2.1 The Atiyah–Kodaira construction, globally

The bundles under study in this paper are a refinement of a construction first investigated by Kodaira [Reference KodairaKod67]. Shortly thereafter, Atiyah independently developed the same construction [Reference AtiyahAti69] and so this class of examples is known as the ‘Atiyah–Kodaira construction’. Our treatment in this paragraph follows the presentation in [Reference MoritaMor01, § 4.3].

Fix a positive integer $m>1$ . Let $X$ be a compact Riemann surface of genus $g_{0}\geqslant 2$ . Let $p:Y\rightarrow X$ be a cyclic unbranched covering with deck group $\langle \unicode[STIX]{x1D70E}\rangle \cong \mathbb{Z}/m\mathbb{Z}$ . For $0\leqslant i\leqslant m-1$ , define the locus

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{i}:=\{(y,\unicode[STIX]{x1D70E}^{i}y)\mid y\in Y\}\subset Y\times Y.\end{eqnarray}$$

Let $p^{\prime }:B\rightarrow Y$ be the unbranched regular covering corresponding to the homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}(Y){\twoheadrightarrow}H_{1}(Y;\mathbb{Z}/m\mathbb{Z}).\end{eqnarray}$$

Define $\widetilde{\unicode[STIX]{x1D6E4}}\subset B\times Y$ as the preimage of $\bigcup _{i=0}^{m-1}\unicode[STIX]{x1D6E4}_{i}$ under $p^{\prime }\times \operatorname{id}:B\times Y\rightarrow Y\times Y$ . Since $p:Y\rightarrow X$ is a regular unbranched covering, $\widetilde{\unicode[STIX]{x1D6E4}}$ intersects each fiber $\{b\}\times Y$ in exactly $m$ distinct points. An analysis involving the Künneth formulas for $B\times Y$ and $Y\times Y$ (see [Reference MoritaMor01, § 4.3] for details) shows that

(4) $$\begin{eqnarray}[\widetilde{\unicode[STIX]{x1D6E4}}]=0\quad \text{in }H_{2}(B\times Y;\mathbb{Z}/m\mathbb{Z}).\end{eqnarray}$$

Viewing $\widetilde{\unicode[STIX]{x1D6E4}}$ as a divisor on $B\times Y$ , (4) implies that there is a line bundle $L\in \operatorname{Pic}(B\times Y)$ such that $mL=[\widetilde{\unicode[STIX]{x1D6E4}}]$ . Let $\unicode[STIX]{x1D70B}:E(L)\rightarrow B\times Y$ denote the projection from the total space of $L$ . There is a map $f$ of line bundles

Here $s$ is the section with divisor $\widetilde{\unicode[STIX]{x1D6E4}}$ and the restriction of $f$ to each fiber has model $z\mapsto z^{m}$ . The Atiyah–Kodaira construction is the algebraic surface

$$\begin{eqnarray}E^{\text{nn}}(X,m):=(f^{-1}\,\circ \,s)(B\times Y);\end{eqnarray}$$

the superscript $\text{nn}$ , as before, stands for ‘non-normal’ and will be explained in the following paragraph. By construction, this is an $m$ -fold cyclic branched covering of $B\times Y$ branched along  $\widetilde{\unicode[STIX]{x1D6E4}}$ . As remarked above, $\widetilde{\unicode[STIX]{x1D6E4}}$ intersects each fiber $\{y_{0}\}\times Y$ in exactly $m$ distinct points. Restricted to some such fiber, the branched covering $E^{\text{nn}}(X,m)\rightarrow B\times Y$ restricts to an $m$ -fold cyclic branched covering $q:Z\rightarrow Y$ branched at $m$ points. Denote the covering group by $\langle \unicode[STIX]{x1D701}\rangle \cong \mathbb{Z}/m\mathbb{Z}$ . The projection $\unicode[STIX]{x1D70B}:E^{\text{nn}}(X,m)\rightarrow B$ endows $E^{\text{nn}}(X,m)$ with the structure of a Riemann surface bundle over $B$ with fibers diffeomorphic to $Z$ .

2.2 Repairing normality

In the remainder of the section, we will undertake a study of the Atiyah–Kodaira construction within the setting of the theory of surface bundles. Our goal is to give a ‘fiberwise’ description of the construction outlined above. This will highlight the role played by branched coverings of Riemann surfaces. We will see that the covering used in the original construction is not normal and we will pass to a further cover in order to repair this. The main result of the subsection is Lemma 2.2.

By construction, $q:Z\rightarrow Y$ is an $m$ -fold cyclic branched covering and $p:Y\rightarrow X$ is an $m$ -fold cyclic unbranched covering. Let $Z^{\circ }$ denote the subsurface of $Z$ on which $q$ restricts to an unbranched covering and define $Y^{\circ }=q(Z^{\circ })$ . By construction, $Z^{\circ }$ is $Z$ with $m$ points removed and $Y^{\circ }$ is similarly $Y$ with $m$ points removed. Moreover, the $m$ removed points in $Y^{\circ }$ correspond to the $m$ points of intersection of $Y$ with the divisor $\widetilde{\unicode[STIX]{x1D6E4}}$ and, by construction, this is the set $\{(y,\unicode[STIX]{x1D70E}^{i}y)\}$ for some fixed $y\in Y$ and $1\leqslant i\leqslant m$ . It follows that $p:Y\rightarrow X$ restricts to an unbranched covering $p:Y^{\circ }\rightarrow X^{\circ }$ . By the above discussion, $X^{\circ }$ is $X$ with the single point $x=p(y)=p(\unicode[STIX]{x1D70E}^{i}y)$ removed.

The coverings $q:Z^{\circ }\rightarrow Y^{\circ }$ and $p:Y^{\circ }\rightarrow X^{\circ }$ are regular by construction. However, we will see below that the composite $p\,\circ \,q:Z^{\circ }\rightarrow X^{\circ }$ is not regular. This presents serious difficulties for the study of the monodromy of the bundle $E^{\text{nn}}(X,m)\rightarrow B$ . To repair this, we will pass to a further (unbranched) cover $r:W\rightarrow Z$ such that the composite $p\,\circ \,q\,\circ \,r:W\rightarrow X$ becomes a regular (albeit non-abelian) cover.

To describe $W$ , it is helpful to make a more explicit study of $X,Y,Z$ . Let $X$ be a surface of genus $g_{0}$ , represented as a $2g_{0}$ -gon $\unicode[STIX]{x1D6E5}$ with edges $\{e_{1},f_{1},\ldots ,e_{g_{0}},f_{g_{0}}\}$ identified so that the word around $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E5}$ (traversed counterclockwise) reads $e_{1}f_{1}e_{1}^{-1}f_{1}^{-1}\ldots e_{g_{0}}f_{g_{0}}e_{g_{0}}^{-1}f_{g_{0}}^{-1}$ . Let $E_{i}$ be the oriented curve on $X$ represented on $\unicode[STIX]{x1D6E5}$ as a segment connecting the edge labeled $e_{i}$ to the edge $e_{i}^{-1}$ and define $F_{i}$ analogously (see Figure 1). The curves $\{E_{i},F_{i}\}$ furnish a set of geometric representatives for a basis of $H_{1}(X;\mathbb{Z})$ . Via the intersection pairing $(\cdot ,\cdot )$ , this also leads to a basis for $H^{1}(X;\mathbb{Z})$ . Explicitly, a class $v\in H_{1}(X;\mathbb{Z})$ determines the element $(v,\cdot )\in H^{1}(X;\mathbb{Z})$ .

The covering $Y\rightarrow X$

The cover $Y\rightarrow X$ is regular with deck group $\langle \unicode[STIX]{x1D70E}\rangle \cong \mathbb{Z}/m\mathbb{Z}$ . Such covers are classified by order- $m$ elements of $H^{1}(X,\mathbb{Z}/m\mathbb{Z})$ . Relative to the basis for $H^{1}(X;\mathbb{Z})$ given above, we take the cover $Y\rightarrow X$ to correspond to the element $(F_{1},\cdot )\;(\text{mod}\;m)$ . There is an explicit model for $Y$ as a union of $m$ copies of a $2g_{0}$ -gon. For $1\leqslant i\leqslant m$ , let $\unicode[STIX]{x1D6E5}_{i}$ be a copy of the labeled $2g_{0}$ -gon above. Identify $e_{1}$ on $\unicode[STIX]{x1D6E5}_{i}$ with $e_{1}^{-1}$ on $\unicode[STIX]{x1D6E5}_{i+1}$ (interpreting subscripts mod $m$ ) and identify all other edges $\unicode[STIX]{x1D702}$ on $\unicode[STIX]{x1D6E5}_{i}$ with their counterpart $\unicode[STIX]{x1D702}^{-1}$ on $\unicode[STIX]{x1D6E5}_{i}$ . See Figure 1.

Figure 1.

The covering $p:Y\rightarrow X$ , illustrated for $m=3$ and $g_{0}=2$ . Here and throughout, we suppress edge identifications whenever confusion is unlikely.

The homology $H_{1}(Y;\mathbb{Z})$ has the structure of a $\mathbb{Z}[\mathbb{Z}/m\mathbb{Z}]$ -module, which can be described explicitly as follows.

Lemma 2.1. Identify $\mathbb{Z}/m\mathbb{Z}\cong \langle \unicode[STIX]{x1D70E}\rangle$ . Then there is an isomorphism

$$\begin{eqnarray}H_{1}(Y;\mathbb{Z})\cong \mathbb{Z}[\unicode[STIX]{x1D70E}]^{2g_{0}-2}\oplus \mathbb{Z}^{2}\end{eqnarray}$$

of $\mathbb{Z}[\unicode[STIX]{x1D70E}]$ -modules. Explicitly, there are generators $\{E_{i},F_{i}\mid 1\leqslant i\leqslant g_{0}\}\subset H_{1}(Y;\mathbb{Z})$ such that

$$\begin{eqnarray}\mathbb{Z}[\unicode[STIX]{x1D70E}]\langle E_{1},F_{1}\rangle \cong \mathbb{Z}^{2}\end{eqnarray}$$

and

$$\begin{eqnarray}\mathbb{Z}[\unicode[STIX]{x1D70E}]\langle E_{i},F_{j}\mid 2\leqslant i,j\leqslant g_{0}\rangle \cong \mathbb{Z}[\unicode[STIX]{x1D70E}]^{2g_{0}-2}.\end{eqnarray}$$

Proof. Let $p:Y\rightarrow X$ denote the projection. For $2\leqslant i\leqslant m$ , the preimage $p^{-1}(E_{i})$ consists of $m$ components and the same is true for $p^{-1}(F_{j})$ for $1\leqslant j\leqslant m$ . By abuse of notation, we define the curves $E_{i}$ and $F_{j}$ as the component of the appropriate preimage that is contained in the polygon $\unicode[STIX]{x1D6E5}_{1}$ . The preimage $p^{-1}(E_{1})$ has a single component, which we denote simply by $E_{1}$ , continuing to abuse notation. The proof now follows by inspection.◻

The covering $Z\rightarrow Y$

The covering $q:Z\rightarrow Y$ is a $\mathbb{Z}/m\mathbb{Z}$ branched covering with branch locus $L=\{\unicode[STIX]{x1D70E}^{i}(y)\}$ for some $y\in Y$ . As above, set $Y^{\circ }:=Y\setminus L$ . The covering $q$ is classified by some element $\unicode[STIX]{x1D703}\in H^{1}(Y^{\circ };\mathbb{Z}/m\mathbb{Z})$ . The inclusion $Y^{\circ }{\hookrightarrow}Y$ induces the short exact sequence

$$\begin{eqnarray}1\rightarrow K\rightarrow H_{1}(Y^{\circ };\mathbb{Z})\rightarrow H_{1}(Y;\mathbb{Z})\rightarrow 1.\end{eqnarray}$$

The kernel $K$ can be described explicitly as follows. Assume that $y\in Y$ is chosen so as to lie in the interior of $\unicode[STIX]{x1D6E5}_{1}$ . Let $C$ be a small loop encircling $y$ . Then

(5) $$\begin{eqnarray}K\cong \mathbb{Z}\langle \unicode[STIX]{x1D70E}^{i}C\mid 1\leqslant i\leqslant m\rangle /\{C+\unicode[STIX]{x1D70E}C+\cdots +\unicode[STIX]{x1D70E}^{m-1}C=0\}.\end{eqnarray}$$

Assume that $y\in Y$ has been chosen so as to be disjoint from the curves $\unicode[STIX]{x1D70E}^{i}E_{j}$ and $\unicode[STIX]{x1D70E}^{i}F_{j}$ on $Y$ . Then the collection of $\unicode[STIX]{x1D70E}^{i}E_{j},\unicode[STIX]{x1D70E}^{i}F_{j}$ determines a splitting

(6) $$\begin{eqnarray}H_{1}(Y^{\circ };\mathbb{Z})\cong H_{1}(Y;\mathbb{Z})\oplus K.\end{eqnarray}$$

Relative to this splitting, the class $\unicode[STIX]{x1D703}\in H^{1}(Y^{\circ };\mathbb{Z}/m\mathbb{Z})$ that classifies the branched cover $q:Z\rightarrow Y$ is defined so that $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D70E}^{i}C)=1$ and $\unicode[STIX]{x1D703}\equiv 0$ on $H_{1}(Y;\mathbb{Z})$ . As $\unicode[STIX]{x1D703}$ is valued in $\mathbb{Z}/m\mathbb{Z}$ , this determines a well-defined class on $K$ .

Figure 2.

The covering $q:Z\rightarrow Y$ , illustrated for $m=3$ and $g_{0}=2$ . The dashed lines indicate the location of the branch cuts. The arcs labeled $A$ and $B$ indicate how the branch cuts are glued.

The cover $q:Z\rightarrow Y$ can be described explicitly by using branch cuts. For $1\leqslant j\leqslant m-1$ , let $\unicode[STIX]{x1D6FE}_{j}$ be the oriented arc beginning at $\unicode[STIX]{x1D70E}^{j}y\in \unicode[STIX]{x1D6E5}_{j}$ that crosses $e_{1}$ onto $\unicode[STIX]{x1D6E5}_{j+1}$ and ends at $\unicode[STIX]{x1D70E}^{j+1}y\in \unicode[STIX]{x1D6E5}_{j+1}$ . Take $m$ copies of $Y\setminus \bigcup \{\unicode[STIX]{x1D6FE}_{j}\}$ , labeled $Y_{1},\ldots ,Y_{m}$ . To construct $Z$ , glue the right-hand side of $\unicode[STIX]{x1D6FE}_{j}$ on sheet $Y_{i}$ to the left-hand side of $\unicode[STIX]{x1D6FE}_{j}$ on sheet $Y_{i+j}$ (as usual, interpret all subscripts mod $m$ ). The covering group of $q:Z\rightarrow Y$ is isomorphic to $\mathbb{Z}/m\mathbb{Z}$ ; let $\unicode[STIX]{x1D701}$ be a generator. It is straightforward to check that this construction really does determine the cover determined by $\unicode[STIX]{x1D703}$ . See Figure 2. Note that in this figure, the points deleted in passing to $Z^{\circ }$ (and $Y^{\circ }$ ) are depicted by the small circles at the center of each polygon. Hence, Figure 2 is also a depiction of the unbranched covering $q:Z^{\circ }\rightarrow Y^{\circ }$ .

Repairing regularity: the covering $W\rightarrow Z$

Having fixed this model for $Z$ , one sees explicitly the non-regularity of the covering $p\,\circ \,q:Z^{\circ }\rightarrow X^{\circ }$ . Consider the curve $F_{1}\subset X^{\circ }$ . Then $p^{-1}(F_{1})\subset Y^{\circ }$ has $m$ components, one on each polygon $\unicode[STIX]{x1D6E5}_{i}$ . The component contained in $\unicode[STIX]{x1D6E5}_{i}$ is denoted $F_{1,i}$ . One sees that $q^{-1}(F_{1,1})$ has $m$ components, while $q^{-1}(F_{1,2})$ has one component. This prevents the $\unicode[STIX]{x1D70E}$ -action on $Y$ from lifting to $Z$ and so $Z\rightarrow X$ is not a normal cover. Despite this, one can repair the regularity by passing to a further cyclic cover.

Lemma 2.2. Let $r:W\rightarrow Z$ be the cyclic unbranched covering classified by the element $\unicode[STIX]{x1D6FC}\in H^{1}(Z;\mathbb{Z}/m\mathbb{Z})$ defined as

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}=((p\,\circ \,q)^{-1}(E_{1}),\cdot ).\end{eqnarray}$$

Let the covering group for $r:W\rightarrow Z$ be denoted $\langle \unicode[STIX]{x1D70F}\rangle \cong \mathbb{Z}/m\mathbb{Z}$ . Then

$$\begin{eqnarray}p\,\circ \,q\,\circ \,r:W^{\circ }\rightarrow X^{\circ }\end{eqnarray}$$

is a regular covering with $m^{3}$ sheets. Moreover, the covering group $H$ admits an explicit presentation via

(7) $$\begin{eqnarray}H\cong \langle \unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F},\unicode[STIX]{x1D701}\mid [\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]=\unicode[STIX]{x1D701},~\unicode[STIX]{x1D701}\text{ central},~\unicode[STIX]{x1D70E}^{m}=\unicode[STIX]{x1D70F}^{m}=\unicode[STIX]{x1D701}^{m}=1\rangle =\mathscr{H}(\mathbb{Z}/m\mathbb{Z}).\end{eqnarray}$$

Here $\mathscr{H}(\mathbb{Z}/m\mathbb{Z})$ denotes the Heisenberg group over $\mathbb{Z}/m\mathbb{Z}$ .

Proof. We will first define an auxiliary covering $\unicode[STIX]{x1D70B}:U^{\circ }\rightarrow X^{\circ }$ which is regular with covering group $H$ by construction; then we will exhibit an isomorphism $U^{\circ }\cong W^{\circ }$ of covers of $X^{\circ }$ .

We describe $U^{\circ }\rightarrow X^{\circ }$ in terms of a homomorphism $h:\unicode[STIX]{x1D70B}_{1}(X^{\circ })\rightarrow H$ . The fundamental group $\unicode[STIX]{x1D70B}_{1}(X^{\circ })$ is a free group of rank $2g_{0}$ and admits a presentation of the form

(8) $$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}(X^{\circ })=\langle e_{1},f_{1},\ldots ,e_{g_{0}},f_{g_{0}},c\mid [e_{1},f_{1}]\cdots [e_{g_{0}},f_{g_{0}}]=c\rangle .\end{eqnarray}$$

Geometrically, the elements $e_{i}$ (respectively $f_{i}$ ) correspond to loops crossing the edge $E_{i}$ (respectively $F_{i}$ ) of $\unicode[STIX]{x1D6E5}$ and the element $c$ corresponds to a loop that encircles the deleted point in $X^{\circ }$ counterclockwise. Define

$$\begin{eqnarray}h:\unicode[STIX]{x1D70B}_{1}(X^{\circ })\rightarrow H\end{eqnarray}$$

via $h(e_{1})=\unicode[STIX]{x1D70E}$ , $h(f_{1})=\unicode[STIX]{x1D70F}$ , $h(c)=\unicode[STIX]{x1D701}$ , with all other generators mapped to the identity $1\in H$ . It is immediate from the presentations (7) and (8) that $h$ is well defined.

The coverings $\unicode[STIX]{x1D70B}:U^{\circ }\rightarrow X^{\circ }$ and $p\circ q\circ r:W^{\circ }\rightarrow X^{\circ }$ correspond to subgroups $\unicode[STIX]{x1D70B}_{1}(U^{\circ }),\unicode[STIX]{x1D70B}_{1}(W^{\circ })$ of $\unicode[STIX]{x1D70B}_{1}(X^{\circ })$ . To show that $U^{\circ }\cong W^{\circ }$ are isomorphic as covers of $X^{\circ }$ , it suffices to show that $\unicode[STIX]{x1D70B}_{1}(U^{\circ })=\unicode[STIX]{x1D70B}_{1}(W^{\circ })$ as subgroups of $\unicode[STIX]{x1D70B}_{1}(X^{\circ })$ . To this end, define

$$\begin{eqnarray}h^{\prime }:\unicode[STIX]{x1D70B}_{1}(X^{\circ })\rightarrow H\end{eqnarray}$$

by $h^{\prime }(e_{1})=\unicode[STIX]{x1D70E}$ , with all other generators sent to $1\in H$ . It is clear that $\ker (h^{\prime })=\unicode[STIX]{x1D70B}_{1}(Y^{\circ })$ . It is elementary to verify that $\unicode[STIX]{x1D70B}_{1}(Y^{\circ })$ admits a presentation with generators

$$\begin{eqnarray}\{e_{1}^{m},f_{1}\}\cup \{e_{1}^{i}e_{j}e_{1}^{-i},e_{1}^{i}f_{j}e_{1}^{-i}\mid 0\leqslant i\leqslant i-1,2\leqslant j\leqslant g_{0}\}\cup \{e_{1}^{i}ce_{1}^{-i}\mid 0\leqslant i\leqslant m-1\}\end{eqnarray}$$

and a single relation that expresses $\prod _{i=1}^{m}e_{1}^{m-i}ce_{1}^{i-m}$ as a product of commutators of the remaining generators.

It follows that the map

$$\begin{eqnarray}h^{\prime \prime }:\unicode[STIX]{x1D70B}_{1}(Y^{\circ })\rightarrow H\end{eqnarray}$$

for which

$$\begin{eqnarray}h^{\prime \prime }(f_{1})=\unicode[STIX]{x1D70F},\quad h^{\prime \prime }(e_{1}^{i}ce_{1}^{-i})=\unicode[STIX]{x1D701}\end{eqnarray}$$

(and all other generators sent to $1\in H$ ) is well defined. A comparison with the explicit description of the regular covering $q\,\circ \,r:W^{\circ }\rightarrow Y^{\circ }$ shows that $\ker (h^{\prime \prime })=\unicode[STIX]{x1D70B}_{1}(W^{\circ })$ . On the other hand, there is a description of $H$ as a semidirect product

$$\begin{eqnarray}H\cong \langle \unicode[STIX]{x1D70F},\unicode[STIX]{x1D701}\rangle \rtimes \langle \unicode[STIX]{x1D70E}\rangle .\end{eqnarray}$$

From this, one sees that $\ker (h^{\prime \prime })=\ker (h)=\unicode[STIX]{x1D70B}_{1}(U^{\circ })$ . The result follows.◻

2.3 The normalized Atiyah–Kodaira construction

Above we gave a global construction of the manifold $E^{\text{nn}}(X,m)$ . In this paragraph, we describe a finite cover of this space that we call the normalized Atiyah–Kodaira construction.

Proof. As remarked above, unbranched $\mathbb{Z}/m\mathbb{Z}$ -coverings of a topological space $S$ are classified by $H^{1}(S;\mathbb{Z}/m\mathbb{Z})$ . The claims of the proposition will follow from the construction of an element $\widetilde{\unicode[STIX]{x1D6FC}}\in H^{1}(E^{\text{nn}}(X,m)^{\prime };\mathbb{Z}/m\mathbb{Z})$ such that the pullback of $\widetilde{\unicode[STIX]{x1D6FC}}$ to $H^{1}(Z;\mathbb{Z}/m\mathbb{Z})$ is the element $\unicode[STIX]{x1D6FC}$ of Lemma 2.2.

We first observe that the branching locus $\widetilde{\unicode[STIX]{x1D6E4}}$ of $E^{\text{nn}}(X,m)$ is a disjoint union of $m$ sections $B\rightarrow B\times Y$ , so $E^{\text{nn}}(X,m)\rightarrow B$ admits a section. Consequently, the 5-term exact sequence for $E^{\text{nn}}(X,m)\rightarrow B$ degenerates, yielding a splitting

$$\begin{eqnarray}H^{1}(E^{\text{nn}}(X,m);\mathbb{Z}/m\mathbb{Z})\cong H^{1}(B;\mathbb{Z}/m\mathbb{Z})\oplus H^{1}(Z;\mathbb{Z}/m\mathbb{Z})^{\unicode[STIX]{x1D70B}_{1}(B)}.\end{eqnarray}$$

As $H^{1}(Z;\mathbb{Z}/m\mathbb{Z})$ is finite, there is some finite-index subgroup $\unicode[STIX]{x1D70B}_{1}(B^{\prime })\leqslant \unicode[STIX]{x1D70B}_{1}(B)$ such that $\unicode[STIX]{x1D6FC}\in H^{1}(Z;\mathbb{Z}/m\mathbb{Z})$ is $\unicode[STIX]{x1D70B}_{1}(B^{\prime })$ -invariant. Define $E^{\text{nn}}(X,m)^{\prime }$ to be the pullback of $E^{\text{nn}}(X,m)$ along the cover $B^{\prime }\rightarrow B$ . Then the 5-term sequence for $E^{\text{nn}}(X,m)^{\prime }\rightarrow B^{\prime }$ shows that there exists a class $\widetilde{\unicode[STIX]{x1D6FC}}$ with the required properties.◻

2.4 The homology of $W$

In preparation for the latter stages of the argument, we will need to understand $H_{1}(W;\mathbb{Z})$ and $H_{1}(W;\mathbb{Q})$ , especially as representations of the covering group $H=\mathscr{H}(\mathbb{Z}/m\mathbb{Z})$ . The central results appear as Lemmas 2.9, 2.10, and 2.11. Rational coefficients will be assumed unless otherwise specified. At a glance, the arguments of the section might appear needlessly complicated; indeed, if we were only interested in the character of $H_{1}(W)$ as an $H$ -representation, then we could obtain this indirectly using the Chevalley–Weil theorem. However, for our purposes we will require an explicit set of generators for $H_{1}(W)$ as a $\mathbb{Q}[H]$ -module, for which a character-theoretic argument does not suffice.

Our description of $H_{1}(W)$ will be derived in two steps. Let $s:V\rightarrow X$ denote the unbranched $(\mathbb{Z}/m\mathbb{Z})^{2}$ -covering associated to the homomorphism

(9) $$\begin{eqnarray}h:\unicode[STIX]{x1D70B}_{1}(X)\rightarrow H/\langle \unicode[STIX]{x1D701}\rangle \cong (\mathbb{Z}/m\mathbb{Z})^{2}\end{eqnarray}$$

given by $h(e_{1})=\unicode[STIX]{x1D70E}$ and $h(f_{1})=\unicode[STIX]{x1D70F}$ .

Lemma 2.4. Identify $\mathbb{Q}[(\mathbb{Z}/m\mathbb{Z})^{2}]\cong \mathbb{Q}[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]$ . There is an isomorphism

$$\begin{eqnarray}H_{1}(V)\cong \mathbb{Q}[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]^{2g_{0}-2}\oplus \mathbb{Q}^{2}\end{eqnarray}$$

of $\mathbb{Q}[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]$ -modules. Explicitly, there are generators $\{E_{i},F_{i}\mid 1\leqslant i\leqslant g_{0}\}\subset H_{1}(V)$ such that

$$\begin{eqnarray}\mathbb{Q}[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]\langle E_{1},F_{1}\rangle \cong \mathbb{Q}^{2}\end{eqnarray}$$

and

$$\begin{eqnarray}\mathbb{Q}[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]\langle E_{i},F_{j}\mid 2\leqslant i,j\leqslant g_{0}\rangle \cong \mathbb{Q}[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]^{2g_{0}-2}.\end{eqnarray}$$

The surface $W$ arises as a $\mathbb{Z}/m\mathbb{Z}$ branched covering $t:W\rightarrow V$ . By construction, the covering $s\circ t:W\rightarrow X$ coincides with $p\circ q\,\circ \,r$ . We write $\mathbb{Q}[H]=\mathbb{Q}[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F},\unicode[STIX]{x1D701}]$ with the understanding that $\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F},\unicode[STIX]{x1D701}$ are subject to the relations in $H$ . Under this identification, the cover $t:W\rightarrow V$ has deck group $\langle \unicode[STIX]{x1D701}\rangle$ .

The homology of the branched cover requires a more delicate analysis than in the unbranched case and will require some preliminary ideas.

For example, the covering $q:Z\rightarrow Y$ depicted in Figure 2 is in planar form relative to the disk $D$ formed by a regular neighborhood of $\unicode[STIX]{x1D6FE}_{1}\cup \unicode[STIX]{x1D6FE}_{2}$ .

Referring again to Figure 2, the two preimages of $\unicode[STIX]{x1D6FE}_{1}$ in $Y_{1}$ and $Y_{2}$ together comprise a $G$ -curve.

When $f:\unicode[STIX]{x1D6F4}\rightarrow \unicode[STIX]{x1D6F4}^{\prime }$ is in planar form, $H_{1}(\unicode[STIX]{x1D6F4})$ has a simple description in terms of $H_{1}(\unicode[STIX]{x1D6F4}^{\prime })$ and a system of $G$ -curves.

Lemma 2.8. Let $f:\unicode[STIX]{x1D6F4}\rightarrow \unicode[STIX]{x1D6F4}^{\prime }$ be a $\mathbb{Z}/m\mathbb{Z}$ -branched covering in planar form relative to $D$ ; identify $\mathbb{Z}/m\mathbb{Z}\cong \langle \unicode[STIX]{x1D701}\rangle$ . Let $\{\unicode[STIX]{x1D6FE}_{i}\}$ be a collection of arcs as in Definition 2.6 such that $\{[\unicode[STIX]{x1D6FE}_{i}]\}$ generates $H_{1}(D,L)$ . Then there is a surjective map of $\mathbb{Q}[\unicode[STIX]{x1D701}]$ -modules

$$\begin{eqnarray}g:\mathbb{Q}[\unicode[STIX]{x1D701}]H_{1}(\unicode[STIX]{x1D6F4}^{\prime })\oplus \mathbb{Q}[\unicode[STIX]{x1D701}]\langle \{G_{\unicode[STIX]{x1D6FE}_{i}}\}\rangle \rightarrow H_{1}(\unicode[STIX]{x1D6F4}).\end{eqnarray}$$

Moreover, $g$ is injective when restricted to $\mathbb{Q}[\unicode[STIX]{x1D701}]H_{1}(\unicode[STIX]{x1D6F4}^{\prime })$ .

Proof. As $f$ is in planar form, the Mayer–Vietoris sequence provides an exact sequence

$$\begin{eqnarray}H_{1}(f^{-1}(\unicode[STIX]{x2202}D))\rightarrow H_{1}(f^{-1}(\unicode[STIX]{x1D6F4}^{\prime }\setminus D))\oplus H_{1}(f^{-1}(D))\rightarrow H_{1}(\unicode[STIX]{x1D6F4})\rightarrow 1.\end{eqnarray}$$

Again, since $f$ is in planar form, $f^{-1}(\unicode[STIX]{x1D6F4}^{\prime }\setminus D)$ consists of $m$ disjoint copies of $\unicode[STIX]{x1D6F4}^{\prime }\setminus D$ , acted on in the obvious way by the deck group $\langle \unicode[STIX]{x1D701}\rangle$ . Thus, $H_{1}(f^{-1}(\unicode[STIX]{x1D6F4}^{\prime }\setminus D))\cong \mathbb{Q}[\unicode[STIX]{x1D701}]H_{1}(\unicode[STIX]{x1D6F4}^{\prime })$ . It remains to be seen that $H_{1}(f^{-1}(D))$ is generated as a $\mathbb{Q}[\unicode[STIX]{x1D701}]$ -module by $\{[G_{\unicode[STIX]{x1D6FE}_{i}}]\}$ under the assumption that $\{[\unicode[STIX]{x1D6FE}_{i}]\}$ generate $H_{1}(D,L)$ .

Let $D^{\circ }$ denote the disk $D$ with a small neighborhood of each branch point $p_{i}\in L$ for $1\leqslant i\leqslant k$ removed. Thus, $D^{\circ }$ is a sphere with $k+1$ boundary components. We describe a cell structure on $D^{\circ }$ . The zero-skeleton is given by

$$\begin{eqnarray}(D^{\circ })^{(0)}=\{v_{1},\ldots ,v_{k},w\},\end{eqnarray}$$

with each $v_{i}$ on the boundary component associated to $p_{i}$ and $w\in \unicode[STIX]{x2202}D$ . Next, take

$$\begin{eqnarray}(D^{\circ })^{(1)}=\{c_{1},\ldots ,c_{k},e_{1},\ldots ,e_{k},d\},\end{eqnarray}$$

with both ends of $c_{i}$ attached to $v_{i}$ , each $e_{i}$ connecting $v_{i}$ and $w$ , and both ends of $d$ attached to $w$ . Then $(D^{\circ })^{(2)}$ consists of a single $2$ -cell attached in the obvious way.

The above cell structure lifts to a $\mathbb{Q}[\unicode[STIX]{x1D701}]$ -equivariant cell structure on $f^{-1}(D^{\circ })$ . The boundary maps on the $1$ -cells are $\mathbb{Q}[\unicode[STIX]{x1D701}]$ -linear and are given by

$$\begin{eqnarray}\unicode[STIX]{x2202}(c_{i})=(\unicode[STIX]{x1D701}-1)c_{i},\quad \unicode[STIX]{x2202}(e_{i})=v_{i}-w,\quad \unicode[STIX]{x2202}(d)=0.\end{eqnarray}$$

A priori, one knows that $H_{0}(f^{-1}(D^{\circ }))=\mathbb{Q}$ . Thus, $\unicode[STIX]{x2202}:C_{1}(f^{-1}(D^{\circ }))\rightarrow C_{0}(f^{-1}(D^{\circ }))$ has corank one as a map of $\mathbb{Q}$ -vector spaces. A dimension count then shows that $Z_{1}(f^{-1}(D^{\circ }))$ has dimension $mk+1$ over $\mathbb{Q}$ . An argument in elementary linear algebra then implies that $Z_{1}(f^{-1}(D^{\circ }))$ , and hence $H_{1}(f^{-1}(D^{\circ }))$ , is generated over $\mathbb{Q}[\unicode[STIX]{x1D701}]$ by the set

$$\begin{eqnarray}\bigg\{\biggl(\mathop{\sum }_{j=1}^{m}\unicode[STIX]{x1D701}^{j}\biggr)c_{i}\mid 1\leqslant i\leqslant k\bigg\}\cup \{(\unicode[STIX]{x1D701}-1)(e_{1}-e_{i})+(c_{i}-c_{1})\mid 2\leqslant i\leqslant k\}\cup \{d\}.\end{eqnarray}$$

Topologically, the inclusion map $f^{-1}(D^{\circ })\rightarrow f^{-1}(D)$ attaches $k$ disks along the boundary components encircling the branch points $p_{i}$ . The boundary of these disks is represented by the classes $\{(\sum _{j=1}^{m}\unicode[STIX]{x1D701}^{j})c_{i}\mid 1\leqslant i\leqslant k\}$ . It follows that $H_{1}(f^{-1}(D))$ is generated over $\mathbb{Q}[\unicode[STIX]{x1D701}]$ by the set

$$\begin{eqnarray}\{(\unicode[STIX]{x1D701}-1)(e_{1}-e_{i})+(c_{i}-c_{1})\mid 2\leqslant i\leqslant k\}\cup \{d\}.\end{eqnarray}$$

The summand spanned by $[d]$ clearly corresponds to $H^{1}(f^{-1}(\unicode[STIX]{x2202}D))$ . The result will now follow from the description of a surjective map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:\mathbb{Q}[\unicode[STIX]{x1D701}]H_{1}(D,L)\rightarrow \mathbb{Q}[\unicode[STIX]{x1D701}]\langle (\unicode[STIX]{x1D701}-1)(e_{1}-e_{i})+(c_{i}-c_{1})\mid 2\leqslant i\leqslant k\rangle \leqslant H_{1}(f^{-1}(D))\end{eqnarray}$$

such that $\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6FE})=G_{\unicode[STIX]{x1D6FE}}$ for all arcs $\unicode[STIX]{x1D6FE}$ connecting two points of $L$ .

Using the cell structure on $D^{\circ }$ described above (which can be extended to a cell structure on $D$ by adding $k$ additional $2$ -cells), the set $L$ is identified with the set $\{v_{1},\ldots ,v_{k}\}$ . Then a generating set for $H_{1}(D,L)$ consists of the $k-1$ elements $e_{1}-e_{i}$ for $2\leqslant i\leqslant k$ . Define $\unicode[STIX]{x1D70B}$ by setting

$$\begin{eqnarray}\unicode[STIX]{x1D70B}(e_{1}-e_{i})=(\unicode[STIX]{x1D701}-1)(e_{1}-e_{i})+(c_{i}-c_{1}).\end{eqnarray}$$

It is evident from the construction that $\unicode[STIX]{x1D70B}(e_{1}-e_{i})=G_{e_{1}-e_{i}}$ and that $[G_{\unicode[STIX]{x1D6FE}_{1}+\unicode[STIX]{x1D6FE}_{2}}]=[G_{\unicode[STIX]{x1D6FE}_{1}}]+[G_{\unicode[STIX]{x1D6FE}_{2}}]$ . The result now follows by linearity.◻

We now apply Lemma 2.8 to the branched covering $t:W\rightarrow V$ . Continuing to abuse notation, we let $E_{i}\subset W$ denote a single component of the preimage $(s\,\circ \,t)^{-1}(E_{i})$ and define $F_{i}$ similarly. Recalling the construction of $X$ in terms of the polygon $\unicode[STIX]{x1D6E5}$ , we observe that $V$ can be constructed from $m^{2}$ copies of $\unicode[STIX]{x1D6E5}$ indexed by elements $(i,j)\in (\mathbb{Z}/m\mathbb{Z})^{2}$ . Each $\unicode[STIX]{x1D6E5}_{i,j}$ has a marked point $p_{i,j}$ corresponding to the unique branch point for $t$ contained in $\unicode[STIX]{x1D6E5}_{i,j}$ . We define $\unicode[STIX]{x1D6FE}_{h}$ to be the arc starting at $p_{0,0}$ that crosses $e_{1}$ onto $\unicode[STIX]{x1D6E5}_{1,0}$ and ends at $p_{1,0}$ . Likewise, $\unicode[STIX]{x1D6FE}_{v}$ is defined to be the arc starting at $p_{0,0}$ that crosses $f_{1}$ onto $\unicode[STIX]{x1D6E5}_{0,1}$ and ends at $p_{0,1}$ . Then the curves $G_{h}$ and $G_{v}$ on $W$ are defined by

(10) $$\begin{eqnarray}G_{h}:=G_{\unicode[STIX]{x1D6FE}_{h}},\quad G_{v}:=G_{\unicode[STIX]{x1D6FE}_{v}}\end{eqnarray}$$

in the sense of Definition 2.6.

Lemma 2.9. The simple closed curves

$$\begin{eqnarray}\{E_{i},F_{i}\mid 1\leqslant i\leqslant g_{0}\}\cup \{G_{h},G_{v}\}\end{eqnarray}$$

generate $H_{1}(W)$ as a $\mathbb{Q}[H]$ -module. Moreover, the submodule spanned by $E_{i},F_{i}$ for $2\leqslant i\leqslant g_{0}$ determines a free $\mathbb{Q}[H]$ -module of rank $2g_{0}-2$ .

Proof. In order to apply Lemma 2.8, it is necessary to identify a disk $D$ relative to which $t:W\rightarrow V$ is in planar form. It is straightforward to verify that a small regular neighborhood of the collection of arcs

$$\begin{eqnarray}\{\unicode[STIX]{x1D70E}^{i}\unicode[STIX]{x1D70F}^{j}\unicode[STIX]{x1D6FE}_{h}\mid 0\leqslant i\leqslant m-2,0\leqslant j\leqslant m-1\}\cup \{\unicode[STIX]{x1D70F}^{j}\unicode[STIX]{x1D6FE}_{v}\mid 0\leqslant j\leqslant m-2\}\end{eqnarray}$$

is a disk $D\subset V$ containing all points $p_{i,j}$ . Moreover, the components of $s^{-1}(E_{1}),s^{-1}(F_{1})$ passing through $\unicode[STIX]{x1D6E5}_{0,0}$ are disjoint from $D$ and the same is true for the entire preimage $s^{-1}(E_{j}),s^{-1}(F_{j})$ for $j\geqslant 2$ . In particular, these curves generate $H_{1}(V)$ over $\mathbb{Q}[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]$ .

Applying Lemma 2.8, we now see that $H_{1}(W)$ is generated as a $\mathbb{Q}[H]$ -module by $\{E_{i},F_{i}\mid 1\leqslant i\leqslant g_{0}\}$ , along with a certain subset of curves in the $\mathbb{Q}[H]$ -orbit of $G_{h}$ and $G_{v}$ . This proves the first claim. The second claim follows from the second assertion of Lemma 2.8 in combination with Lemma 2.4.◻

Define the element

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{na}=(m-(1+\unicode[STIX]{x1D701}+\cdots +\unicode[STIX]{x1D701}^{m-1}))\in \mathbb{Z}[H];\end{eqnarray}$$

over $\mathbb{Q}$ , this determines the projection onto the summand of $\mathbb{Q}[H]$ spanned by irreducible $H$ -representations that do not factor through the abelianization $H^{\operatorname{ab}}$ . (We call such representations non-abelian; cf. § 8.)

Lemma 2.10. For $d=2g_{0}-1$ , there is an isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{na}H_{1}(W;\mathbb{Q})\cong \unicode[STIX]{x1D6F1}_{na}\mathbb{Q}[H]^{d}\end{eqnarray}$$

of  $\unicode[STIX]{x1D6F1}_{na}\mathbb{Q}[H]$ -modules.

Proof. This follows readily from the determination of the $\mathbb{Q}[H]$ -module structure of $H_{1}(W;\mathbb{Q})$ via the method of Chevalley–Weil. This in turn follows from an elaboration of the method of Lemma 2.8. Place a cell structure on $X^{\circ }$ as follows: there are two zero-cells $v,w$ ; $2g_{0}+2$ one-cells $a_{1},b_{1},\ldots ,a_{g_{0}},b_{g_{0}},c,e$ ; and one two-cell $F$ . Each cell $a_{i},b_{i}$ has both ends attached to $v$ , the cell $c$ has both ends attached to $w$ , and $e$ connects $v$ to $w$ . The two-cell $F$ is attached in the obvious way (this does not need to be described in detail).

This gives rise to the chain complex $C_{\bullet }(X^{\circ })$ computing $H_{1}(X^{\circ };\mathbb{Q})$ :

$$\begin{eqnarray}\mathbb{Q}\langle F\rangle \rightarrow \mathbb{Q}\langle a_{1},\ldots ,b_{g_{0}},c,e\rangle \rightarrow \mathbb{Q}\langle v,w\rangle \rightarrow 0.\end{eqnarray}$$

Lifting this cell structure along the covering map $W^{\circ }\rightarrow X^{\circ }$ , we arrive at an $H$ -equivariant cell structure on $W^{\circ }$ . On the level of chain complexes,

$$\begin{eqnarray}C_{\bullet }(W^{\circ })=\mathbb{Q}[H]\otimes _{\mathbb{Q}}C_{\bullet }(X^{\circ }).\end{eqnarray}$$

We wish to determine the character $\unicode[STIX]{x1D712}(H_{1}(W^{\circ };\mathbb{Q}))$ . This can be obtained by taking the Euler characteristic of the chain complex $C_{\bullet }(W^{\circ })$ , viewed as a virtual character of $H$ . Since every $\mathbb{Q}[H]$ -module is semisimple,

$$\begin{eqnarray}\unicode[STIX]{x1D712}(C_{\bullet }(W^{\circ }))=\unicode[STIX]{x1D712}(H_{\bullet }(W^{\circ };\mathbb{Q})).\end{eqnarray}$$

This provides the following equality of characters:

$$\begin{eqnarray}\unicode[STIX]{x1D712}(C_{0}(W^{\circ }))-\unicode[STIX]{x1D712}(C_{1}(W^{\circ }))+\unicode[STIX]{x1D712}(C_{2}(W^{\circ }))=\unicode[STIX]{x1D712}(H_{0}(W^{\circ };\mathbb{Q}))-\unicode[STIX]{x1D712}(H_{1}(W^{\circ };\mathbb{Q}))+\unicode[STIX]{x1D712}(H_{2}(W^{\circ };\mathbb{Q})).\end{eqnarray}$$

By construction, each $C_{i}(W^{\circ })$ is a free $\mathbb{Q}[H]$ -module on 2, $2g_{0}+2$ , one generators, respectively. On the right-hand side, we observe that $H_{0}(W^{\circ };\mathbb{Q})\cong \mathbb{Q}$ and $H_{2}(W^{\circ };\mathbb{Q})=0$ . Altogether, this determines the character of $H_{1}(W^{\circ };\mathbb{Q})$ completely and furnishes an isomorphism

$$\begin{eqnarray}H_{1}(W^{\circ };\mathbb{Q})\cong \mathbb{Q}[H]^{2g_{0}-1}\oplus \mathbb{Q}.\end{eqnarray}$$

To determine $H_{1}(W;\mathbb{Q})$ as a $\mathbb{Q}[H]$ -representation, we exploit the $H$ -equivariant exact sequence

(11) $$\begin{eqnarray}1\rightarrow K\rightarrow H_{1}(W^{\circ };\mathbb{Q})\rightarrow H_{1}(W;\mathbb{Q})\rightarrow 1\end{eqnarray}$$

induced by the inclusion map $W^{\circ }\rightarrow W$ . The kernel $K$ is the subspace of $H_{1}(W^{\circ };\mathbb{Q})$ spanned by loops around the punctures of $W^{\circ }$ . The cover $W\rightarrow X$ factors through the intermediate cover $V$ and the covering $V\rightarrow X$ is unbranched. Consequently, the punctures of $W^{\circ }$ are in one-to-one correspondence with the elements of the covering group $H^{\operatorname{ab}}\cong (\mathbb{Z}/m\mathbb{Z})^{2}$ and this bijection intertwines the action of the deck group $H$ with multiplication by $H^{\operatorname{ab}}$ .

On the level of $H_{1}(W^{\circ };\mathbb{Q})$ , this implies that $K$ is spanned by the $H$ -orbit of a single puncture $[c]$ and that $[c]$ is stabilized by $[H,H]=\langle \unicode[STIX]{x1D701}\rangle$ . The $\mathbb{Q}$ -span of $h[c]$ for $h\in H^{\operatorname{ab}}$ is subject to the single relation

$$\begin{eqnarray}\mathop{\sum }_{h\in H^{\operatorname{ab}}}h[c]=0.\end{eqnarray}$$

In other words, there is an isomorphism of $\mathbb{Q}[H]$ -modules

$$\begin{eqnarray}K\cong \mathbb{Q}[H^{\operatorname{ab}}]/\mathbb{Q}[1];\end{eqnarray}$$

here $\mathbb{Q}[1]$ denotes the trivial submodule.

Taking the Euler characteristics of the short exact sequence (11), we determine the character of $H_{1}(W;\mathbb{Q})$ and find that

(12) $$\begin{eqnarray}H_{1}(W;\mathbb{Q})\cong \mathbb{Q}^{2}\oplus \mathbb{Q}[H]^{2g_{0}-2}\oplus \mathbb{Q}[H]/\mathbb{Q}[H^{\operatorname{ab}}].\end{eqnarray}$$

Finally, to determine $\unicode[STIX]{x1D6F1}_{na}H_{1}(W;\mathbb{Q})$ as a $\unicode[STIX]{x1D6F1}_{na}\mathbb{Q}[H]$ -module, we multiply both sides of (12) by $\unicode[STIX]{x1D6F1}_{na}$ and obtain

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{na}H_{1}(W;\mathbb{Q})\cong \unicode[STIX]{x1D6F1}_{na}\mathbb{Q}[H]^{2g_{0}-1}.\Box\end{eqnarray}$$

Below, we record some properties of the intersection form $(\cdot ,\cdot )$ on $H_{1}(W)$ in the basis specified by Lemma 2.9.

The monodromy group

Theorem C is concerned with arithmetic properties of the monodromy of the bundle $E(X,m)\rightarrow B^{\prime }$ . In (13) below, we define the monodromy groups in question. Throughout, $E\rightarrow B$ will denote an arbitrary bundle (in particular, the base space $B$ is not the particular base of the Atiyah–Kodaira bundle). For a proof of Proposition 3.1 below (as well as general background on surface bundles), see [Reference Farb and MargalitFM12, § 5.6.1]. Recall that the mapping class group

$$\begin{eqnarray}\operatorname{Mod}(\unicode[STIX]{x1D6F4}_{g}):=\unicode[STIX]{x1D70B}_{0}\operatorname{Diff}(\unicode[STIX]{x1D6F4}_{g})\end{eqnarray}$$

is the group of diffeomorphisms of $\unicode[STIX]{x1D6F4}_{g}$ , up to isotopy.

Proposition 3.1 (Existence of monodromy representation).

Let $B$ be any paracompact Hausdorff space and $\unicode[STIX]{x1D6F4}_{g}$ a closed oriented surface of genus $g\geqslant 2$ . Associated to any $\unicode[STIX]{x1D6F4}_{g}$ -bundle $\unicode[STIX]{x1D70B}:E\rightarrow B$ is a homomorphism called the monodromy representation

$$\begin{eqnarray}\unicode[STIX]{x1D707}:\unicode[STIX]{x1D70B}_{1}(B,b)\rightarrow \operatorname{Mod}(\unicode[STIX]{x1D6F4}_{g}),\end{eqnarray}$$

well defined up to conjugacy. Informally, $\unicode[STIX]{x1D707}$ records how a local identification of the fiber $\unicode[STIX]{x1D70B}^{-1}(b)\cong \unicode[STIX]{x1D6F4}_{g}$ changes as the fiber is transported around loops in $B$ .

We will also require a version of Proposition 3.1 for a $\unicode[STIX]{x1D6F4}_{g}$ -bundle with $k$ disjoint sections. In this case, the monodromy takes values in

$$\begin{eqnarray}\operatorname{Mod}(\unicode[STIX]{x1D6F4}_{g},k):=\unicode[STIX]{x1D70B}_{0}\operatorname{Diff}(\unicode[STIX]{x1D6F4}_{g},k),\end{eqnarray}$$

where $\operatorname{Diff}(\unicode[STIX]{x1D6F4}_{g},k)$ is the group of diffeomorphisms that preserve a $k$ -element set of marked points on $\unicode[STIX]{x1D6F4}_{g}$ . Sometimes when there is only one marked point $p\in \unicode[STIX]{x1D6F4}$ , we write $\operatorname{Mod}(\unicode[STIX]{x1D6F4},p)$ instead of $\operatorname{Mod}(\unicode[STIX]{x1D6F4},1)$ .

While understanding this mapping class group-valued monodromy will be essential in the ensuing analysis, our ultimate goal is to understand an algebraic ‘approximation’ to $\unicode[STIX]{x1D707}$ .

Definition 3.2 (The symplectic representation).

Let

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}:\operatorname{Mod}(\unicode[STIX]{x1D6F4}_{g})\rightarrow \operatorname{Aut}(H_{1}(\unicode[STIX]{x1D6F4}_{g};\mathbb{Z}),(\cdot ,\cdot ))\end{eqnarray}$$

be the homomorphism induced by the action of $\operatorname{Homeo}(\unicode[STIX]{x1D6F4}_{g})$ on $H_{1}(\unicode[STIX]{x1D6F4}_{g};\mathbb{Z})$ . Here $\operatorname{Aut}(H_{1}(\unicode[STIX]{x1D6F4}_{g};\mathbb{Z}),(\cdot ,\cdot ))$ denotes the group of automorphisms of $H_{1}(\unicode[STIX]{x1D6F4}_{g};\mathbb{Z})$ preserving the algebraic intersection pairing $(\cdot ,\cdot )$ . This group is isomorphic to the symplectic group $\operatorname{Sp}(2g,\mathbb{Z})$ . For $E\rightarrow B$ a $\unicode[STIX]{x1D6F4}_{g}$ bundle with monodromy representation $\unicode[STIX]{x1D707}$ , the symplectic representation is the composition

$$\begin{eqnarray}\unicode[STIX]{x1D70C}:=\unicode[STIX]{x1D6F9}\,\circ \,\unicode[STIX]{x1D707}.\end{eqnarray}$$

For the remainder of the paper, we fix the notation $\unicode[STIX]{x1D707}_{X,m}$ for the monodromy of $E(X,m)$ , as well as

$$\begin{eqnarray}\hat{\unicode[STIX]{x1D6E4}}_{X,m}:=\operatorname{Im}(\unicode[STIX]{x1D707}_{X,m})\end{eqnarray}$$

and

(13) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{X,m}:=\operatorname{Im}(\unicode[STIX]{x1D70C}_{X,m}).\end{eqnarray}$$

Often $X$ and/or $m$ will be implicit and we will write simply $\hat{\unicode[STIX]{x1D6E4}},\unicode[STIX]{x1D6E4}$ .

Fiberwise coverings and monodromy

As described above, the Atiyah–Kodaira bundle is constructed as a fiberwise branched covering. In this paragraph we establish some basic facts concerning the structure of monodromy representations of such bundles. Throughout this paragraph, we fix the following setup: let $\unicode[STIX]{x1D6F4}\rightarrow \unicode[STIX]{x1D6F4}^{\prime }$ be a regular covering of Riemann surfaces with finite deck group $G$ , possibly branched. Suppose that $\unicode[STIX]{x1D70B}:E\rightarrow B$ is a $\unicode[STIX]{x1D6F4}$ -bundle, $\unicode[STIX]{x1D70B}^{\prime }:E^{\prime }\rightarrow B$ is a $\unicode[STIX]{x1D6F4}^{\prime }$ -bundle, and there is a fiberwise branched covering $E\rightarrow E^{\prime }$ . The monodromy representations for $\unicode[STIX]{x1D70B}:E\rightarrow B$ and $\unicode[STIX]{x1D70B}^{\prime }:E^{\prime }\rightarrow B$ will be denoted $\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}^{\prime }$ , respectively.

Proofs for Lemmas 3.3–3.6 to follow can be found in [Reference LooijengaLoo97] and [Reference Grunewald, Larsen, Lubotzky and MalesteinGLLM15].

The action of $G$ on $\unicode[STIX]{x1D6F4}$ endows $H_{1}(\unicode[STIX]{x1D6F4};\mathbb{Q})$ with the structure of a $\mathbb{Q}[G]$ -module. Since $G$ is finite, $H_{1}(\unicode[STIX]{x1D6F4};\mathbb{Q})$ is a semisimple module and thus there is a decomposition

(14) $$\begin{eqnarray}H_{1}(\unicode[STIX]{x1D6F4};\mathbb{Q})\cong \bigoplus _{U_{i}}U_{i}^{m_{i}},\end{eqnarray}$$

where the sum runs over the simple $\mathbb{Q}[G]$ -modules $U_{i}$ . The summands $U_{i}^{m_{i}}$ are known as isotypic factors. The classical Chevalley–Weil theorem gives a complete description of each multiplicity $m_{i}$ , as long as one has a complete list of the simple $\mathbb{Q}[G]$ -modules (or their characters). The latter can be worked out in theory, but can be tedious in practice. Up through 7, we will not need to know the $m_{i}$ (nor even the $U_{i}$ ) explicitly. We will see that the mere existence of the decomposition (14) has consequences for the study of the monodromy $\unicode[STIX]{x1D707}$ . (Later in § 9 we will need to know something about the decomposition for $G$ the Heisenberg group: we establish the necessary facts in § 8.)

In light of Lemma 3.3, in the remainder of the paragraph we will assume that $\unicode[STIX]{x1D707}$ is valued in $\operatorname{Mod}(\unicode[STIX]{x1D6F4})^{G}$ . The following shows that such an assumption has strong consequences for the symplectic monodromy representation $\unicode[STIX]{x1D70C}:\unicode[STIX]{x1D70B}_{1}(B)\rightarrow \operatorname{Aut}(H_{1}(\unicode[STIX]{x1D6F4}),(\cdot ,\cdot ))$ .

Definition 3.4 (Reidemeister pairing).

Let $G\leqslant \operatorname{Mod}(\unicode[STIX]{x1D6F4})$ be a finite subgroup. The Reidemeister pairing relative to $G$ is the form

$$\begin{eqnarray}\langle \cdot ,\cdot \rangle _{G}:H_{1}(\unicode[STIX]{x1D6F4};\mathbb{Z})\times H_{1}(\unicode[STIX]{x1D6F4};\mathbb{Z})\rightarrow \mathbb{Z}[G]\end{eqnarray}$$

defined by

$$\begin{eqnarray}\langle x,y\rangle _{G}=\mathop{\sum }_{g\in G}(x,g\cdot y)g.\end{eqnarray}$$

If the group $G$ is implicit, we will write simply $\langle \cdot ,\cdot \rangle$ .

Lemma 3.6. Let $\unicode[STIX]{x1D719}\in \operatorname{Mod}(\unicode[STIX]{x1D6F4})^{G}$ be given. Then $\unicode[STIX]{x1D6F9}(\unicode[STIX]{x1D719})\in \operatorname{Aut}(H_{1}(\unicode[STIX]{x1D6F4};\mathbb{Z}),(\cdot ,\cdot ))$ preserves the Reidemeister pairing $\langle \cdot ,\cdot \rangle _{G}$ . Moreover, $\unicode[STIX]{x1D719}$ preserves each isotypic factor. Thus, $\unicode[STIX]{x1D6F9}(\unicode[STIX]{x1D719})$ belongs to the subgroup

$$\begin{eqnarray}\mathop{\prod }_{U_{i}}\operatorname{Aut}_{G}(U_{i}^{m_{i}},\langle \cdot ,\cdot \rangle _{G})\leqslant \operatorname{Aut}(H_{1}(\unicode[STIX]{x1D6F4};\mathbb{Q}),(\cdot ,\cdot )).\end{eqnarray}$$

The arguments in §§ 4 and 5 make use of some of the explicit structure of the Reidemeister pairing $\langle \cdot ,\cdot \rangle _{H}$ on $H_{1}(W)$ . We record these here for later use.

Point-pushing diffeomorphisms

In this section and the next, we carry out a topological analysis of the monodromy map $\unicode[STIX]{x1D707}_{X,m}$ . In this section, we describe the monodromy of simple elements in the language of point-pushing diffeomorphisms and work out some general formulas. These formulas are the main objective of the section and appear as Lemmas 4.12 and 4.14. In the subsequent section, we will apply these results to analyze a specific class of monodromy elements in preparation for the algebraic portion of the argument as carried out in §§ 6–9.

Definition 4.1. Let $\unicode[STIX]{x1D6F4}$ be a closed surface with marked point $p$ and let $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6F4},p)$ be a based loop. There is an isotopy $\unicode[STIX]{x1D6F1}_{t}(\unicode[STIX]{x1D6FE})$ of $\unicode[STIX]{x1D6F4}$ that ‘pushes’ $p$ along the path $\unicode[STIX]{x1D6FE}$ at unit speed. The point-push map $P(\unicode[STIX]{x1D6FE})\in \operatorname{Mod}(\unicode[STIX]{x1D6F4},p)$ is defined by

$$\begin{eqnarray}P(\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D6F1}_{1}(\unicode[STIX]{x1D6FE}).\end{eqnarray}$$

Suppose now that $\unicode[STIX]{x1D6FE}$ determines a simple closed curve on $\unicode[STIX]{x1D6F4}$ . Let $\unicode[STIX]{x1D6FE}_{L},\unicode[STIX]{x1D6FE}_{R}$ be the left-hand, respectively, right-hand sides of $\unicode[STIX]{x1D6FE}$ , viewed as simple closed curves on $\unicode[STIX]{x1D6F4}\setminus \{p\}$ .

Fact 4.2. For a simple closed curve $c$ on $\unicode[STIX]{x1D6F4}$ , let $T_{c}$ denote the left-handed Dehn twist about $c$ . For $\unicode[STIX]{x1D6FE}$ a simple closed curve based at $p$ ,

$$\begin{eqnarray}P(\unicode[STIX]{x1D6FE})=T_{\unicode[STIX]{x1D6FE}_{L}}T_{\unicode[STIX]{x1D6FE}_{R}}^{-1}\in \operatorname{Mod}(\unicode[STIX]{x1D6F4},p)\end{eqnarray}$$

From a topological point of view, the monodromy $\unicode[STIX]{x1D707}$ is closely related to point-pushing maps. Recall from Proposition 2.3 our construction of the $W$ -bundle $E(X,m)\rightarrow B^{\prime }$ . The branched covering $W\rightarrow X$ factors as

$$\begin{eqnarray}s\,\circ \,t:W\rightarrow V\rightarrow X,\end{eqnarray}$$

with $s:V\rightarrow X$ an unramified $(\mathbb{Z}/m\mathbb{Z})^{2}$ -covering and $t:W\rightarrow V$ a ramified $\mathbb{Z}/m\mathbb{Z}$ -covering branched over $s^{-1}(p)$ for some point $p\in X$ . Following the ‘global’ treatment of $E^{\text{nn}}(X,m)$ given in § 2, we see that $E(X,m)$ arises as a $\mathbb{Z}/m\mathbb{Z}$ fiberwise branched covering

$$\begin{eqnarray}E(X,m)\rightarrow B^{\prime }\times V\end{eqnarray}$$

of the product bundle $B^{\prime }\times V\rightarrow B^{\prime }$ . The branch locus $\unicode[STIX]{x1D6E5}\subset B^{\prime }\times V$ can be described as follows. Let $\unicode[STIX]{x1D6E5}_{X}\subset X\times X$ denote the diagonal. There is a natural covering map

$$\begin{eqnarray}Q:B^{\prime }\times V\rightarrow X\times X\end{eqnarray}$$

(as both $B^{\prime }$ and $V$ arise as unbranched covers of $X$ ) and $\unicode[STIX]{x1D6E5}=Q^{-1}(\unicode[STIX]{x1D6E5}_{X})$ .

This description makes the connection with point-pushing maps apparent. Indeed, one can view the bundle $B^{\prime }\times V\rightarrow B^{\prime }$ as a trivial $V$ -bundle equipped with $m^{2}$ disjoint sections corresponding to $\unicode[STIX]{x1D6E5}$ . As mentioned after Proposition 3.1, there is a monodromy representation

$$\begin{eqnarray}\unicode[STIX]{x1D707}^{\prime }:\unicode[STIX]{x1D70B}_{1}(B^{\prime })\rightarrow \operatorname{Mod}(V,m^{2}).\end{eqnarray}$$

The monodromy about some loop $\unicode[STIX]{x1D6FE}\in B^{\prime }$ can be described in terms of point-pushing maps. Under the covering map $B^{\prime }\rightarrow X$ , the loop $\unicode[STIX]{x1D6FE}\subset B^{\prime }$ determines a loop on $X$ . Taking the preimage of $\unicode[STIX]{x1D6FE}$ under $s:V\rightarrow X$ , one obtains $m^{2}$ parameterized loops $\unicode[STIX]{x1D6FE}_{i,j}(t)$ on $V$ such that for each fixed $t$ , the $m^{2}$ points $\{\unicode[STIX]{x1D6FE}_{i,j}(t)\}$ are distinct. The monodromy $\unicode[STIX]{x1D707}^{\prime }(\unicode[STIX]{x1D6FE})$ is thus a simultaneous multipush along the curves $\unicode[STIX]{x1D6FE}_{i,j}$ . More generally, we can apply this construction to any loop $\unicode[STIX]{x1D6FE}\subset X$ , not merely those $\unicode[STIX]{x1D6FE}$ that lift to $B^{\prime }$ .

Given any covering $\unicode[STIX]{x1D6F4}\rightarrow \unicode[STIX]{x1D6F4}^{\prime }$ of surfaces, basic topology implies that there is a finite-index subgroup $\operatorname{LMod}\leqslant \operatorname{Mod}(\unicode[STIX]{x1D6F4}^{\prime })$ that lifts to $\unicode[STIX]{x1D6F4}$ , in the sense that there is a homomorphism $\ell :\operatorname{LMod}\rightarrow \operatorname{Mod}(\unicode[STIX]{x1D6F4})$ . The following lemma is immediate from the global topological construction of $E(X,m)$ given above.

Lemma 4.3. Let $\unicode[STIX]{x1D707}^{\prime }:\unicode[STIX]{x1D70B}_{1}(B^{\prime })\rightarrow \operatorname{Mod}(V,m^{2})$ be the simultaneous multipush map described above and let $\operatorname{LMod}\leqslant \operatorname{Mod}(V,m^{2})$ be the subgroup admitting a lift $\ell :\operatorname{LMod}\rightarrow \operatorname{Mod}(W,m^{2})$ . Then $\unicode[STIX]{x1D707}^{\prime }\mid _{\unicode[STIX]{x1D70B}_{1}(B^{\prime })}\leqslant \operatorname{LMod}$ . Consequently, there is a factorization

$$\begin{eqnarray}\unicode[STIX]{x1D707}=\ell \,\circ \,\unicode[STIX]{x1D707}^{\prime }.\end{eqnarray}$$

Lifting Dehn twists

The preceding analysis gives a satisfactory description of $\unicode[STIX]{x1D707}^{\prime }(\unicode[STIX]{x1D6FE})$ . To study $\unicode[STIX]{x1D707}$ , it therefore remains to understand the lifting map $\ell$ . We will be especially interested in a study of lifting Dehn twists $T_{c}$ . Our treatment here follows [Reference LooijengaLoo97, § 3.1], but there are some crucial differences arising from the fact that we are studying branched coverings. Recalling that a branched covering of compact Riemann surfaces becomes unbranched after deleting the branch locus, we formulate the results here for unbranched coverings of possibly non-compact Riemann surfaces.

The following lemma appears in [Reference LooijengaLoo97, § 3.1]. We will analyze when the reverse implication fails to hold in the next subsection.

Lemma 4.5. Let $q:\unicode[STIX]{x1D6F4}\rightarrow \unicode[STIX]{x1D6F4}^{\prime }$ be an unbranched cyclic $m$ -fold covering, classified by $f\in H^{1}(\unicode[STIX]{x1D6F4}^{\prime };\mathbb{Z}/m\mathbb{Z})$ . Let $c$ be a simple closed curve on $\unicode[STIX]{x1D6F4}^{\prime }$ . Then the Dehn twist power $T_{c}^{d}$ lifts to $\operatorname{Mod}(\unicode[STIX]{x1D6F4})$ if (but not necessarily only if) the equation $d\cdot f(c)=0$ holds in $\mathbb{Z}/m\mathbb{Z}$ .

The preimage $q^{-1}(c)$ has $[\mathbb{Z}/m\mathbb{Z}:\langle f(c)\rangle ]$ components, in correspondence with the set of cosets $(\mathbb{Z}/m\mathbb{Z})/\langle f(c)\rangle$ . Let $\widetilde{c}$ denote one such component. For $d=|f(c)|$ , where $|f(c)|$ denotes the order in $\mathbb{Z}/m\mathbb{Z}$ , there is a distinguished lift

$$\begin{eqnarray}\widetilde{T_{c}^{d}}=\mathop{\prod }_{g\in (\mathbb{Z}/m\mathbb{Z})/\langle f(c)\rangle }T_{g\cdot \widetilde{c}}.\end{eqnarray}$$

We wish to describe $\unicode[STIX]{x1D6F9}(\widetilde{T_{c}^{d}})\in \operatorname{Sp}(H_{1}(\unicode[STIX]{x1D6F4};\mathbb{Z}))$ . The formula is best expressed using the Reidemeister pairing $\langle \cdot ,\cdot \rangle$ described in § 3.

Proposition 4.6 (Cf. [Reference LooijengaLoo97, (3.1)]).

With all notation as above, $\unicode[STIX]{x1D6F9}(\widetilde{T_{c}^{d}})\in \operatorname{Sp}(H_{1}(\unicode[STIX]{x1D6F4};\mathbb{Z}))$ is given by

(15) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}(\widetilde{T_{c}^{d}})(x)=x+d^{-1}\langle x,\widetilde{c}\rangle [\widetilde{c}].\end{eqnarray}$$

Lifting separating Dehn twists

We return to the setting of Lemma 4.5. Our objective is to understand when a Dehn twist power $T_{c}^{d}$ lifts to a mapping class on $\unicode[STIX]{x1D6F4}$ even when the equation $d\cdot f(c)=0$ fails to hold. This phenomenon is a consequence of the degeneracy of the intersection pairing on a non-compact Riemann surface.

Proof. Choose $p\in \unicode[STIX]{x1D6F4}^{\prime }$ such that $T_{c}(p)=p$ . By covering space theory, $T_{c}$ lifts to $\unicode[STIX]{x1D6F4}$ if and only if $T_{c}$ preserves the subgroup $\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6F4})=\ker (f)$ of $\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6F4}^{\prime },p)$ . Since the cover $\unicode[STIX]{x1D6F4}\rightarrow \unicode[STIX]{x1D6F4}^{\prime }$ is abelian, it suffices to show that the action of $T_{c}$ on $H_{1}(\unicode[STIX]{x1D6F4}^{\prime },\mathbb{Z})$ is trivial. The result now follows, since the action of a Dehn twist $T_{c}$ on homology is given by the transvection formula

$$\begin{eqnarray}x\mapsto x+(x,c)[c]\end{eqnarray}$$

and $(x,c)=0$ for all $x$ since $c$ is separating.◻

The diffeomorphism $\widetilde{T_{c}}$ is ambiguously defined: there are $m$ distinct lifts of $T_{c}$ to $\unicode[STIX]{x1D6F4}$ , each differing by an element of the covering group for $q:\unicode[STIX]{x1D6F4}\rightarrow \unicode[STIX]{x1D6F4}^{\prime }$ . To fix a choice, observe that since $c$ is separating, there is a decomposition $\unicode[STIX]{x1D6F4}^{\prime }=\unicode[STIX]{x1D6F4}_{L}^{\prime }\cup \unicode[STIX]{x1D6F4}_{R}^{\prime }$ with $\unicode[STIX]{x2202}(\unicode[STIX]{x1D6F4}_{L}^{\prime })=\unicode[STIX]{x2202}(\unicode[STIX]{x1D6F4}_{R}^{\prime })=c$ . Choosing an orientation of $c$ distinguishes $\unicode[STIX]{x1D6F4}_{L}^{\prime }$ by the condition that $\unicode[STIX]{x1D6F4}_{L}^{\prime }$ lies to the left of $c$ . There is exactly one lift of $T_{c}$ to $\unicode[STIX]{x1D6F4}$ such that $q^{-1}(\unicode[STIX]{x1D6F4}_{L}^{\prime })$ is pointwise fixed: we take this as our definition of the distinguished lift $\widetilde{T_{c}}$ .

The aim of this subsection is again to determine $\unicode[STIX]{x1D6F9}(\widetilde{T_{c}})$ . There are various possibilities, depending on the value of $f(c)$ . This value depends on a choice of orientation on $c$ , which we fix once and for all. Our primary case of interest is when $f(c)=1\in \mathbb{Z}/m\mathbb{Z}$ .

Lemma 4.8. Let $c\subset \unicode[STIX]{x1D6F4}^{\prime }$ be separating and suppose that $f(c)=1$ . Necessarily $q^{-1}(c)$ is a single separating curve on $\unicode[STIX]{x1D6F4}$ that gives a decomposition $\unicode[STIX]{x1D6F4}=\unicode[STIX]{x1D6F4}_{L}\cup \unicode[STIX]{x1D6F4}_{R}$ . Then

$$\begin{eqnarray}H_{1}(\unicode[STIX]{x1D6F4};\mathbb{Z})\cong H_{1}(\unicode[STIX]{x1D6F4}_{L};\mathbb{Z})\oplus H_{1}(\unicode[STIX]{x1D6F4}_{R};\mathbb{Z}).\end{eqnarray}$$

Relative to this decomposition, $\unicode[STIX]{x1D6F9}(\widetilde{T_{c}})$ acts on $H_{1}(\unicode[STIX]{x1D6F4}_{L};\mathbb{Z})$ trivially, and on $H_{1}(\unicode[STIX]{x1D6F4}_{R};\mathbb{Z})$ by $\unicode[STIX]{x1D701}$ , where $\unicode[STIX]{x1D701}$ is the generator of the covering group of $q:\unicode[STIX]{x1D6F4}\rightarrow \unicode[STIX]{x1D6F4}^{\prime }$ .

Lifting a point-push

We now fix our attention on the branched covering $t:W\rightarrow V$ . Let $b\in L\subset V$ be a branch point and let $\unicode[STIX]{x1D6FE}$ be a simple closed loop on $V$ based at $b$ that is disjoint from $L\setminus \{b\}$ . As above, $\unicode[STIX]{x1D6FE}$ determines curves $\unicode[STIX]{x1D6FE}_{L},\unicode[STIX]{x1D6FE}_{R}\subset V^{\circ }$ . We seek a formula for $(\unicode[STIX]{x1D6F9}\,\circ \,\ell )(P(\unicode[STIX]{x1D6FE}^{k}))$ for $k$ such that $P(\unicode[STIX]{x1D6FE}^{k})$ lifts to $W^{\circ }$ .

Since $\unicode[STIX]{x1D6FE}_{L}\cup \unicode[STIX]{x1D6FE}_{R}$ bounds an annulus on $V$ containing $b$ , there is an equality in $H_{1}(V^{\circ };\mathbb{Z}/m\mathbb{Z})$ of the form

$$\begin{eqnarray}[\unicode[STIX]{x1D6FE}_{R}]=[\unicode[STIX]{x1D6FE}_{L}]+C,\end{eqnarray}$$

where, as above, $C$ denotes the homology class of a small loop encircling $b$ counterclockwise. Recall that the unbranched covering $t:W^{\circ }\rightarrow V^{\circ }$ is classified by $\unicode[STIX]{x1D703}\in H^{1}(V^{\circ };\mathbb{Z}/m\mathbb{Z})$ , where $\unicode[STIX]{x1D703}(C)=1$ . It now follows from the discussion of the preceding section that $P(\unicode[STIX]{x1D6FE}^{k})=T_{\unicode[STIX]{x1D6FE}_{L}}^{k}T_{\unicode[STIX]{x1D6FE}_{R}}^{-k}$ lifts to $Z^{\circ }$ if $m\mid k$ .

Lemma 4.9. Suppose that $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D6FE}_{L})=0$ . Then $(\unicode[STIX]{x1D6F9}\circ \ell )(P(\unicode[STIX]{x1D6FE}^{m}))$ is given by

$$\begin{eqnarray}(\unicode[STIX]{x1D6F9}\,\circ \,\ell )(P(\unicode[STIX]{x1D6FE}^{m}))(x)=x+(m-(1+\unicode[STIX]{x1D701}+\cdots +\unicode[STIX]{x1D701}^{m-1}))\langle x,\widetilde{\unicode[STIX]{x1D6FE}}_{L}\rangle _{t}[\widetilde{\unicode[STIX]{x1D6FE}}_{L}].\end{eqnarray}$$

Here $\langle \cdot ,\cdot \rangle _{t}$ denotes the Reidemeister pairing with respect to the $\langle \unicode[STIX]{x1D701}\rangle$ -covering $t:W\rightarrow V$ .

Proof. A formula for $(\unicode[STIX]{x1D6F9}\,\circ \,\ell )(P(\unicode[STIX]{x1D6FE}^{m}))$ can be found by applying the results of the previous section. Suppose first that $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D6FE}_{L})=0$ . Then $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D6FE}_{R})=1$ . Thus, $t^{-1}(\unicode[STIX]{x1D6FE}_{L})$ consists of $m$ disjoint components, while $t^{-1}(\unicode[STIX]{x1D6FE}_{R})$ is a single curve. Moreover, the annulus bounded by $\unicode[STIX]{x1D6FE}_{L},\unicode[STIX]{x1D6FE}_{R}$ on $V$ lifts to a surface with these $m+1$ boundary components. On the level of homology, this implies that

$$\begin{eqnarray}(1+\unicode[STIX]{x1D701}+\cdots +\unicode[STIX]{x1D701}^{m-1})[\widetilde{\unicode[STIX]{x1D6FE}}_{L}]=[\widetilde{\unicode[STIX]{x1D6FE}}_{R}].\end{eqnarray}$$

From Proposition 4.6,

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(\widetilde{T_{\unicode[STIX]{x1D6FE}_{L}}^{m}})(x)=x+m\langle x,\widetilde{\unicode[STIX]{x1D6FE}}_{L}\rangle _{t}[\widetilde{\unicode[STIX]{x1D6FE}}_{L}],\end{eqnarray}$$

while

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(\widetilde{T_{\unicode[STIX]{x1D6FE}_{R}}^{-m}})(x)=x-m^{-1}\langle x,\widetilde{\unicode[STIX]{x1D6FE}}_{R}\rangle _{t}[\widetilde{\unicode[STIX]{x1D6FE}}_{R}].\end{eqnarray}$$

As $[\widetilde{\unicode[STIX]{x1D6FE}}_{R}]=(1+\cdots +\unicode[STIX]{x1D701}^{m-1})[\widetilde{\unicode[STIX]{x1D6FE}}_{L}]$ , the skew-Hermitian property of the Reidemeister pairing and the equation $(1+\cdots +\unicode[STIX]{x1D701}^{m-1})^{2}=m(1+\cdots +\unicode[STIX]{x1D701}^{m-1})$ imply the formula.◻

Monodromy of clean elements

Our analysis of $\unicode[STIX]{x1D707}$ hinges on a study of a special class of elements of $\unicode[STIX]{x1D70B}_{1}(X,x_{0})$ .

If $\unicode[STIX]{x1D6FE}$ is clean, then $s^{-1}(\unicode[STIX]{x1D6FE})\subset V$ consists of $m^{2}$ disjoint components which are permuted by the covering group $\langle \unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\rangle \cong (\mathbb{Z}/m\mathbb{Z})^{2}$ of $s:V\rightarrow X$ . Choosing a distinguished lift $\widetilde{\unicode[STIX]{x1D6FE}}$ , the monodromy $\unicode[STIX]{x1D707}^{\prime }(\unicode[STIX]{x1D6FE})$ then consists of $m^{2}$ point-push maps about the disjoint curves $\unicode[STIX]{x1D70E}^{i}\unicode[STIX]{x1D70F}^{j}\widetilde{\unicode[STIX]{x1D6FE}}$ .

Lemma 4.12. Suppose that $\unicode[STIX]{x1D6FE}$ is clean. Then $\unicode[STIX]{x1D6FE}^{m}$ lifts to an element of $\operatorname{Mod}(W)$ , in the sense that $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D6FE}^{m})$ is defined. For any $x\in H_{1}(W)$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6FE}^{m})(x)=x+(m-(1+\unicode[STIX]{x1D701}+\cdots +\unicode[STIX]{x1D701}^{m-1}))\langle x,\widetilde{\unicode[STIX]{x1D6FE}_{L}}\rangle _{s\,\circ \,t}[\widetilde{\unicode[STIX]{x1D6FE}_{L}}].\end{eqnarray}$$

Here $\langle \cdot ,\cdot \rangle _{s\,\circ \,t}$ denotes the Reidemeister pairing with respect to the $H$ -covering $s\,\circ \,t:W\rightarrow X$ .

Proof. The monodromy $\unicode[STIX]{x1D70C}$ factors as $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D6F9}\,\circ \,\ell \,\circ \,\unicode[STIX]{x1D707}^{\prime }$ . Topologically, $\unicode[STIX]{x1D707}^{\prime }(\unicode[STIX]{x1D6FE})$ consists of the simultaneous point-push maps about the $m^{2}$ disjoint curves $\unicode[STIX]{x1D70E}^{i}\unicode[STIX]{x1D70F}^{j}\widetilde{\unicode[STIX]{x1D6FE}}$ . By the $(\mathbb{Z}/m\mathbb{Z})^{2}$ -symmetry, this is given by

$$\begin{eqnarray}\unicode[STIX]{x1D707}^{\prime }(\unicode[STIX]{x1D6FE})=\mathop{\prod }_{(i,j)\in (\mathbb{Z}/m\mathbb{Z})^{2}}(\unicode[STIX]{x1D70E}^{i}\unicode[STIX]{x1D70F}^{j})\cdot P(\widetilde{\unicode[STIX]{x1D6FE}}).\end{eqnarray}$$

The result now follows from applying Lemma 4.9:

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6FE}^{m})(x) & = & \displaystyle \mathop{\prod }_{(i,j)\in (\mathbb{Z}/m\mathbb{Z})^{2}}(\unicode[STIX]{x1D70E}^{i}\unicode[STIX]{x1D70F}^{j})(\unicode[STIX]{x1D6F9}\,\circ \,\ell )(P(\widetilde{\unicode[STIX]{x1D6FE}}^{m}))(x)\nonumber\\ \displaystyle & = & \displaystyle x+\mathop{\sum }_{(i,j)\in (\mathbb{Z}/m\mathbb{Z})^{2}}(\unicode[STIX]{x1D70E}^{i}\unicode[STIX]{x1D70F}^{j})\cdot (m-(1+\cdots +\unicode[STIX]{x1D701}^{m-1}))\langle x,\widetilde{\unicode[STIX]{x1D6FE}_{L}}\rangle _{t}[\widetilde{\unicode[STIX]{x1D6FE}_{L}}]\nonumber\\ \displaystyle & = & \displaystyle x+(m-(1+\cdots +\unicode[STIX]{x1D701}^{m-1}))\langle x,\widetilde{\unicode[STIX]{x1D6FE}_{L}}\rangle _{s\,\circ \,t}[\widetilde{\unicode[STIX]{x1D6FE}_{L}}].\Box \nonumber\end{eqnarray}$$

There is a subclass of clean elements which will be of particular importance.

Lemma 4.14. Suppose that $\unicode[STIX]{x1D6FE}$ is $W$ -separating. Then $\unicode[STIX]{x1D6FE}$ (and not merely $\unicode[STIX]{x1D6FE}^{m}$ ) lifts to an element of $\operatorname{Mod}(W)$ . Such $\unicode[STIX]{x1D6FE}$ induces a decomposition

$$\begin{eqnarray}H_{1}(W)=H_{1}(W^{\prime })\oplus H_{1}(W^{\prime \prime })\end{eqnarray}$$

with $W^{\prime }$ the subsurface of $W$ lying to the left of $t^{-1}(\unicode[STIX]{x1D6FE}_{R})$ . Relative to this, $\unicode[STIX]{x1D6FE}$ acts on $x\in H_{1}(W)$ via the formula

$$\begin{eqnarray}x\mapsto \left\{\begin{array}{@{}ll@{}}x,\quad & x\in H_{1}(W^{\prime }),\\ \unicode[STIX]{x1D701}^{-1}\cdot x,\quad & x\in H_{1}(W^{\prime \prime }).\end{array}\right.\end{eqnarray}$$

Proof. As discussed above,

(16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}^{\prime }(\unicode[STIX]{x1D6FE}) & = & \displaystyle \mathop{\prod }_{(i,j)\in (\mathbb{Z}/m\mathbb{Z})^{2}}(\unicode[STIX]{x1D70E}^{i}\unicode[STIX]{x1D70F}^{j})\cdot P(\widetilde{\unicode[STIX]{x1D6FE}})\end{eqnarray}$$

(17) $$\begin{eqnarray}\displaystyle & = & \displaystyle \mathop{\prod }_{(i,j)\in (\mathbb{Z}/m\mathbb{Z})^{2}}(\unicode[STIX]{x1D70E}^{i}\unicode[STIX]{x1D70F}^{j})\cdot T_{\widetilde{\unicode[STIX]{x1D6FE}_{L}}}T_{\widetilde{\unicode[STIX]{x1D6FE}_{R}}}^{-1}.\end{eqnarray}$$

The assumption implies that $\widetilde{\unicode[STIX]{x1D6FE}_{L}}$ and $\widetilde{\unicode[STIX]{x1D6FE}_{R}}$ are both separating on $V$ and hence on $V^{\circ }$ . By Lemma 4.7 (as applied to the unbranched covering $t:W^{\circ }\rightarrow V^{\circ }$ ), both $T_{\widetilde{\unicode[STIX]{x1D6FE}_{L}}}$ and $T_{\widetilde{\unicode[STIX]{x1D6FE}_{R}}}^{-1}$ lift to elements of $\operatorname{Mod}(W)$ . As $\unicode[STIX]{x1D6FE}$ is clean, $\unicode[STIX]{x1D703}(\widetilde{\unicode[STIX]{x1D6FE}_{L}})=0$ by assumption. It follows that $\widetilde{\unicode[STIX]{x1D6FE}_{L}}$ lifts to a collection of $m$ curves on $W$ , each of which is separating by assumption. Thus, the action of the lift of $T_{\widetilde{\unicode[STIX]{x1D6FE}_{L}}}$ on $H_{1}(W)$ is trivial.

Since $\unicode[STIX]{x1D703}(\widetilde{\unicode[STIX]{x1D6FE}_{L}})=0$ , it follows that $\unicode[STIX]{x1D703}(\widetilde{\unicode[STIX]{x1D6FE}_{R}})=1$ . Lemma 4.8 can therefore be applied to give a formula for the action of $T_{\widetilde{\unicode[STIX]{x1D6FE}_{R}}}^{-1}$ on $H_{1}(W)$ . The claimed formula now follows by combining this and (17).◻

Curve-arc sums

In order to make the subsequent work with (fairly elaborate) simple closed curves as painless as possible, we introduce here two operations on curves.

The first of these is the curve-arc sum procedure. Let $\unicode[STIX]{x1D6F4}$ be a surface and $\unicode[STIX]{x1D6FE}_{1},\unicode[STIX]{x1D6FE}_{2}$ be disjoint oriented simple closed curves on $\unicode[STIX]{x1D6F4}$ . Let $\unicode[STIX]{x1D6FC}$ be an arc on $\unicode[STIX]{x1D6F4}$ beginning at some point on the left-hand side of $\unicode[STIX]{x1D6FE}_{1}$ and ending on the left-hand side of $\unicode[STIX]{x1D6FE}_{2}$ that is otherwise disjoint from $\unicode[STIX]{x1D6FE}_{1}\cup \unicode[STIX]{x1D6FE}_{2}$ . The curve-arc sum of $\unicode[STIX]{x1D6FE}_{1}$ and $\unicode[STIX]{x1D6FE}_{2}$ along $\unicode[STIX]{x1D6FC}$ is the simple closed curve $\unicode[STIX]{x1D6FE}_{1}+_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FE}_{2}$ defined pictorially in Figure 3.

De-crossing

The second operation we will require is de-crossing. Suppose that $\unicode[STIX]{x1D6FE}\subset \unicode[STIX]{x1D6F4}$ is non-simple, with a self-intersection at $p\in \unicode[STIX]{x1D6F4}$ . Suppose that $S\subset \unicode[STIX]{x1D6F4}$ is a subsurface with $S\cong \unicode[STIX]{x1D6F4}_{1,1}$ such that $S\,\cap \,\unicode[STIX]{x1D6FE}$ contains only the self-intersection at $p$ . Then the de-crossing of $\unicode[STIX]{x1D6FE}$ along $S$ is the curve $DC(\unicode[STIX]{x1D6FE},S)$ with one fewer self-intersection depicted in Figure 4.

Figure 3.

The curve-arc sum (the orientations on $\unicode[STIX]{x1D6FE}_{1},\unicode[STIX]{x1D6FE}_{2}$ are implicit).

Figure 4.

De-crossing of $\unicode[STIX]{x1D6FE}$ along $S$ . The two local strands of $\unicode[STIX]{x1D6FE}$ have been depicted in different stroke styles for clarity.

In practice, the portion of $S$ connecting $p$ to the rest of $S$ can be quite long and thin. Where the clarity of a figure dictates, this will sometimes be depicted as an arc connecting $p$ to some genus 1 subsurface.

Non-simple curves will arise as the image of simple curves under covering maps. Suppose that $f:\unicode[STIX]{x1D6F4}\rightarrow \unicode[STIX]{x1D6F4}^{\prime }$ is a regular covering of surfaces with deck group $G$ . Let $p\in \unicode[STIX]{x1D6F4}^{\prime }$ be given and identify the fiber $f^{-1}(p)$ with the set $G\cdot p$ . If $\unicode[STIX]{x1D6FE}\subset \unicode[STIX]{x1D6F4}$ passes through points $g\cdot p,h\cdot p\in \unicode[STIX]{x1D6F4}$ , then the image $f(\unicode[STIX]{x1D6FE})$ will have a double point at $p$ . In this situation, we say that $\unicode[STIX]{x1D6FE}$ has local branches in sheets $g,h$ . The following lemma records some properties of the de-crossing procedure in this context.