2.1 Four equivalence relations on paths and loops on maps

We recall briefly here standard notions and define some notations of topological discretisation of a surface.

This definition might seem very abstract at first sight, but it is actually simple: it merely says that a graph is made of edges and vertices, and that each edge is incident to either 1 vertex (the edge is then called a loop) or 2 vertices. Let ${\mathbb {G}}=(V,E,I)$ be a graph, and $e_1,e_2\in E$ be two distinct edges.

1. 1. If there is $v\in V$ such that $(v,e_1)\in I$ and $(v,e_2)\in I$ , then $e_1$ and $e_2$ are called adjacent.

2. 2. If there are $v_1,v_2\in V$ such that $(v_i,e_j)\in I$ for all $1\leq i,j\leq 2$ , then $e_1$ and $e_2$ form a double edge.

More generally, if n edges share the same incidence vertices, they form a multiple edge, and ${\mathbb {G}}$ is called a multigraph. A topological map M on a surface $\Sigma $ is a finite multigraph ${\mathbb {G}}=(V,E,I)$ together with an embedding $\theta :{\mathbb {G}}\to \Sigma $ such that

• • the images of two distinct vertices $v_1,v_2\in V$ by $\theta $ are distinct points of $\Sigma $ ,

• • for any edge $e\in E$ , there is an edge $e^{-1}\in E$ such that $\theta _{e^{-1}}=\theta _e^{-1}$ ,

• • the images of edges $e\in E$ are continuous curves $\theta _e:[0,1]\to \Sigma $ with endpoints $\underline e = \theta _e(0)$ and $\overline e = \theta _e(1)$ such that $\theta _e$ and $\theta _{e'}$ can only intersect at their endpoints (unless $e'\in \{e,e^{-1}\}$ ),

• • the complement of the skeleton $\mathrm {Sk}({\mathbb {G}})=\bigcup _{e\in E}\theta _e$ of ${\mathbb {G}}$ in $\Sigma $ is split in one or several connected components that are all homeomorphic to discs and represent the faces of the map.

An orientation of the map is the choice of a subset $E_+\subset E$ such that for any $e\in E$ , $\vert \{e,e^{-1}\}\cap E_+\vert =1$ . The orientation of $\Sigma $ also induces an orientation of the faces as follows: a face f is positively oriented if its boundary $\partial f$ is represented by $e_1\cdots e_n$ , where $e_1,\ldots ,e_n$ are the edges constituting $\partial f$ in positive order. It is negatively oriented if its boundary is represented by $e_n^{-1}\cdots e_1^{-1}$ . We denote by F (resp. $F_+$ ) the set of all faces with both orientations (resp. the positively oriented faces), and for any $f\in F_+$ , we denote by $f^{-1}\in F$ the same face with reverse orientation.

Remark 2.2. With our conventions, each unoriented edge and unoriented face is counted twice; therefore, Euler’s formula reads

$$\begin{align*}\vert V\vert -\tfrac12\vert E\vert + \tfrac12\vert F\vert = \vert V\vert - \vert E_+\vert + \vert F_+\vert = 2-2g \end{align*}$$

if ${\mathbb {G}}$ is embedded in a surface of genus g. See Figure 5 for an illustration.

Figure 5

A map embedded in the torus, with $\vert V\vert = 1$ , $\vert E_+\vert = 3$ and $\vert F_+\vert = 2$ . The edges named a are glued together, and same for the edges named b. There are two positively oriented faces, with respective boundaries $ea^{-1}b$ and $ab^{-1}e^{-1}$ ; the orientations are represented by the counterclockwise green arrows. Euler’s formula is indeed satisfied.

From now on, we will denote by ${\mathbb {G}}=(V,E,F)$ a topological map, and $\theta $ and $\Sigma $ will be implied. Given ${\mathbb {G}}=(V,E,F)$ , one can describe a CW-complex corresponding to the map: its vertices are 0-cells, its edges 1-cells and its faces 2-cells. Besides, the skeleton of the map is exactly the skeleton of the complex. We will describe the corresponding chain and cochain complexes in the next section, as well as their (co)homology. In this section, we rather focus on topological features of maps and algebraic properties of loops in a map.

A map with boundary is a map $(V,E,F)$ together with a proper subset B of F such that the closures of 2-cells associated to any pair of distinct elements of B do not intersect. A path in ${\mathbb {G}}$ Footnote ²⁸ is either a single vertex or a finite string of edges $e_1\ldots e_n$ with $n\ge 1$ such that for all $k\in \{1,\ldots ,n-2\}$ , $\underline {e}_{k+1}=\overline {e}_k.$ We say it is constant in the first case and set $|\gamma |=0$ , while in the second, we denote by $\overline \gamma = \overline e_n$ and $\underline \gamma = \underline e_1$ its endpoint and starting point, and by $|\gamma |=n$ its length. A loop of ${\mathbb {G}}$ is a path $\gamma $ with $\underline \gamma =\overline \gamma .$ A loop $\ell $ is based at a vertex v when $\underline {\ell }=v.$ We say it is simple when all vertices of $\ell $ occur only once $\ell $ , but $\underline {\ell }$ which occurs exactly twice. We write, respectively, $\mathrm {P}({\mathbb {G}})$ and $\mathrm {L}({\mathbb {G}})$ for the set of paths and loops of ${\mathbb {G}}.$ The respective sets of paths starting from a vertex $v\in V$ are denoted by $\mathrm {P}_v({\mathbb {G}})$ and $\mathrm {L}_v({\mathbb {G}}).$ Whenever $\alpha $ and $\beta $ are two paths with $\overline \alpha =\underline \beta $ , $\alpha \beta $ denotes their concatenation, while $\alpha ^{-1}$ is the path run in reverse direction, with the convention that $\gamma _1\alpha \gamma _2=\alpha $ when $\gamma _1$ and $\gamma _2$ are constant paths at $\underline \alpha $ and $\overline \alpha $ . We say that $\beta $ is a subpath of $\delta \in \mathrm {P}({\mathbb {G}})$ and write $\beta \prec \delta $ , if there are paths $\alpha $ and $\gamma $ with $\delta =\alpha \beta \gamma .$

Homeomorphic loops: When two maps ${\mathbb {G}},{\mathbb {G}}'$ yield homeomorphic CW complexes, they induce a bijection between cells of same dimension. Denote by $\Phi : E\to E'$ the associated bijection between edges of ${\mathbb {G}}$ and ${\mathbb {G}}'$ and the associated bijection between $\mathrm {P}({\mathbb {G}})$ and $\mathrm {P}({\mathbb {G}}').$ Consider two paths $\alpha $ and $\beta $ within maps ${\mathbb {G}}_1$ and ${\mathbb {G}}_2$ . We say that $\alpha $ and $\beta $ are homeomorphic and write

$$ \begin{align*}\alpha\sim_\Sigma\beta\end{align*} $$

if there are maps ${\mathbb {G}}$ and ${\mathbb {G}}'$ finer than, respectively, ${\mathbb {G}}_1$ and ${\mathbb {G}}_2$ such that ${\mathbb {G}}$ and ${\mathbb {G}}'$ are homeomorphic, with induced bijection $\Phi :\mathrm {P}({\mathbb {G}})\to \mathrm {P}({\mathbb {G}}')$ such that

$$ \begin{align*}\Phi(\alpha)=\beta.\end{align*} $$

Cyclically equivalent loops: We say that two loops are cyclically equivalent when one can be obtained from the other by cyclically permuting its edges. By convention, two constant loops are cyclically equivalent if they have equal base-point. This defines an equivalence relation $\sim _c$ on $\mathrm {L}({\mathbb {G}})$ . An element of the quotient $\mathrm {L}_c({\mathbb {G}})=\mathrm {L}({\mathbb {G}})/\sim _c$ is called an unrooted loop.

Reduced loops: A path $\gamma '$ is obtained by insertion of an edge in a path $\gamma $ if $\gamma =\gamma _1\gamma _2$ and $\gamma '=\gamma _1ee^{-1}\gamma _2$ with $\gamma _1,\gamma _2$ two subpaths of $\gamma $ and e an edge such that $\overline {\gamma _1}=\underline e=\underline {\gamma _2}.$ Vice versa, we say in this situation that $\gamma $ is obtained by erasing of an edge of $\gamma '$ . Two paths are said to have the same reduction if a finite sequence of erasures and insertions of edges transforms one into the other. This defines an equivalence relation $\sim _r$ on $\mathrm {P}({\mathbb {G}})$ , and we write $\mathrm {RP}({\mathbb {G}})=\mathrm {P}({\mathbb {G}})/\sim _r$ , $\mathrm {RP}_v({\mathbb {G}})=\mathrm {P}_v({\mathbb {G}})/\sim _r$ and $\mathrm {RL}_v({\mathbb {G}})=\mathrm {L}_v({\mathbb {G}})/\sim _r$ for any $v\in V$ . The reduction of a path $\gamma \in \mathrm {P}({\mathbb {G}})$ is the unique path of minimal length in its $\sim _r$ -equivalence class. We say that two loops are $\sim _{r,c}$ -equivalent if one can be obtained from the other by iterated cyclic permutations, insertions and erasures of edges.

Lassos and discrete homotopy: For any face $f\in F$ , its boundary can be identified with an unrooted loop $\partial f.$ When $r\in V$ is a vertex of $\partial f,$ we write $\partial _r f$ for the loop in the $\sim _c$ -class of $\partial f$ with $\underline {\partial _r f}= r.$ When $F_{*}$ is a subset of $F,$ a $F_{*}$ -lasso is a loop of the form $\alpha \partial _r f \alpha ^{-1},$ where f is an oriented face belonging up to orientation to $F_{*}$ and $\alpha \in \mathrm {P}({\mathbb {G}})$ is a path such that $r=\overline \alpha $ is a vertex of $\partial f.$ When $\gamma \in \mathrm {P}({\mathbb {G}})$ , $\gamma '$ is obtained by lasso insertion from $\gamma $ if $\gamma =\gamma _1\gamma _2$ for some paths $\gamma _1,\gamma _2\in \mathrm {P}({\mathbb {G}})$ and $\gamma '= \gamma _1 \ell \gamma _2,$ where $\ell $ is a lasso with $\overline {\gamma }_1=\underline {\ell }=\underline {\gamma }_2.$ Conversely, $\gamma '$ is said to be obtained from $\gamma $ by lasso erasure. We say that two paths are discrete homotopic if there is a finite sequence of lassos or edge erasures and insertions transforming one into the other. This defines an equivalence relation $\sim _h$ on $\mathrm {P}({\mathbb {G}})$ which is also well defined on $\mathrm {RP}({\mathbb {G}}).$ Moreover, two paths of ${\mathbb {G}}$ are discrete homotopic if and onlyFootnote ²⁹ if their images in $\Sigma _{\mathbb {G}}$ are homotopic with fixed endpoints. For any $v\in V,$ we denote the quotient $\mathrm {P}_v({\mathbb {G}})/\sim _h$ and $\mathrm {L}_v({\mathbb {G}})/\sim _h$ by $\tilde V_v$ and $\pi _{1,v}({\mathbb {G}}).$ When $F_{*}\subset F,$ we say that two paths of ${\mathbb {G}}$ are $F_{*}$ -homotopic if there is a finite sequence of $F_{*}$ -lassos or edge erasures and insertions transforming one into the other. This defines an equivalence relation on $\mathrm {P}({\mathbb {G}})$ denoted by $\sim _{F_{*}}\!.$ When K is a closed, compact, contractible subset of $\Sigma _{\mathbb {G}}$ given by the closure of the union of images of $F_{*}$ , for any pair of paths $\gamma _1,\gamma _2\in \mathrm {P}({\mathbb {G}})$ whose image in $\Sigma _{\mathbb {G}}$ is included in K and with same endpoints,Footnote ³⁰ $\gamma _1\sim _{F_{*}}\gamma _2.$

The group of reduced loops and the fundamental group: For any vertex $v\in V$ , we define a group by endowing $\mathrm {RL}_v({\mathbb {G}})$ with the multiplication given by concatenation and the inverse map given by reversing the orientation of loops. The group $\pi _{1,v}({\mathbb {G}})$ is the quotient of $\mathrm {RL}_v({\mathbb {G}})$ by the normal subgroup generated by lassos based at $v.$ Since two loops of ${\mathbb {G}}$ are discrete homotopic if and only if their images in $\Sigma _{\mathbb {G}}$ are homotopic, the group $\pi _{1,v}({\mathbb {G}})$ is isomorphic to the fundamental group of the surface $\Sigma _{\mathbb {G}}$ . For any group G, let us write $[a,b]=aba^{-1}b^{-1},\ \forall a,b\in G.$ Then $\pi _{1,v}({\mathbb {G}})$ is isomorphic to the surface group

$$ \begin{align*}\Gamma_g={\langle} x_1,y_1,\ldots, x_g,y_g| [x_1,y_1]\ldots [x_g,y_g]{\rangle}.\end{align*} $$

We will also consider, for $r\geq 1$ , the group

$$\begin{align*}\Gamma_{r,g}={\langle}z_1,\ldots,z_{r}, x_1,y_1,\ldots, x_g,y_g\ |\ z_1\ldots z_{r} = [x_1,y_1]\ldots [x_g,y_g]{\rangle}. \end{align*}$$

Refining maps: When ${\mathbb {G}}'=(V',E',F')$ and ${\mathbb {G}}=(V,E,F)$ are two maps, ${\mathbb {G}}'$ is finer than ${\mathbb {G}}$ if $(V,E)$ is a subgraph of $(V',E')$ and $\Sigma _{{\mathbb {G}}'}=\Sigma _{\mathbb {G}}$ , so that we can identify V and E with subsets of, respectively, $V'$ and $\mathrm {P}({\mathbb {G}}),$ while any face of ${\mathbb {G}}$ is the union of faces of ${\mathbb {G}}'.$ Conversely, we say that ${\mathbb {G}}$ is coarser then ${\mathbb {G}}'$ .

Dual map: When ${\mathbb {G}}=(V,E,F)$ is a map of genus g with surface $\Sigma _{\mathbb {G}},$ we define its dual map as follows: we put a vertex $f^{*}$ inside each face $f\in F$ , and for each edge $e\in E$ that separates two faces $f_1$ and $f_2$ , we draw a new edge $e^{*}$ that intersects it in its midpoint and connects the vertices $f_1^{*}$ and $f_2^{*}$ . There is a bijection $V^{*}\simeq F$ , $E^{*}\simeq E$ and $F^{*}\simeq V$ , and a dual edge inherits the orientation from the edges it crosses as follows: if $e^{*}$ crosses $e\in E_+$ from the right,Footnote ³² then $e^{*}\in E_+^{*}$ . See Figure 6 for an illustration. In particular, we see that if $e=(\underline e,\overline e)$ is an edge and $e^{*}=(\underline {e^{*}},\overline {e^{*}})$ is the dual edge, then we have the following facts:

(16) $$ \begin{align} e^{*}\in\partial \underline{e},\ (e_{*}^{-1})\in\partial\overline{e},\ e\in\partial\overline{e^{*}},\ e^{-1}\in\partial\underline{e^{*}}. \end{align} $$

Figure 6

On the left: a map embedded in the sphere (in plain lines), and its dual map (in dashed lines). On the right: the orientation convention of an edge and its dual. We have $ \underline {e}, \overline {e}\in V=F^{*}$ and $ \underline {e^{*}}, \overline {e^{*}}\in F=V^{*}$ .

Cut of a map: When ${\mathbb {G}}=(V,E,F)$ is a map and $\ell $ is a simple loop of ${\mathbb {G}},$ with dual edges $E_\ell ^{*}$ , we say that $\ell $ is separating if the graph $(F,E^{*}\setminus E_\ell ^{*})$ has exactly two connected components $(F_1,E_1^{*})$ and $(F_2,E_2^{*})$ . Consider $i\in \{1,2\}.$ Denote by $E_i$ the union of $E_\ell $ with the set of edges of ${\mathbb {G}}$ dual to $E_i^{*}$ , and by $V_i$ the vertices of ${\mathbb {G}}$ endpoints of edges in $E_i$ . We then define a map with one boundary component by setting ${\mathbb {G}}_i=(V_i,E_i,F_i\sqcup \{f_{i,\infty }\})$ , where $\{f_i,\infty \}$ is the label of a boundary face with boundary $\ell .$ We say that the pair of maps with boundary $({\mathbb {G}}_1,\{f_{1,\infty }\}),({\mathbb {G}}_2,\{f_{2,\infty }\})$ is the cut of ${\mathbb {G}}$ along $\ell .$ We say that the cut is essential if $\ell $ is not contractible. A cut is essential if and only if the maps ${\mathbb {G}}_1$ and ${\mathbb {G}}_2$ have genus larger or equal to $1.$ Both cases are illustrated in Figure 7 below. When a map is cut, the lemma 2.3 can be specified as follows.

Figure 7

Two examples of map cuts. In the first line, a genus 2 map is cut along $\ell $ into two genus 1 maps with boundary $\ell $ ; hence, it is an essential cut. In the second line, a genus 1 map is cut into a genus 1 and a genus 0 maps, with boundary $\ell $ . It is therefore not essential.

2.2 Discrete homology, winding function and Makeenko–Migdal vectors

We recall here some elementary results about the homology of topological maps and discuss their relation to Makeenko–Migdal vectors introduced in [Reference Lévy43, Reference Dahlqvist and Norris21, Reference Hall33]. It will lead us to a construction of the winding function, as well as a characterisation of the Makeenko–Migdal vectors, which, as we recall, encode the deformations that are allowed by Makeenko–Migdal equations. In the sequel, R will denote a ring that is either $\mathbb {R}$ or $\mathbb {Z}$ , unless specified otherwise. We shall start with a general property of finitely generated free modules.

Proof. The first point is obvious: we set $\langle e_i,e_j\rangle =\delta _{ij}$ , and we extend the form by bilinearity. For the second point, set

$$\begin{align*}\Phi:\left\lbrace\begin{array}{ccc} A & \longrightarrow & \mathrm{Hom}(A,R)\\ x & \longmapsto & (y\mapsto \langle x,y\rangle) \end{array}\right. \end{align*}$$

and notice that it is indeed an isomorphism.

We can present the homology of a topological map from several equivalent ways, and we will present three of them.

Definition 2.6. Let ${\mathbb {G}}=(V,E,F)$ be a topological map. We define its associated chain complex by the sequence

$$\begin{align*}0\longrightarrow C_2({\mathbb{G}};R)\mathrel{\mathop{\kern 0pt\longrightarrow}\limits_{}^{\partial}} C_1({\mathbb{G}};R)\mathrel{\mathop{\kern 0pt\longrightarrow}\limits_{}^{\partial}} C_0({\mathbb{G}};R)\longrightarrow 0, \end{align*}$$

where $C_0({\mathbb {G}};R)$ (resp. $C_1({\mathbb {G}};R)$ , $C_2({\mathbb {G}};R)$ ) is the free R-module generatedFootnote ³³ by V (resp. E, F). The boundary operator is defined by linear extension of the boundary operator in the underlying surface, defined by

$$ \begin{align*} \partial e = & \overline e - \underline e,\ \forall e\in E,\\ \partial f = & \sum_{e\in\partial f} e,\ \forall f\in F. \end{align*} $$

Let ${\mathbb {G}}=(V,E,F)$ be a topological map, and let ${\mathbb {G}}^{*}=(V^{*},E^{*},F^{*})$ be its dual map. For any $v\in V=F^{*}$ , we define its boundary $\partial v$ as the cycle $e_1^{*}\cdots e_n^{*}$ of dual edges constituting the positively oriented boundary of the face v.

Definition 2.7. Let ${\mathbb {G}}=(V,E,F)$ be a topological map. Its associated cochain complex is defined by the sequence

$$\begin{align*}0\longleftarrow \Omega^2({\mathbb{G}},R)\mathrel{\mathop{\kern 0pt\longleftarrow}\limits_{}^{d}} \Omega^1({\mathbb{G}},R)\mathrel{\mathop{\kern 0pt\longleftarrow}\limits_{}^{d}} \Omega^0({\mathbb{G}},R)\longleftarrow 0, \end{align*}$$

where $\Omega ^k({\mathbb {G}},R)=\mathrm {Hom}(C_k({\mathbb {G}};R),R)$ for any $0\leq k\leq 2$ , and d is the dual of the boundary operator:

$$ \begin{align*} df(e) = & f(\partial e) = f(\overline e) - f(\underline e), \ \forall e\in E, \ \forall f\in\Omega^0({\mathbb{G}},R),\\ d\omega(f) = & \omega(\partial f) = \sum_{e\in\partial f} \omega(e),\ \forall f\in F,\ \forall \omega\in\Omega^1({\mathbb{G}},R). \end{align*} $$

The elements of $\Omega ^k({\mathbb {G}},R)$ are called R-valued k-forms on ${\mathbb {G}}$ .

Proof. Let us first note that, as free modules, we have indeed canonical isomorphisms $\Phi :C_k({\mathbb {G}}^{*};R)\cong \Omega ^{2-k}({\mathbb {G}},R)$ for $0\leq k\leq 2$ . They are explicitely given as follows:

$$\begin{align*}f_v:v'\in V\mapsto \delta_{vv'},\ \forall v\in F_+^{*}\simeq V, \end{align*}$$

$$\begin{align*}\omega_e:e'\in E_+\mapsto \delta_{ee'},\ \forall e^{*}\in E_+^{*}\simeq E_+, \end{align*}$$

$$\begin{align*}\mu_f:f'\in F_+\mapsto \delta_{ff'},\ \forall f\in V^{*}\simeq F_+. \end{align*}$$

Using (16) in conjunction with the definitions of $\partial $ and d, one can easily find that the diagram

commutes, which proves the isomorphism.

We shall denote by $\mu _{*}$ the constant 2-form defined by $\mu _{*}(f)=1$ for all $f\in F_+$ . Thanks to Proposition 2.5, there is for any $0\leq k\leq 2$ a canonical isomorphism $\Phi :C_k({\mathbb {G}};R)\mathrel {\mathop {\kern 0pt\longrightarrow }\limits _{}^{\cong }}\Omega ^k({\mathbb {G}};R)$ , represented by the applications $v\mapsto f_v$ , $e\mapsto \omega _e$ and $f\mapsto \mu _f$ used in the proof of Proposition 2.8.

We define $d^{*}$ as the adjoint of d on the cochain complex of ${\mathbb {G}}$ , meaning that

$$ \begin{align*} d^{*}\omega = & \sum_{e\in E_+}\omega(e)f_{\partial e}=\sum_{e\in E_+}\sum_{v\in \partial e} \omega(e)f_v,\ \forall \omega\in\Omega^1({\mathbb{G}},R),\\ d^{*}\mu = & \sum_{f\in F_+} \mu(f)\omega_{\partial f} = \sum_{f\in F_+}\sum_{e\in\partial f}\mu(f)\omega_e,\ \forall \mu\in\Omega^2({\mathbb{G}},R). \end{align*} $$

In particular, $d^{*}:\Omega ^1({\mathbb {G}},R)\to \Omega ^0({\mathbb {G}},R)$ is the divergence operator in R. Kenyon’s terminology [Reference Kenyon38]. Let $e\in E_+$ be an oriented edge, and $e^{*}\in E_+^{*}$ be the dual edge (i.e., the faces $f,f'\in F_+$ such that $e\in \partial f$ and $e^{-1}\in \partial f$ satisfy $f'=\underline {e^{*}}$ and $f=\overline {e^{*}}$ ). Then we have, for any $\mu \in \Omega ^2({\mathbb {G}},R)\cong C_0({\mathbb {G}}^{*};R)$ ,

$$\begin{align*}d^{*}\mu(e) = \mu(f)-\mu(f') = \langle \mu,\partial e^{*}\rangle. \end{align*}$$

We obtain an isomorphism of chain complexes given by the following commutative diagram:

Equipped with these chain complexes, we can then do some (basic) homology. Let us introduce a few notations, following, for instance, the conventions of [Reference Kassel and Lévy35]. We set

• • $\diamondsuit _1=\ker (d^{*}:\Omega ^1({\mathbb {G}},R))\simeq \ker (\partial :C_1({\mathbb {G}};R)\to C_2({\mathbb {G}};R)$ the module of cycles,

• • $\bigstar _1^{*}=d^{*}(\Omega ^2({\mathbb {G}},R))\simeq \partial (C_2({\mathbb {G}};R))$ the module of boundaries,

• • $\bigstar _1=d(\Omega ^0({\mathbb {G}},R))\simeq \partial (C_2({\mathbb {G}}^{*};R))$ the module of coboundaries.

All these modules are equipped with the bilinear form $\langle \cdot ,\cdot \rangle $ from Proposition 2.5, associated to a given basis.

Definition 2.9. Let ${\mathbb {G}}$ be a topological map. Its first homology module is defined as the R-module

$$ \begin{align*} H_1({\mathbb{G}};R) = \diamondsuit_1/\bigstar_1^{*}. \end{align*} $$

When $\ell $ is a loop of ${\mathbb {G}}$ , its R-homology $[\ell ]_R$ is the image of the element $\omega _\ell $ in $H_1({\mathbb {G}};R).$ For any $n\ge 2, $ its $\mathbb {Z}_n$ -homology $[\ell ]_{\mathbb {Z}_n}$ is the element $1\otimes [\ell ]_{\mathbb {Z}}\in H_1({\mathbb {G}};\mathbb {Z}_n)=\mathbb {Z}_n\otimes _{\mathbb {Z}} H_1({\mathbb {G}};\mathbb {Z}).$

Note that by the universal coefficient theorem for homology, the change of ring commutes with the homology

$$\begin{align*}H_1({\mathbb{G}};R)=R\otimes_{\mathbb{Z}} H_1({\mathbb{G}};\mathbb{Z}), \end{align*}$$

even if we take $R=\mathbb {Z}_n$ .

Proof. Let us start by showing that $\diamondsuit _1=\bigstar _1^\perp $ . If $\omega \in \diamondsuit _1$ , then for any $f\in \Omega ^0({\mathbb {G}},R)$ , we have

$$\begin{align*}\langle \omega, df\rangle = \langle d^{*}\omega, f\rangle = 0 \end{align*}$$

and $\omega \in \bigstar _1^\perp $ . Conversely, remark that a free basis of $\bigstar _1$ is given by $(df_v,v\in V)$ , so that for any $\omega \in \bigstar _1^\perp $ , we have

$$\begin{align*}\langle d^{*}\omega, f_v\rangle = \langle \omega,df_v\rangle = 0, \end{align*}$$

and $\omega \in \diamondsuit _1$ .

Now let us prove that $\diamondsuit _0=\diamondsuit _1$ . If $\ell =e_1\cdots e_n$ is a loop in ${\mathbb {G}}$ , then

$$\begin{align*}d^{*}\omega_\ell = \sum_{i=1}^n\sum_{e\in E_+}\omega_{e_i}(e)f_{\partial e} =\sum_{i=1}^nf_{\partial e_i} = 0, \end{align*}$$

so that $\diamondsuit _0\subset \diamondsuit _1$ . Let $\omega \in \diamondsuit _0^\perp $ be a 1-form. We define a 0-form $f_\omega \in \Omega ^0({\mathbb {G}},R)$ by setting

$$\begin{align*}f_\omega(v) = \sum_{i=1}^n \omega(e_i), \end{align*}$$

where $e_1\cdots e_n$ is a path in ${\mathbb {G}}$ starting from a given reference vertex $v_0$ and ending at v. The fact that it does not depend on the path follows from the fact that $\omega \perp \omega _\ell $ for any loop $\ell $ . We see that for any $e\in E$ , $df_\omega (e)=\omega (e)$ ; therefore, we have the inclusion $\diamondsuit _0^\perp \subset \bigstar _1=\diamondsuit _1^\perp $ . We finally get that $\diamondsuit _0=\diamondsuit _1$ . The direct sum decomposition follows from the standard decomposition of a module into a submodule and its orthogonal, provided that the bilinear form is not degenerate on this submodule, which is trivially the case here.

Proposition 2.11. The R-module

$$\begin{align*}\mathcal{H}_1=(\bigstar_1^{*})^\perp\cap\diamondsuit_1 \end{align*}$$

is isomorphic to $H_1({\mathbb {G}};R)$ .

Proof. Recall that $\bigstar _1^{*}\subset \diamondsuit _1$ thanks to the property of the chain complex. It follows from the direct sum decomposition

$$\begin{align*}\diamondsuit_1 = \bigstar_1^{*}\oplus\mathcal{H}_1 \end{align*}$$

that for any $\omega \in \diamondsuit _1$ , there is a unique couple $(\omega _0,\mu )\in \mathcal {H}_1\times \Omega ^2({\mathbb {G}},R)$ such that $\omega =\omega _1+d^{*}\mu $ . It is then straightforward to check that the map $[\omega ]\mapsto \omega _0$ is the isomorphism that we were looking for.

The winding number of a planar loop $\ell =e_1\cdots e_n$ around a point is an integer that counts how many times the loop cycles around the point; in particular, we see that in the case of a topological map, it defines a function $n_\ell \in \Omega ^2({\mathbb {G}},\mathbb {Z})$ that counts how many times the loop cycles around each face. One can see that it is equivalent to require that $d^{*}n_\ell (e_i) = 1$ for any i such that $e_i\in E_+$ , and $-1$ for any i such that $e_i^{-1}\in E_+$ . It sums up as

$$\begin{align*}d^{*}n_\ell = \omega_\ell. \end{align*}$$

Is it possible to get such a construction for compact orientable surfaces? The general answer is not exactly, because ‘bad’ things can happen when $\ell $ has a nontrivial homology, but it is still possible when $[\ell ]=0$ , as stated by the following lemma.

Proof. Points $1$ and $2$ are standard results, and their proof can be found in Chapter 2 of [Reference Stillwell64]. We shall prove the last point. Recall that

$$\begin{align*}\diamondsuit_1 = \bigstar_1^{*}\oplus\mathcal{H}_1, \end{align*}$$

and that for any loop $\ell $ , the 1-form $\omega _\ell $ is in $\diamondsuit _1$ . Hence, there is a unique pair $(h_\ell ,n_\ell )$ with $h_\ell \in \mathcal {H}_1$ and $n_\ell \in \Omega ^2({\mathbb {G}},R)$ such that $\omega _\ell =h_\ell +d^{*}n_\ell $ . If $[\ell ]_R=0$ , it means that $h_\ell =0$ and $\omega _\ell =d^{*}n_\ell $ , as expected.

An example of winding number is depicted in Figure 8.

Figure 8

A representant of the winding number function with $c\in R$ , for a loop $\ell $ of null homology, on a map of genus $2$ . The loop is drawn in green, and the value on each positively oriented face is displayed on each $2$ -cell.

Let $\ell $ be a based loop of a topological map ${\mathbb {G}}=(V,E,F)$ which uses each non-oriented edge at most once and each vertex at most twice. We denote by $E_\ell $ the subset of edges $e\in E$ such that $\ell $ runs through e or $e^{-1}$ . Whenever a vertex v is visited twice, the four outgoing edges at v visited by $\ell $ can be ordered $e_1,e_2,e_3,e_4$ respecting the counterclockwise, cyclic ordering of the orientation of the map. There are six possible configurations, described by the order of appearance of the edges in $\ell $ . For instance, if $\ell $ is based at $\overline {e_1}$ and starts with $e_1^{-1}$ , these configurations correspond to the orders $(e_1^{-1},e_2,e_3,e_4^{-1})$ , $(e_1^{-1},e_2,e_4^{-1},e_3)$ , $(e_1^{-1},e_3,e_2^{-1},e_4)$ , $(e_1^{-1},e_3,e_4^{-1},e_2)$ , $(e_1^{-1},e_4,e_2^{-1},e_3)$ and $(e_1^{-1},e_4,e_3^{-1},e_2)$ . Up to rotations, they fall into the three generic configurations illustrated in Figure 9. We say that $\ell $ is tame if it is only of the first type of Figure 9 – that is, if it can be split into two loops that meet at the intersection point. See Figure 10 for an example.

Figure 9

The three main types of transverse simple intersections. In each case, the dotted paths might be arbitrarily complicated and have multiple intersections outside $\{e_1,e_2,e_3,e_4\}$ . Only the first case corresponds to a tame loop.

The set $V_\ell $ of vertices visited twice by $\ell $ is then called the (transverse) intersection points of $\ell .$

Definition 2.13. Let $\ell $ be a tame loop in a map ${\mathbb {G}}$ . The Makeenko–Migdal vector at $v\in V_\ell $ is the 2-form

(18) $$ \begin{align} \mu_v=d(\omega_{e_1} )+ d(\omega_{e_3})=-d(\omega_{e_2})-d(\omega_{e_4}). \end{align} $$

We denote by $\mathfrak {m}_\ell $ the $\mathbb {R}$ -vector space generated by $\{\mu _v, v\in V_\ell \}$ and $\{d\omega _e, e\notin E_\ell \}$ .

The Makeenko–Migdal vectors are a way to encode algebraically the previously defined Makeenko–Migdal deformations; see, in particular, Figure 1.

Figure 10

A tame loop in a graph with one vertex and $2$ faces. The value of $\mu _v$ is displayed on each face in blue.

Lemma 2.14. Let $\ell $ be a tame loop of a map ${\mathbb {G}}$ . Then

$$ \begin{align*}\mathfrak{m}_\ell=\left\{\begin{array}{ll} \{\alpha\in \Omega^2({\mathbb{G}},R): {\langle}\alpha,\mu_{*}{\rangle} =0 \}& \text{ if }[\ell]_R\not=0, \\ &\\ \{\alpha\in \Omega^2({\mathbb{G}},R): {\langle}\alpha,\mu_{*}{\rangle} = {\langle}\alpha, n_\ell {\rangle}=0 \}& \text{ if }[\ell]_R=0.\end{array}\right.\end{align*} $$

Proof of Lemma 2.14.

Let us first remark that the above construction is invariant by the following appropriate subdivisions. Let us call subdivision of an oriented face $f_{*}$ , the operation of adding two new vertices on its boundary and adding an edge e connecting them; the new map ${\mathbb {G}}'$ has $2$ new vertices, $1$ more edge and $1$ more face, with in place of $f_{*}$ , two faces $f_1$ and $f_2$ with the same orientation induced from $f_{*}$ , while any other face is identified with a face of ${\mathbb {G}}$ . The map ${\mathbb {G}}'$ being finer than ${\mathbb {G}}$ , $\ell $ can be identified with a tame loop of ${\mathbb {G}}'$ that we denote by the same letter. Consider the map $P:\Omega ^2({\mathbb {G}}',R)\to \Omega ^2({\mathbb {G}},R)$ with $P(\varphi )(f)=\varphi (f')$ whenever a face f of ${\mathbb {G}}$ is identified with a face of ${\mathbb {G}}'$ and $\varphi (f_1)+\varphi (f_2)$ when $f=f_{*}.$ On one hand, $P(d\omega _e)=0$ , and P maps all other vectors of the defining generating family of $\mathfrak {m}_\ell '$ to the generating family of $\mathfrak {m}_\ell .$ Therefore, $P(\mathfrak {m}_\ell ')=\mathfrak {m}_\ell $ . As $P: \{d\omega _e\}^\bot \to \Omega ^2({\mathbb {G}},R)$ is an isometry, while $d\omega _e\in \mathfrak {m}_\ell '\cap \ker (P)$ , $P({\mathfrak {m}_\ell '}^\bot )=\mathfrak {m}_\ell ^\bot .$ On the other hand, $P(\mu _{*}')=\mu _{*}$ , and when $[\ell ]_R=0$ , $P(n_{\ell }')=n_\ell .$ We conclude that it is enough to prove the claim for any subdivision of ${\mathbb {G}}.$

We can then w.l.o.g. assume that $\ell $ and the paths $a_1,b_1,\ldots , a_g,b_g$ of Lemma 2.12 do not share any edge in common. Under this assumption, let us set $\mathscr {S}=\{\ell ,a_1,b_1,\ldots , a_g,b_g\}$ and denote by $\mathrm {T}(\mathscr {S})$ the set of oriented edges e such that an element of $\mathscr {S}$ runs through e or $e^{-1}.$ Let $\eta $ be the permutation of the edges E such that $ \eta (e^{-1})=\eta (e)^{-1}$ for any edge $e\in E$ , with $2+4g$ nontrivial cycles associated to elements of $\mathscr {S}$ forgetting the base point. More precisely, for each $\gamma \in \{\ell ,a_1,b_1,\ldots , a_g,b_g\}$ with $\gamma =e_1\ldots e_n$ , $(e_1,\ldots ,e_n)$ and $(e_1^{-1}, \ldots , e_n^{-1})$ are cycles of $\eta ,$ whereas $\eta (e)=e$ for any $e\not \in \mathrm {T}(\mathscr {S}).$ For any $\omega \in \Omega ^1({\mathbb {G}},R),$ setting

$$ \begin{align*}\eta.\omega=\omega \circ\eta^{-1}\end{align*} $$

defines a 1-form. We claim that for any oriented edge $e\in \mathrm {T}(\mathscr {S})$ ,

$$ \begin{align*}\alpha_e=d\omega_e- d (\eta.\omega_e) \in \mathfrak{m}_\ell.\end{align*} $$

Indeed, it is nonzero only when $\gamma \in \mathscr {S}$ runs through e or $e^{-1}$ , in which case, it follows from (18) that $\alpha _e$ is a Makeenko–Migdal vector at, respectively, $\overline e$ or $ \underline e$ .

Let us now consider $\beta \in \mathfrak {m}_\ell ^\bot \cap \{\mu _{*}\}^\bot .$ Then

$$\begin{align*}{\langle}\beta,\alpha_e{\rangle}= {\langle}\beta,(d-d\circ\eta)\omega_e{\rangle}=0,\ \forall e\in \mathrm{T}(\mathscr{S}), \end{align*}$$

whereas ${\langle }\beta , d\omega _e{\rangle }=0,\ \forall e\not \in \mathrm {T}(\mathscr {S})$ , so that

$$\begin{align*}d^{*}\beta= (d\circ\eta)^{*}(\beta)=\eta^{-1}\circ d^{*}\beta \text{ and }{\langle}d^{*}\beta,\omega_e{\rangle}=0,\ \forall e\not\in \mathrm{T}(\mathscr{S}). \end{align*}$$

It follows that

$$\begin{align*}d^{*}\beta= c\omega_\ell+\sum_{i=1}^g( a_i\omega_{a_i}+b_i\omega_{b_i}), \text{ for some }c,a_1,b_1,\ldots,a_g,b_g\in \mathbb{R}. \end{align*}$$

Using the decomposition $\diamondsuit _1= \bigstar _1^{*}\oplus \mathcal {H}_1,$ we find

$$ \begin{align*}d^{*}\beta= cd^{*} n_\ell \text{ and } c h_\ell+\sum_{i=1}^g( a_i\omega_{a_i}+b_i\omega_{b_i})=0.\end{align*} $$

Since $\beta \in \hat {\mathfrak {m}}_\ell ^\bot ,$ for any edge e such that $e,e^{-1}$ do not belong to $\ell ,$ we obtain that ${\langle }d^{*}\beta ,\omega _e {\rangle }=0.$ In particular, $a_i=b_i=0$ for all i and $d^{*}\beta = c\omega _\ell .$ Since $\beta \in \mu _{*}^\bot ,$ it follows that either $[\ell ]_R=0$ and $\beta =cn_\ell $ or $c=0$ and $\beta =0.$ We conclude that either $[\ell ]_R=0$ and $\mathfrak {m}_\ell ^\bot \cap \{\mu _{*}\}^\bot =\mathbb {R}. n_\ell $ , or $[\ell ]_R\not =0$ and $\mathfrak {m}_\ell ^\bot \cap \{\mu _{*}\}^\bot = \{0\}.$

2.3 Regular polygon tilings of the universal cover, tiling-length of a tame loop and geodesic loops

To simplify the presentation, we shall work only with surfaces of genus g obtained by a standard quotient of $4g$ polygons. We fix here notations and definitions relative to the universal cover of such maps. We refer to [Reference Birman and Series7] for more details.

Regular maps and regular loops: A $2g$ -bouquet map is a map $(V,E,F)$ with $1$ vertex v, $1$ face and $2g$ edges, so that for $f\in F$ , there are $2g$ oriented edges $a_1,b_1,\ldots , a_g,b_g\in E$ corresponding to distinct edges, with $\partial _v f=[a_1,b_1]\ldots [a_g,b_g].$ A $2g$ -bouquet map can be obtained by labelling the edges of a $4g$ -polygon counterclockwise $e_1,\ldots ,e_{4g}$ and gluing $e_{i+4k}$ to $e_{i+4k+1}$ for all $0\le k\le g-1$ $i\in \{1,2\}$ ,; cf. Figure 11.

Figure 11

Three representations of a 4-bouquet. From left to right: as a polygon whose sides are identified pairwise (and whose vertices are all identified), as a graph embedded in a surface of genus 2, and as the skeleton of the corresponding CW complex.

A regular map is a pair given by a map ${\mathbb {G}}=(V,E,F)$ and a $2g$ -bouquet map ${\mathbb {G}}_g$ , such that ${\mathbb {G}}$ is finer than ${\mathbb {G}}_g$ . Each edge of ${\mathbb {G}}_g$ is uniquely decomposed as a concatenation of edges of ${\mathbb {G}}.$ Let $\partial E\subset E $ be the set of edges appearing in these concatenations. We then denote by $\partial V$ the set of endpoints of edges of $\partial E$ and $\mathring {V}=V\setminus \partial V.$ When $({\mathbb {G}},{\mathbb {G}}_g)$ is a regular map, we refine the notion of tame loops defined in the previous section as follows. A loop $\ell \in \mathrm {L}({\mathbb {G}})$ is regular whenever it is tame and none of its edges belong to $\partial E$ and $\underline {\ell }\in \mathring {V}$ . In particular, its intersection points satisfy $V_\ell \subset \mathring {V}.$

Universal cover of a regular map: Let $({\mathbb {G}},{\mathbb {G}}_g)$ be a regular map with ${\mathbb {G}}=(V,E,F).$ When $g=1,$ consider the closed square $P_1$ with vertices coordinates in $\{-\frac {1}{2},\frac 1 2\}$ and the tiling of $\mathbb {R}^2$ by translation of $P_1$ by $\mathbb {Z}^2.$ When $g\ge 2,$ consider a tiling of the Poincaré hyperbolic disc $\mathbb {H}$ by a family of closed regular $4g$ -polygons of $\mathbb {H}$ whose sides do not intersect $0$ , and denote by $P_1$ the polygon among them enclosing $0.$ The group $\Gamma _g$ can be identified with $\mathbb {Z}^2$ when $g=1$ and with a subgroup of Möbius transformations that acts properly by isometry on $\mathbb {H}$ when $g\geq 2$ . The group $\Gamma _g$ acts freely on the set of tiles, and for each $h\in \Gamma _g,$ there is a unique tile $P_h$ with $h\cdot 0$ belonging to the interior of $P_h.$ Let us define $\tilde \Sigma _{\mathbb {G}}$ as $\mathbb {R}^2$ when $g=1$ and $\mathbb {H}$ when $g\ge 2.$ The quotient of $\tilde \Sigma _{\mathbb {G}}$ by $\Gamma _g$ is homeomorphic to $ \Sigma _{\mathbb {G}}$ , and we denote by $p:\tilde \Sigma _{\mathbb {G}}\to \Sigma _{\mathbb {G}}$ the quotient mapping. There is a unique CW decomposition of $\tilde \Sigma _{\mathbb {G}}$ such that the restriction of p to the interior of each cell of $\tilde \Sigma _{\mathbb {G}}$ is a homeomorphism onto the interior of a cell of $\Sigma _{\mathbb {G}}$ labelled by an element of $V,E$ or F. We denote a labelling of the cells of this CW complex by $\tilde {\mathbb {G}}=(\tilde V,\tilde E,\tilde F)$ and call $\tilde {\mathbb {G}}$ a universal cover of ${\mathbb {G}}.$ There is a natural map from $\tilde V,\tilde E,\tilde F$ to, respectively, $V,E$ and F that we also denote by $p.$ The map $\tilde {\mathbb {G}}$ is finer than the universal cover $\tilde {\mathbb {G}}_g=(\tilde V_g,\tilde E_g,\tilde F_g)$ of ${\mathbb {G}}_g,$ where faces $\tilde F_g$ can be identified with polygons $(P_g)_{g\in \Gamma _g},$ and $\Gamma _g$ acts free transitively on $\tilde V_g.$ As for maps, the pair $(\tilde V, \tilde E)$ can be identified with a graph, and we denote by $\mathrm {P} (\tilde {\mathbb {G}})$ its set of paths. For each path $\gamma =e_1\ldots e_n\in \mathrm {P}({\mathbb {G}})$ and $\tilde v\in p^{-1}(v_0),$ the lift of $\gamma $ from $\tilde v$ is the unique path $\tilde \gamma =\tilde e_1\ldots , \tilde e_n\in \mathrm {P}(\tilde {\mathbb {G}})$ with $\underline {\tilde \gamma }=\tilde v$ and $p(\tilde e_k)=e_k$ for all $1\le k\le n.$ Viceversa, when $\tilde \gamma =(\tilde e_1,\ldots ,\tilde e_n)\in \mathrm {P}(\tilde {\mathbb {G}}),$ its projection is the path $p(\gamma )=(p(\tilde e_1),\ldots ,p(\tilde e_n))\in \mathrm {P}({\mathbb {G}}).$ Its image in $\mathrm {RP}({\mathbb {G}})$ does not depend on the $\sim _r$ equivalence class $[\gamma ]$ of $\gamma ;$ we denote it by $p([\gamma ])\in \mathrm {RP}({\mathbb {G}}).$ When $\tilde v\in \tilde V$ and $v=p(\tilde v),$ the group $\mathrm {RL}_{\tilde v}(\tilde {\mathbb {G}}) $ of reduced loop of $(\tilde V,\tilde E)$ based at $\tilde v$ allows to complete the diagram of Lemma 2.3 in the following way. The proof is standard and left to the reader.

Tile decomposition: For all $h\in \Gamma _g,$ we denote by $D_h\subset \tilde V , D^{*}_h\subset \tilde F $ and $ \mathring {D_h}\subset \tilde V$ the subsets of vertices and faces of ${\mathbb {G}}$ , whose image in $\tilde \Sigma _{{\mathbb {G}}}$ is included, respectively, in $P_h$ and its interior $\mathring {P_h}$ . The projection $\mathring {D}$ of $\mathring {D}_h$ does not depend on $h\in \Gamma .$ When $U\subset \tilde F$ and $E_c\subset \tilde E,$ we denote by $U\setminus E_c$ the subgraph of the graph of $\tilde {\mathbb {G}}^{*}$ , where all faces from $\tilde F\setminus U$ and all edges dual to $E_c$ are removed. Let us consider the oriented graph with vertices $\Gamma _g$ such that there is an edge between a and b if and only if $P_{a}$ and $P_b$ share a side. The action of $\Gamma _g$ on $\mathbb {H}$ induces a free, transitive, isometric action on this graph, and we denote by $|h|_{\Gamma _g}$ the distance between any $h\in \Gamma _g$ and $1$ . For any nonconstant regular loop $\ell =(e_1,\ldots ,e_n)\in \mathrm {L}({\mathbb {G}}),$ we call

$$ \begin{align*}|\ell|_D=n-1- \#\left\{1\le i\le n-1:\exists h\in \Gamma_g \text{ with } \{\underline{e_i},\overline{e_i},\overline{e_{i+1}}\} \subset D_h \right\}\end{align*} $$

the tiling length ofFootnote ³⁵ $\ell .$ See Figure 12.

Figure 12

On the left: a regular loop with base v in a regular map of genus 1. On the right: its lift on the universal cover, with base $\tilde {v}$ . It has a tiling length of 4, as it crosses four times the boundaries of tiles (at the purple crosses).

There is then a unique tuple $\gamma _0,\ldots , \gamma _{|\ell |_D}$ of paths of ${\mathbb {G}},$ such that for any lift $\tilde \ell $ of $\ell $ , there are lifts $\tilde \gamma _0,\ldots , \tilde \gamma _{|\gamma |_D}$ of $\gamma _0,\ldots , \gamma _{|\ell |_D}$ such that

(19) $$ \begin{align} \tilde\ell=\tilde\gamma_0\ldots \tilde\gamma_{|\gamma|_D}, \end{align} $$

and for all $0\le k\le |\ell |_D$ , there are $h_0,h_1,\ldots , h_{|\ell |_{|D|}}\in \Gamma _g$ such that allFootnote ³⁶ vertices of $\tilde \gamma _k$ belong to $D_{h_k},$ while $\ell _\Gamma =(h_0,\ldots , h_{|\ell |_D})$ is a path in $\Gamma _g$ . We call $\ell _D=\gamma _{|\ell |_D}\gamma _0$ the initial strand of $\ell .$ We call $\ell _\Gamma $ the tiling path of $\ell $ and set

$$ \begin{align*}|\ell|_{\Gamma}= |h_{|\ell|_D}|_{\Gamma_g}.\end{align*} $$

A loop $\ell _1$ of $({\mathbb {G}},{\mathbb {G}}_g)$ is called an inner loop of $\ell $ if $\ell _1$ is regular and included in $\mathring {D}$ , and $\ell _1\prec \ell .$ We then say that $\underline {\ell _1}$ is a contractible intersection point of $\ell $ and denote by $V_{c,\ell }$ the set of such points. A proper loop is a regular loop $\ell $ with $\#V_{c,\ell }=0.$

A path $\gamma \in \mathrm {P}({\mathbb {G}})$ is said to be geodesic when its embedding in the surface is the restriction of a geodesic of the surface.Footnote ³⁷ A path in $\Gamma _g$ is geodesic if it is the tiling path of a geodesic path of a regular map.

2.4 Shortening homotopy sequence

We define here operations on regular loops allowing to decrease their tiling length. We say that a sequence $\ell _1,\ldots ,\ell _n$ is a shortening homotopy sequence from $\ell _1$ to $\ell _n$ if $\ell _1,\ldots ,\ell _n$ are regular loops such that $|\ell _1|_D\ge \ldots \ge |\ell _n|_D$ , and for all $1\le l<n,$

$$ \begin{align*}\#V_{c,l}=\#V_{c,l+1}=0 \text{ or } \#V_{c,l}>\#V_{c,l+1},\end{align*} $$

while there is a regular map $(V,E,F)$ with $\ell _{l},\ell _{l+1}\in \mathrm {P}({\mathbb {G}})$ and a subset of faces $K_l\varsubsetneq F$ , with

$$ \begin{align*}\ell_l\sim_{K_l} \ell_{l+1}.\end{align*} $$

The aim of this section is to prove the following.