We give definitions of moduli spaces of framed, r-Spin and Pin surfaces. We apply earlier work of the author to show that each of these moduli spaces exhibits homological stability, and we identify the stable integral homology with that of certain infinite loop spaces in each case. We further show that these moduli spa…
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48 results for “r-spin Mapping Class Group”
We give a combinatorial model for r-spin surfaces with parametrised boundary based on Novak (2015). The r-spin structure is encoded in terms of -valued indices assigned to the edges of a polygonal decomposition. This combinatorial model is designed for our state sum construction of two-dimensional topolog…
Higher spin mapping class groups and strata of Abelian differentials over Teichm{ü}ller spacemath.GT
For , we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli space. Building off of work of the first author (arXiv:1901.05482), we find that the …
Constructs chiral rational homology spheres with hyperbolic groups.
problem Existence of strongly chiral rational homology spheres with hyperbolic fundamental groups.
method Construction of rational homology spheres using -spins and investigation of self-map degrees.
result Strongly chiral rational homology spheres with hyperbolic fundamental groups constructed.
Study of monodromy and vanishing cycles for complete intersection curves.
problem Computing topological monodromy of complete intersection curves.
method Innovative tools for studying monodromy of tensor products of very ample line bundles, induction on multi-degree.
result Answer given by the r-spin mapping class group associated to the maximal root of the adjoint line bundle.
We introduce the stack of r-spin maps. These are stable maps into a variety V from n-pointed algebraic curves of genus g, with the additional data of an r-spin structure on the curve. We prove that this stack is a Deligne-Mumford stack, and we define analogs of the Gromov-Witten classes associated to these spaces. We s…
Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfacesmath.AG
Monodromy and vanishing cycles computed for ample linear systems on simply connected surfaces.
problem Characterizing curves that can be vanishing cycles in degenerations of linear systems.
method Computing mapping class group-valued monodromy and identifying it with r-spin mapping class groups.
result Identifies simple closed curves as vanishing cycles and provides characterizations of discriminants and Lefschetz fibrations.
We give a review of the quantum singularity theory of Fan-Jarvis-Ruan and the r-spin theory of Jarvis-Kimura-Vaintrob and describe the work of Abramovich-Jarvis showing that for the singularity A_{r-1} = x^r the stack of A_{r-1}-curves of is canonically isomorphic to the stack of r-spin curves. We prove that the A_{r-1…
Generalised spin structures, or r-spin structures, on a 2-dimensional orbifold Σare r-fold fibrewise connected coverings (also called r-th roots) of its unit tangent bundle STΣ. We investigate such structures on hyperbolic orbifolds. The conditions on r for such structures to exist are given. The action of the diffeomo…
A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this case the vector spaces agree with the state spaces of the correspondi…
The purpose of this note is introduce a new axiom (called the Descent Axiom) in the theory of -spin cohomological field theories. This axiom explains the origin of gravitational descendants in this theory. Furthermore, the Descent Axiom immediately implies the Vanishing Axiom, explicating the latter (which has no a …
Novel approach for large genus intersection number asymptotics.
problem Computing intersection numbers in large genus.
method Resurgent analysis of n-point functions with quantum curve.
result Extension of Aggarwal's results and new -spin and Theta-class intersection numbers.
The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. What makes this study tractable is that there is a surjective map from the hyperelliptic mapping class group to a mapping class group of a punctured sphere. The more general family of superelliptic mapping…
Classifies manifolds with dense conjugacy classes in their mapping class groups.
problem Classifying manifolds based on conjugacy classes in their mapping class groups.
method Analyzing connected orientable 2-manifolds and their mapping class groups.
result Mapping class groups of certain manifolds have dense conjugacy classes.
Explicit presentations found for asymptotically rigid mapping class groups.
problem Understanding the structure of asymptotically rigid mapping class groups.
method Using a graph of groups structure, we compute explicit presentations.
result Computed explicit presentations for asymptotically rigid mapping class groups of surfaces.
The article explores the mapping class group using unicellular maps and provides filtrations.
problem Understanding the structure of the mapping class group.
method Using unicellular maps and surgeries, the article describes the mapping class group.
result Provides filtrations of the mapping class group.
Study of hyperelliptic mapping class groups with applications and profinite completions.
problem Understanding hyperelliptic mapping class groups and their properties.
method Defined and studied hyperelliptic mapping class groups, applied theory to counterexamples, and examined profinite completions.
result Found a counterexample to a conjecture about mapping class groups and extended congruence subgroup property.
Overview of infinite surface mapping class groups.
problem Understanding mapping class groups of infinite surfaces.
method Survey of recent research findings.
result Recent developments in mapping class groups of infinite surfaces.
In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show th…
Continuous epimorphisms between certain mapping class groups are induced by homeomorphisms.
problem Understanding continuous epimorphisms between specific mapping class groups.
method Analyzing subgroups of mapping class groups of infinite-genus 2-manifolds with no planar ends.
result Continuous epimorphisms are induced by homeomorphisms for the specified subgroups.
Paper shows mapping class groups are not extremely amenable except for specific cases.
problem Determining when mapping class groups are extremely amenable.
method Utilized Kechris-Pestov-Todorčević machinery.
result Big mapping class groups are not extremely amenable unless the surface is a sphere or a once-punctured sphere.
This thesis introduces big mapping class groups and their structure.
problem Understanding mapping class groups of infinite-type surfaces.
method Systematic introduction and analysis of structure and topological generation.
result Key differences from finite-type mapping class groups.
The study finds abundant normal generators for mapping class groups.
problem Understanding normal generation in mapping class groups.
method Analyzing restrictions on invariant subsurfaces and Teichmüller spaces.
result Reducible mapping classes can normally generate mapping class groups based on their asymptotic translation lengths.
Finite presentations for mapping class groups of surfaces and surfaces with points/boundaries.
problem Finding finite presentations for balanced superelliptic mapping class groups.
method Construct finite presentations for corresponding liftable mapping class groups in a different generating set.
result Finite presentations for balanced superelliptic mapping class groups of various surfaces.
Study maps surface configurations to Heisenberg homologies for mapping class groups.
problem Understanding Mapping Class Groups of punctured surfaces.
method Action of mapping classes on Heisenberg homologies of surface configurations.
result Representations of Mapping Class Groups derived from Heisenberg homologies.
Three elements generate balanced superelliptic mapping class groups.
problem Generating balanced superelliptic mapping class groups.
method Proving groups are generated by three elements through normalizers and liftable mapping class groups.
result Balanced superelliptic mapping class groups are generated by three elements.
We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that ma…
Two elements generate extended mapping class groups of certain surfaces.
problem Generating extended mapping class groups with specific elements.
method Analyzing finite order elements and isotopy classes of homeomorphisms.
result Extended mapping class groups of certain surfaces are generated by two elements of finite order.
We prove that both the hyperelliptic mapping class group and the extended hyperelliptic mapping class group are generated by two torsion elements. We also compute the index of the subgroup of the hyperelliptic mapping class group which is generated by involutions and we prove that the extended hyperelliptic mapping cla…
Computes mapping class groups of 4-manifolds with boundary.
problem Computing mapping class groups for 4-manifolds with boundary.
method Topological and smooth methods applied to compact, simply connected 4-manifolds.
result Description of topological and stable smooth mapping class groups.
Proves asymptotic mapping class groups of Cantor manifolds are of type F_infinity.
problem Finiteness properties of asymptotic mapping class groups.
method General theorem deducing asymptotic mapping class groups of Cantor manifolds are of type F_infinity under certain hypotheses.
result Asymptotic mapping class groups of Cantor manifolds are of type F_infinity.
Article provides conditions for embedding braid groups into mapping class groups.
problem Embedding braid groups into mapping class groups.
method Necessary and sufficient condition for embedding finite index subgroups.
result Conditions for embedding braid groups into mapping class groups.
New descriptions of a subgroup in mapping class groups.
problem Understanding the normal subgroup generated by genus bounding pair maps.
method Using Chillingworth and Casson-Morita invariants.
result Two descriptions of the subgroup generated by genus bounding pair maps.
Infinitely many unique ways to generate a surface's mapping class group.
problem Identifying unique ways to generate mapping class groups.
method Analyzing Nielsen equivalence classes of two-element generators.
result There are infinitely many unique ways to generate a surface's mapping class group.
Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.
problem Understanding asymptotic dimension of big mapping class groups of infinite type surfaces.
method Analyzing big mapping class groups with coarsely bounded generating sets and essential shifts.
result Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.
The paper studies liftable mapping class groups of cyclic covers of spheres.
problem Understanding liftable mapping class groups of cyclic covers of spheres.
method Derived finite generating sets, provided algorithms, determined isomorphism classes, derived presentations, and calculated normalizers and centralizers.
result Presentations and isomorphism classes of liftable mapping class groups for various covers.
Study mapping class groups of infinite type surfaces with noncompact boundaries.
problem Classify pure mapping class groups of infinite type surfaces.
method Developed a method to cut surfaces into simpler ones and combined recent results.
result Complete classification of perfect and uniformly perfect pure mapping class groups.
New findings on generating mapping class groups with specific torsion elements.
problem Understanding the structure of mapping class groups through torsion elements.
method Analyzing the orders of torsion elements required to generate mapping class groups of surfaces of varying genus.
result For specific genera, mapping class groups can be generated by two torsion elements of particular orders.
We show that the mapping class group of a compact orientable surface with higher complexity has the following extreme rigidity in the sense of measure equivalence: if the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernel. Moreover, we describe all lattice emb…
The paper computes the mapping class group of certain 6-manifolds.
problem Computing the mapping class group of specific 6-manifolds.
method Algebraic properties and homology groups of the mapping class group were determined.
result The mapping class group is residually finite and virtually torsion-free.
We show that finite index subgroups of the handlebody group are rigid in their ambient mapping class group: any injective map of a finite index subgroup of the genus handlebody group into the genus mapping class group is conjugation by a mapping class group element. On the other hand, we construct an injection …
We show that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups, induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses automorphisms of free groups with boundaries which play the role of mapping class groups o…
The paper proves mapping class groups of closed surfaces are simply connected at infinity.
problem Understanding connectivity at infinity for mapping class groups of surfaces.
method Proved a general simply connected at infinity result for finitely presented groups.
result All mapping class groups of closed surfaces of genus ≥ 3 are simply connected at infinity.
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.
Study the relationship between orbit braid group and equivariant mapping class group on surfaces.
problem Understanding the relationship between mapping class groups and braid groups with group actions.
method Using the fibration and exact sequence.
result The conclusion is closely connected with the braid group of the quotient space.
Study shows mapping class groups are one-ended for surfaces with at least one end.
problem Analyzing the number of ends in mapping class groups of surfaces.
method Proving the associated translatable curve graph is one-ended, quasi-isometric to the mapping class group.
result Mapping class groups are one-ended for surfaces with at least one end of discrete type.
On the large scale geometry of big mapping class groups of surfaces with a unique maximal endmath.GT
Characterizes and analyzes the large scale geometry of big mapping class groups of surfaces.
problem Analyzing the large scale geometry of big mapping class groups of surfaces with a unique maximal end.
method Building on previous work, the paper characterizes and analyzes the large scale geometry of big mapping class groups of surfaces with a unique maximal end.
result Proves that any locally CB big mapping class group is CB generated and gives an explicit criterion for determining which big mapping class groups are CB generated.
The paper explores the twisted Rokhlin property in mapping class groups of surfaces.
problem Classifying surfaces whose mapping class groups have the twisted Rokhlin property.
method Generalizing the Rokhlin property to the twisted version, the authors classify surfaces based on their mapping class groups' properties.
result The mapping class groups of connected orientable infinite-type surfaces without boundaries have the twisted Rokhlin property, while those of other surfaces do not.