In this note, I discuss in some detail the dual version of the ribbon graph decomposition of the moduli spaces of Riemann surfaces with boundary and marked points, which I introduced in math.AG/0402015, and used in math.QA/0412149 to construct open-closed topological conformal field theories. This dual version of the ribbon graph decomposition is a compact orbi-cell complex with a natural weak homotopy equivalence to the moduli space.
The fusion number of a ribbon knot is the minimal number of 1-handles needed to construct a ribbon disk. The strong homotopy fusion number of a ribbon knot is the minimal number of 2-handles in a handle decomposition of a ribbon disk complement. We demonstrate that these invariants behave completely differently under cabling by showing that the (p,1)-cable of any ribbon knot with fusion number one has strong homotopy fusion number one and fusion number p. Our main tools are Juhász-Miller-Zemke's bound on fusion number coming from the torsion order of knot Floer homology and Hanselman-Watson's …
We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new operad governing associative algebras with involution. This operad is Koszul and we identify the dual dg operad governing A-infinity algebras with involution in terms of Mobius graphs which are a generalisation of ribbon graphs. We then generalise open topological conformal field theories to open Klein topological conformal field theories and give a generat…
In this article we describe cell decompositions of the moduli space of Riemann surfaces and their relationship to a Hurwitz problem. The cells possess natural linear structures and with respect to this they can be described as rational convex polytopes which come equipped with natural integer points and a volume form. We show how to effectively calculate the number of lattice points and the volumes over all the cells and that these calculations contain deep information about the moduli space.
This paper studies compactifications of moduli spaces involving closed Riemann surfaces. The first main result identifies the homeomorphism types of these compactifications. The second main result introduces orbicell decompositions on these spaces using semistable ribbon graphs extending the earlier work of Looijenga.
This survey covers earlier work of the author as well as recent work on Riemann's moduli space, its canonical cell decomposition and compactification, and the related operadic structure of arc complexes.
The moduli space of Riemann surfaces with at least two punctures can be decomposed into a cell complex by using a particular family of ribbon graphs called Nakamura graphs. We distinguish the moduli space with all punctures labelled from that with a single labelled puncture. In both cases, we describe a cell decomposition where the cells are parametrised by graphs or equivalence classes of finite sequences (tuples) of permutations. Each cell is a convex polytope defined by a system of linear equations and inequalities relating light-cone string parameters, quotiented by the automorphism group …
This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of homotopy algebras and new constructions of 'quantum invariants' of manifolds inspired by ideas originating from physics. We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We generalise open topological conformal field theories to open Klein topological conformal field theories and c…
Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichmüller space of punctured surfaces. We extend this cell decomposition to the moduli space of decorated strictly convex projective structures of finite volume on punctured surfaces. The proof uses Fock and Goncharov's $\mathcal{A}$-coordinates for doubly decorated structures. In addition, we describe a s…
We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Goncharov, the action of the mapping class group as well as algorithmic real algebraic geometry. We also show that the decorated moduli space of framed strictly convex real projective structures of finite volume on the thrice-punctured sphere has a natural cell decomposition.
We consider (local) parametrizations of Teichmuller space $T_{g,n}$ (of genus $g$ hyperbolic surfaces with $n$ boundary components) by lengths of $6g-6+3n$ geodesics. We find a large family of suitable sets of $6g-6+3n$ geodesics, each set forming a special structure called "admissible double pants decomposition". For admissible double pants decompositions containing no double curves we show that the lengths of curves contained in the decomposition determine the point of $T_{g,n}$ up to finitely many choices. Moreover, these lengths provide a local coordinate in a neighborhood of all points of…
We present a simplified formulation of open intersection numbers, as an alternative to the theory initiated by Pandharipande, Solomon and Tessler. The relevant moduli spaces consist of Riemann surfaces (either with or without boundary) with only interior marked points. These spaces have a combinatorial description using a generalization of ribbon graphs, with a straightforward compactification and corresponding intersection theory. Crucially, the generating functions for the two different constructions of open intersection numbers are identical. In particular, our construction provides a compl…
We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and Lie types in computations of the cohomology of the outer automorphism group of a free group. Another consequence is an explicit computation of dualizing sheaves on spaces of metric graphs, thus characterizing to which extent these spaces are different from oriented orbifolds. We also provide a relation between the cohomology of the…
We associate a moduli space M(G) to a planar trivalent graph G. We proved several decomposition properties of M(G), which implies that the Euler characteristic of M(G) equals to the number of Tait colorings of G when G is bipartite. Then we interpret M(G) as a representation space of the fundamental group of G to SU(3).
The orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. The resolution maps planar metric trees to their underlying abstract representatives, collapsing and folding an explicit geometric decomposition of the moduli space into cubes. This decomposition endows the resolving space with an interesting canonical pseudometric. The second part of this paper defines a related (stacky) resolution of a space of real quadratic forms, and sugg…
A closed hyperbolic surface of genus $g\ge 2$ can be decomposed into pairs of pants along shortest closed geodesics and if these curves are sufficiently short (and with lengths uniformly bounded away from 0), then the geometry of the surface is essentially determined by the combinatorics of the pants decomposition. These combinatorics are determined by a trivalent graph, so we call such surfaces {\em trivalent}. In this paper, in a first attempt to understand the "shape" of the subset $\ts$ of moduli space consisting of surfaces whose systoles fill, we compare it metrically, asymptotically in …
We prove combinatorially the explicit relation between genus filtrated $s$-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich--Penner matrix model (KPMM). The latter is the generating function for volumes of discretized (open) moduli spaces $M_{g,s}^{\mathrm{disc}}$ given by $N_{g,s}(P_1,\dots,P_s)$ for $(P_1,\dots,P_s)\in{\mathbb Z}_+^s$. This generating function therefore enjoys the topological recursion, and we prove that it is simultaneously the generating function for ancestor invariants of a cohomological field theory thus enjoying the Givental dec…
Recently Andrei Teleman considered instanton moduli spaces over negative definite four-manifolds $X$ with $b_2(X) \geq 1$. If $b_2(X)$ is divisible by four and $b_1(X) =1$ a gauge-theoretic invariant can be defined; it is a count of flat connections modulo the gauge group. Our first result shows that if such a moduli space is non-empty and the manifold admits a connected sum decomposition $X \cong X_1 # X_2$ then both $b_2(X_1)$ and $b_2(X_2)$ are divisible by four; this rules out a previously natural appearing source of 4-manifolds with non-empty moduli space. We give in some detail a constru…
In this paper we study the smooth moduli space of closed Riemann surfaces. This smooth moduli is an infinite cover of the usual moduli space $\mathscr{M}_g$ of closed Riemann surfaces, and is identified with the Schottky space of rank $g.$ The main theorem of the paper is: Closed Riemann surfaces are uniformizable by Schottky groups of Hausdorff dimension less than one. This work seem to be the only paper in literature to study question of Riemann surface uniformization and its Hausdorff dimension. We develop new techniques of rational norm of homological marking of Riemann surface and, decomp…
Starting from a Heegaard splitting of a three-manifold, we use Lagrangian Floer homology to construct a three-manifold invariant, in the form of a relatively Z/8-graded abelian group. Our motivation is to have a well-defined symplectic side of the Atiyah-Floer Conjecture, for arbitrary three-manifolds. The symplectic manifold used in the construction is the extended moduli space of flat SU(2)-connections on the Heegaard surface. An open subset of this moduli space carries a symplectic form, and each of the two handlebodies in the decomposition gives rise to a Lagrangian inside the open set. In…
