi }_1,{\xi }_2):=(e^{2\pi i{\xi }_1},e^{2\pi i{\xi }_2}),$ the universal covering of the two-torus $({\mathbb{C}}^{\ast }\mathrm{/}{\mathbb{R}}_{>0}{)}^2$; it is a regular covering with deck group ${\mathbb{Z}}^2$ acting by translation (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings, Deck transformations and the deck-transformation group of a covering, Connected covering spaces are classified by conjugacy classes of fundamental-group subgroups). Define $\tilde{P}:=\{(x,p)\in {\mathbb{R}}^2\times P:\exp (x)={\delta }_P(p)\}$ with the subspace topology and the projection $\tilde{\pi }(x,p):=p$. Then $\tilde{\pi }:\tilde{P}\to P$ is a covering map with deck group ${\mathbb{Z}}^2$ acting by $\chi (r_1,r_2)(x,p):=(x+(r_1,r_2),p),$ the pullback of the universal covering along ${\delta }_P$. This is the source's ${\mathbb{Z}}^2$-cover of $P$.

Bigradings and bigraded curves. For a curve $c$ (unoriented, as in Curves and geometric intersection numbers on the marked disk) the canonical section is $s_c\text{\hspace{0.17em}⁣}:c\setminus \mathrm{\Delta }⟶P,s_c(z):=T_{z}c\mathrm{.}$ It is well defined because a curve meets $\mathrm{\Delta }$ in at most its two endpoints, so $T_{z}c$ exists for every $z\in c\setminus \mathrm{\Delta }$, and it is continuous. A bigrading of $c$ is a continuous lift $\tilde{c}\text{\hspace{0.17em}⁣}:c\setminus \mathrm{\Delta }⟶\tilde{P},\tilde{\pi }\circ \tilde{c}=s_c,$ of the canonical section. Pairs $(c,\tilde{c})$ consisting of a curve and a bigrading are bigraded curves; we often write $\tilde{c}$ in place of $(c,\tilde{c})$. A curve need not admit a bigrading — the obstruction for simple closed curves is computed by the source — and precisely the arcs admit bigradings. An arc with its marked endpoints removed is contractible, so its tangent section lifts. A simple closed curve enclosing $k\ge 1$ marked points has tangent-section monodromy $\pm (2-2k,k)\mathrm{\ne }0$: the tangent makes one full turn and $h$ winds $k$ times, so the two coordinates of ${\delta }_P$ wind $2-2k$ and $k$. Hence its tangent section cannot lift (Khovanov--Seidel, Lemma 3.12, printed p. 24). The bigradings of any fixed arc form a ${\mathbb{Z}}^2$-torsor by uniqueness of path lifting, and are acted on by the deck group $\chi$: $\chi (r_1,r_2)\tilde{c}:=\chi (r_1,r_2)\circ \tilde{c}\mathrm{.}$

The diffeomorphism and mapping-class actions. Let $\mathcal{D}:=\mathrm{Diff}(D,\mathrm{\partial }D;\mathrm{\Delta })$ be the actual orientation-preserving diffeomorphism group; its component group is the mapping class group $G$ used in the marked-disk Definition. For $f\in \mathcal{D}$, the derivative induces $P\to P$, $[v]\mapsto [Df(v)]$. It preserves the monodromy homomorphism: a fibre loop maps to a fibre loop of degree one, and a positively oriented puncture loop maps to the corresponding loop about the permuted puncture. No extra fibre winding occurs because $Df:D\to {\mathrm{G}\mathrm{L}}^{+}(2,\mathbb{R})$ is defined on the whole disk, so its restriction to any loop in $D$ is null-homotopic. The monodromy images of a fibre generator and a puncture generator are $(1,0)$ and $(-2,1)$; they generate ${\mathbb{Z}}^2$, so the pullback cover is connected and its deck group is exactly ${\mathbb{Z}}^2$.

There is a unique deck-equivariant preferred lift $\tilde{f}$ fixing every point of the fibre over one chosen boundary tangent line $T_z\mathrm{\partial }D$. This base tangent line is fixed by the derivative, so the lifting criterion gives the based lift (Lifting criterion for maps from path-connected locally path-connected spaces), and monodromy preservation makes it deck-equivariant. Along the connected boundary tangent section the derivative is the identity; lifting its paths shows that $\tilde{f}$ fixes every fibre over every $T_w\mathrm{\partial }D$. Uniqueness of based lifts (Two lifts from a connected space that agree at one point agree everywhere) gives $\widetilde{fg}=\tilde{f}\tilde{g}$. The action on bigraded curves is parametrization-aware: the bigrading of $f(c)$ at $f(z)$ is $\tilde{f}(\tilde{c}(z))$. An isotopy of actual diffeomorphisms lifts from the identity by homotopy lifting (Existence and uniqueness of homotopy lifts through a covering map) and hence carries bigraded curves through bigraded isotopies; therefore the action on bigraded isotopy classes descends to $G={\pi }_0(\mathcal{D})$. Isotopy. A bigraded isotopy between bigraded curves $({\tilde{c}}_0,{\tilde{c}}_1)$ is an isotopy $c_t$ of curves, together with a continuous family of lifts ${\tilde{c}}_t$ through bigraded curves; the deck action and the $\mathcal{D}$-action take bigraded isotopy classes to bigraded isotopy classes. Two bigraded curves are isotopic when such a family exists, and isotopy relates only bigradings of curves in the same curve-isotopy class.