Deformations of Bordered Surfaces and Convex Polytopes Satyan L. Devadoss, Timothy Heath, and Wasin Vipismakul he moduli space of Riemann surfaces of extension of the work by Liu [16] on the moduli genus g with n marked particles is influ- of J-holomorphic curves and open Gromov-Witten ential in many areasof mathematics and invariants. theoretical physics, ranging from quan- A third field of intersection comes from the tum cohomology to number theory to world of operads and algebraic structures. For Tfluid dynamics. This space has a natural extension instance, Kaufmann, Livernet, and Penner provide by considering surfaces with boundary, appear- several topological and homological operads based ing, for instance, alongside open-closed string on arcs on bordered surfaces [14]. Indeed, the theory and holomorphic curves with Lagrangian natural convex polytopes appearing in our setting submanifolds. One goal of this article is to pro- fitcomfortablyintheframeworkofhighercategory vide an accessible understanding of these moduli theory and the study of A∞ and L∞ structures seen spaces by constructing a stratification based on the from generalizations of associahedra, cyclohedra, pair-of-pants decomposition and the well-known and multiplihedra [8]. phenomenon of bubbling. Along the way, we will An overview of the paper is as follows: We begin classify all such spaces that can be realized as with a review of the definitions of interest, followed convex polytopes, relating them to the famous by construction of the moduli spaces, providing associahedron. details of several low-dimensional examples and There are several (overlapping) fields that touch their stratification. The polytopal spaces are then upon these ideas. Recent work spearheaded by classified, along with a description of the as- Fomin, Shapiro, and Thurston [11] has established sociahedron and cyclohedron polytopes. A new a combinatorial world of cluster algebrasrelatedto polytope is introduced based on the moduli space bordered surfaces with marked points, introduc- of the annulus. Finally, the combinatorial and al- ing notions of triangulated surfaces and tagged gebraic properties of this polytope are explored arc complexes. Another perspective comes from and related to the multiplihedron. symplectic geometry, where an analytic approach has been taken by Fukaya and others [12] to con- Definitions struct moduli spaces of bordered surfaces. Indeed, A smooth connected oriented bordered Riemann this article can be viewed, at a high level, as an surface S of type (g, h) has genus g ≥ 0 with h ≥ 0 Satyan L. Devadoss is associate professor of mathemat- disjoint ordered circles B1,...,Bh for its boundary. ics at Williams College. His email address is satyan. We assume the surface is compact whose bound- [email protected]. ary is equipped with the holomorphic structure Timothy Heath is a graduate student at Columbia Univer- induced by a holomorphic atlas on the surface. sity. His email address is timheath@math. Specifically, the boundary circles will always be columbia.edu. given the orientation induced by the complex Wasin Vipismakul is a graduate student at the Univer- structure. The surface has a marking set M of sity of Texas. His email address is wvipismakul@math. type (n, m) if there are n labeled marked points in utexas.edu. the interior of S (called punctures) and mi labeled 530 Notices of the AMS Volume 58, Number 4 marked points on the boundary component Bi , Let S be a surface without boundary. A decom- where m = m1,...,mh. Throughout the paper, position of (S, M) into pairs of pants is a collection we define m := m1 +···+ mh. of disjoint pairs of pants on S such that the union of their closures covers the entire surface and the Definition. The set (S, M) fulfilling the above re- pairwise intersection of their closures is either quirements is called a marked bordered Riemann empty or a union of marked points and closed surface. We say (S, M) is stable if its automorphism geodesic curves on S. Indeed, all the marked points group is finite. of M appear as boundary components of pairs of pants in any decomposition of S. A disjoint set (a) (b) of curves decomposing the surface into pairs of pants will be realized by a unique set of disjoint geodesics, since there is only one geodesic in each homotopy class. We now extend this decomposition to include marked surfaces with boundary. Consider the complex double (SC, σ ) of a marked bordered Riemann surface (S, M). If P is a decomposition of SC into pairs of pants, then σ (P) is another Figure 1. (a) An example of a marked bordered decomposition into pairs of pants. The following Riemann surface (b) along with its complex is a generalization of the work of Seppälä: double. Lemma 1 (16, Section 4). There exists a decompo- sition of SC into pairs of pants P such that σ (P) = Figure1(a)shows anexampleof (S, M) where S is P and the decomposing curves are simple closed of type (1, 3) and M is of type (3, 1, 2, 0). Indeed, geodesics of SC. any stable markedborderedRiemann surface has a Figure 2(a) shows examples of some of the geo- unique hyperbolic metric such that it is compatible desic arcs from a decomposition of SC, where part with the complex structure, where all the boundary (b) shows the corresponding decomposition for S. circles are geodesics, all punctures are cusps, and Notice that there are three types of decomposing all boundary marked points are half cusps. We geodesics γ. assume all our spaces (S, M) are stable throughout this paper. (1) Involution σ fixes all points on γ: The geodesic must be a boundary curve of S, The complex double SC of a bordered Riemann surface S is the oriented double cover of S without such as the curve labeled x in Figure 2(b). 1 (2) Involution σ fixes no points on γ: The boundary. It is formed by gluing S and its mirror geodesic must be a closed curve on S, such image along their boundaries; see [2] for a detailed as the curve labeled y in Figure 2(b). construction. For example, the disk is the surface (3) Involution σ fixes two points on γ: The of type (0, 1) whose complex double is a sphere, geodesic must be an arc on S, with its whereas the annulus is the surface of type (0, 2) endpoints on the boundary of the surface, whose complex double is a torus. Figure 1(b) shows such as the curve labeled z in Figure 2(b). the complex double of (S, M) from part (a). In the case when S has no boundary, the double SC is simply the trivial disconnected double-cover of S. (a)y (b) The pair (SC, σ ) is called a symmetric Riemann surface, where σ : SC → SC is the antiholomorphic involution. The symmetric Riemann surface with a marking set M of type (n, m) has an involu- x x tion σ together with 2n distinct interior points z z {p1,...,pn, q1,...,qn} such that σ (pi ) = qi , along with m boundary points {b1,...,bm} such that y y σ (bi ) = bi. Definition. A pair of pants is a sphere from which Figure 2. Examples of some geodesic arcs three points or disjoint closed disks have been re- from a pair of pants decomposition of (a) the moved. A pair of pants can be equipped with a complex double and (b) its marked bordered unique hyperbolic structure compatible with the Riemann surface. complex structure such that the boundary curves are geodesics. We assign a weight to each type of decompos- 1The complex double and the Schottky double of a ing geodesic in a pair of pants decomposition, surface coincide, since the surface is orientable. corresponding to the number of Fenchel-Nielsen April 2011 Notices of the AMS 531 coordinates needed to describe the geodesic. The The Moduli Space geodesic of type (2) above has weight two be- Giventhe definitionofa markedborderedRiemann cause it needs two Fenchel-Nielsen coordinates to surface, we are now in position to study its moduli describe it (length and twisting angle), whereas space. For a bordered Riemann surface S of type geodesics of types (1) and (3) have weight one (g, h) with marking set M of type (n, m), we (needing only their length coordinates). denote M(g,h)(n,m) as its compactified moduli space. Lemma 2 (1, Chapter 2). Every pair of pants de- Analytic methods can be used for the construction composition of a marked bordered Riemann sur- of this moduli space that follow from several face (S, M) of type (g, h) with marking set (n, m) important, foundational cases. The topology of has a total weight of the moduli space Mg,n of algebraic curves of genus g with n marked points was provided by (1) 6g + 3h − 6 + 2n + m Abikoff [1]. Later, Seppälä gave a topology for from the weighted decomposing curves. the moduli space of real algebraic curves [18]. Based on the discussion above, we can reformu- Finally, Liu modified this for marked bordered late the decomposing curves on marked bordered Riemann surfaces; the reader is encouraged to Riemann surfaces in a combinatorial setting: consult [16, Section 4] for a detailed treatment of the construction of M(g,h)(n,m). Definition. An arc is a curve on S such that its endpoints are on the boundary of S, it does not Theorem 3 (16, Section 4). The moduli space intersect M nor itself, and it cannot be deformed M(g,h)(n,m) of marked bordered Riemann surfaces arbitrarily close to a point on S or in M. An arc is equipped with a (Fenchel-Nielson) topology that corresponds to a geodesic decomposing curve of is Hausdorff.