The symplectic representation of mapping classes is not surjective for certain types of mapping classes.
problem The surjectivity of the symplectic representation of mapping classes, particularly pseudo-Anosov ones, is not always preserved.
method Explicit construction of symplectic matrices with a bi-Perron leading eigenvalue that cannot be represented by orientable pseudo-Anosov mapping classes.
result The symplectic representation of orientable pseudo-Anosov mapping classes is not surjective.
Infinite order elements found in symplectic mapping class groups.
problem Understanding infinite order elements in symplectic mapping class groups.
method Analyzing symplectically embedded (−1)-tori. result Infinite order elements exist and are shown to be non-trivial.
Study symplectic mapping classes and their relations to surface mapping classes.
problem Comparing symplectic mapping classes with classical ones.
method Analogous configurations of Lagrangian submanifolds in Weinstein domains.
result Relations between Dehn twists in symplectic manifolds imply similar relations in surface mapping classes.
The paper finds symplectic mapping class relations using pencil pairs.
problem Understanding symplectic mapping class relations in 4-manifolds.
method Applying classification results for Fano 3-folds and polarized K3 surfaces to a general methodology of finding pencil pairs.
result Symplectic mapping class relations between products of positive Dehn twists.
Study shows symplectic mapping groups of K3 surfaces are infinitely generated.
problem Understanding symplectic mapping class groups of K3 surfaces.
method Uses Kronheimer's approach and Seiberg-Witten invariants.
result Symplectic mapping class groups of many K3 surfaces are infinitely generated.
Proves naturalness of Symplectic Instanton homology and defines cobordism maps.
problem Naturalness and cobordism maps of Symplectic Instanton homology.
method Proves naturalness and defines maps associated with four-dimensional cobordisms.
result Defines representations of mapping class groups and fundamental groups.
The paper explores the dynamics of composite symplectic Dehn twists with nonuniform hyperbolicity.
problem Understanding the dynamics and properties of composite symplectic Dehn twists.
method Analyzing the form of nonuniform hyperbolicity, growth of Floer cohomology, and classification of symplectic mapping classes.
result Composite symplectic Dehn twists exhibit positive topological entropy and exponential growth in Floer cohomology.
The well-known fact that any genus g symplectic Lefschetz fibration X4→S2 is given by a word that is equal to the identity element in the mapping class group and each of whose elements is given by a positive Dehn twist, provides an intimate relationship between words in the mapping class group and 4-manif…
New relation found in 4D symplectic mapping class group.
problem Relation between Dehn twists in symplectic 4-manifolds.
method Holomorphic curve techniques, symplectic isotopy problem solution.
result Relation between two products of Dehn twists.
Compute group cohomology for mapping class group with non-symplectic coefficients.
problem Compute group cohomology for mapping class group with non-symplectic coefficients.
method Compute the invariant subspace of the rational group ring of a surface, truncated by powers of the augmentation ideal, under the action of the mapping class group.
result First group cohomology computation for the mapping class group with non-symplectic coefficients.
The paper explores stability properties of cohomology groups and norms in symplectic and mapping class groups.
problem Stability properties of bounded cohomology in mapping class groups and symplectic groups.
method Utilizes results from Bestvina and Fujiwara, calculates norms of signature classes, and estimates cohomology norms.
result The bounded cohomology of mapping class groups does not stabilize, while that of symplectic groups does not stabilize via isometries.
Constructs symplectic surface bundles with positive signatures.
problem Symplectic surface bundles over surfaces with positive signatures.
method Constructs symplectic surface bundles with specific genera.
result Determines commutator lengths of new mapping classes.
I construct the real counterparts (which I call Borel-Bott classes) of the R/Z classes constructed in "Characteristic classes in symplectic topology", to appear, in the cohomology of volume-preserving and symplectomorhisms of a compact (symplectic) manifold.I show that, for the symplectic action of the mapping class gr…
Any nontrivial homomorphism from the mapping class group of an orientable surface of genus g≥3 to $\GL(2g,\C)$ is conjugate to the standard symplectic representation. It is also shown that the mapping class group has no faithful linear representation in dimensions less than or equal to 3g−3.
New computations show symplectic groups and mapping class groups have different properties regarding torsion.
problem Comparing properties of symplectic groups and mapping class groups.
method Using K-theory, Weil representations, and quantum representations. result Symplectic groups have uniformly bounded torsion, while mapping class groups have more complex torsion.
Minimal action of mapping class group on character variety.
problem Character variety of Deroin-Tholozan representations.
method Geometric perspective using symplectic structure.
result Infinite mapping class group orbits are dense.
Geometric quantization for symplectic maps via Toeplitz operators.
problem Quantization of symplectic maps and Witten's conjecture.
method Berezin-Toeplitz operators and holomorphic sections over Kähler manifolds.
result Established a semi-classical trace formula for quantum representations of mapping class groups.
Study Dehn-Seidel twists on Lagrangian spheres in K3 surfaces.
problem Understanding configurations of Lagrangian spheres in symplectic K3 surfaces.
method Use Seiberg-Witten theory and tools from symplectic mapping class groups.
result Proves algebraic independence of certain twists and generation results.
In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an…
Cyclotomic polynomials help classify mapping classes on surfaces.
problem Characterizing mapping classes on surfaces using cyclotomic polynomials.
method Investigating characteristic polynomials of integral symplectic matrices and using cyclotomic polynomials to classify them.
result For n≥3, the polynomial φn(x) is realized by a mapping class of algebraically finite type if and only if n has at most two distinct prime divisors. Goldman symplectic form and complex structure compatible on SL(3,R) Hitchin component.
problem Compatibility of Goldman's symplectic form with complex structure on SL(3,R) Hitchin component. method Proof of compatibility between Goldman's symplectic form and Labourie-Loftin complex structure.
result Goldman symplectic form and complex structure determine a pseudo-Kähler structure on SL(3,R) Hitchin component. Graph cohomology solves symplectic problems in surface mapping groups.
problem Compute the symplectic decomposition of Torelli group and its cohomology.
method Graph cohomology and ideas from graph cohomology.
result Effective computation of the symplectic decomposition of the quadratic dual of the lower central series of the Torelli group.
Renormalized circle diffeos with breaks converge to Moebius maps with a symplectic structure.
problem Analyzing the renormalization of circle diffeomorphisms with breaks.
method Proving convergence to invariant piecewise Moebius maps and identifying the renormalization operator with a sub-action of the mapping class group.
result Renormalization identifies with a symplectic form preserved by the mapping class group.
We consider symplectic Floer homology in the lowest nontrivial dimension, that is to say, for area-preserving diffeomorphisms of surfaces. Particular attention is paid to the quantum cap product; we show that it distinguishes the trivial element of the mapping class group from any nontrivial one.
We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes wil…
Obstructions found for closed Fedosov star products on symplectic and Kähler manifolds.
problem Existence of closed Fedosov star products on symplectic and Kähler manifolds.
method Normalized trace of Fedosov star product, cohomology classes, and formal 2-forms.
result Integral invariants attached to symplectic and Kähler manifolds as obstructions to closed Fedosov star products.
In this article, using combinatorial techniques of mapping class groups, we show that a Stein fillable integral homology 3-sphere supported by an open book decomposition with page a 4-holed sphere admits a unique Stein filling up to diffeomorphism. Furthermore, according to a property of deforming symplectic fillin…
Using the existence of certain symplectic submanifolds in symplectic 4-manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal Lefschetz fibrations over surfaces of positive genus. This estimate is then used to deduce that mapping class groups are not uni…
Given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact s…
Maximal representations in symplectic lattices proven for most cases.
problem Understanding maximal representations in symplectic lattices.
method Analyzing mapping class group orbits and continuous deformations of maximal diagonal representations.
result Proof of maximal representations in most lattices of Sp(2n,R).
Given a closed, oriented surface, possibly with boundary, and a mapping class, we obtain sharp lower bounds on the number of fixed points of a surface symplectomorphism (i.e. area-preserving map) in the given mapping class, both with and without nondegeneracy assumptions on the fixed points. This generalizes the Poinca…
The paper extends a theorem about momentum maps to singular symplectic spaces.
problem Extending a theorem about momentum maps to singular symplectic spaces.
method Using integral affine stratification and equivariant locally trivial fibrations, the paper extends the linear variation theorem to singular values of the momentum map.
result Cohomology classes of symplectic forms on reduced spaces vary linearly within strata.
The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We give an algorithm to compute HF_*(f) for f a surface symplectomorphism in a pseudo-Anosov or reducible mapping class, com…
Maximal discs in Anti-de Sitter space linked to Teichmüller space.
problem Characterizing maximal discs in Anti-de Sitter space.
method Introduced and studied maximal discs, identified their parametrization space, and used the Mess map to relate them to Teichmüller space.
result Maximal discs of Weil-Petersson class in Anti-de Sitter space are bijectively parametrized by certain submanifolds of Teichmüller space.
Symplectic manifolds derived from Torelli groups, showing complex structure invariants without holomorphic structures.
problem Deriving symplectic structures from Torelli groups without holomorphic structures.
method Symplectization of mapping tori derived from Torelli group representatives.
result Symplectic manifolds with complex structure invariants but no holomorphic structures.
Study of symplectically flat connections and their functionals on smooth manifolds.
problem Understanding symplectically flat connections and their functionals on smooth manifolds.
method Extend symplectically flat connections to ζ-flat connections, introduce functionals with zeroes as symplectically flat connections, study critical points of these functionals, describe characteristic classes of ζ-flat bundles. result Novel geometric flows and characteristic classes of ζ-flat bundles are described. A group action on a moduli space is shown to be faithful.
problem Injectivity of a homomorphism from mapping class group to symplectic mapping class group.
method Instanton Floer homology, Atiyah-Floer Conjecture, Heegaard Floer strategy.
result The homomorphism is injective for surfaces of genus at least 2.
We introduce the modular class of a Poisson map. We look at several examples and we use the modular classes of Poisson maps to study the behavior of the modular class of a Poisson manifold under different kinds of reduction. We also discuss their symplectic groupoid version, which lives in groupoid cohomology.
We exhibit a finitely generated group $\M$ whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface $\su$ of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g with n bo…
Study shows Hamiltonian diffeomorphisms form a connected component in C0-topology for most symplectic rational surfaces.
problem Understanding the C0-topology of symplectic diffeomorphisms on rational surfaces. method Combining techniques from symplectic mapping class groups and C0-symplectic topology, establishing C0-distance estimates. result Hamiltonian diffeomorphisms form a connected component in C0-topology for all but a few exceptions on rational surfaces. In this paper we compute the homotopy groups of the symplectomorphism groups of the 3-, 4- and 5-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy groups of the compactly supported symplectomorphism groups of the cotangent bundl…
We present an approach to Gromov-Witten invariants that works on arbitrary (closed) symplectic manifolds. We avoid genericity arguments and take into account singular curves in the very formulation. The method is by first endowing mapping spaces from (prestable) algebraic curves into the symplectic manifold with the st…
We calculate the abelianizations of the level L subgroup of the genus g mapping class group and the level L congruence subgroup of the 2g×2g symplectic group for L odd and g≥3.
In this paper, we examine mapping class group relations of some symplectic manifolds. For each n≥1 and k≥1, we show that the 2n-dimensional Weinstein domain W={f=δ}∩B2n+2, determined by the degree k homogeneous polynomial f∈C[z0,…,zn], has a Boothby-Wang type boundary …
Introduces group-valued momentum maps for symplectic fiber bundles.
problem Determining momentum maps for non-classical actions of automorphism groups.
method Develops a general framework for group-valued momentum maps.
result Illustrates the power of group-valued momentum maps with various examples.
We prove that the symplectic group Sp(2n,Z) and the mapping class group ModS of a compact surface S satisfy the R∞ property. We also show that Bn(S), the full braid group on n-strings of a surface S, satisfies the R∞ property in the cases where S is either the compact disk …
The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology theory for symplectomorphisms of surface and a calculation of Seidel's symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and asympltot…
New calculations of topological complexity for symplectic CW-complexes.
problem Calculating topological complexity for symplectic CW-complexes.
method Using atoroidal cohomology classes and CW-complexes, proving topological complexity for symplectic spaces.
result Every atoroidally symplectic CW-complex of dimension 2n has topological complexity 4n.