For a specific class of 4-manifolds, random isometries cannot lift to orientation-preserving diffeomorphisms.
problem When does a finite group of isometries of a specific 4-manifold lift to orientation-preserving diffeomorphisms?
method Combination of equivariant connected-sum constructions, fixed-point theory, finite group actions on surfaces, analytic combinatorics, and previous work.
result Random subgroups of isometries are asymptotically almost never realizable in orientation-preserving diffeomorphisms.
Develops Floer cohomology for 4-manifolds with involutions and links.
problem Floer cohomology for 4-manifolds with involutions and links.
method Floer cohomology framework for spin^c 4-manifolds with involution and oriented links.
result Proves Frøyshov-type inequalities relating 4-manifold topological quantities and homology cobordism invariants.
Study shows non-cyclic groups of diffeomorphisms can't act on certain 3-manifolds.
problem Realization of finite groups as diffeomorphisms on specific 3-manifolds.
method Analysis of group actions on connected sums of S2imesS1. result No non-cyclic subgroup of twist subgroup can be realized by diffeomorphisms.
Criterion for realizing groups on Enriques manifolds.
problem Realizing groups on Enriques manifolds.
method Using recent developments in Birman-Hilden theory and Nielsen realization for hyper-Kähler manifolds.
result Numerical criterion for realizing groups on Enriques manifolds.
Dehn twists on K3-type 4-manifolds are not homotopy coherently Nielsen realizable.
problem Homotopy coherent Nielsen realization problem for Dehn twists on 4-manifolds
method Using family Seiberg-Witten theory
result Failure of the classical Nielsen realization problem in K3-type 4-manifolds
Solves Nielsen realization problem for hyper-Kähler manifolds.
problem Realization problem for hyper-Kähler manifolds.
method Uses same invariant as for K3 surfaces and determines representation of mapping class group.
result Representation of mapping class group admits a section on its image for some deformation types.
Study shows non-spin 4-manifolds where smooth Nielsen realization fails.
problem Existence of non-spin 4-manifolds with non-realizable mapping class groups.
method Investigation of multi-twists, projective twists, and multi-reflections.
result Examples of non-spin 4-manifolds with non-realizable mapping class groups.
Groups of homotopy equivalences of graphs help realize compact subgroups.
problem Realizing compact subgroups of homotopy equivalences of graphs.
method Introduced a Polish group topology on the group of proper homotopy equivalences and proved the Nielsen Realization theorem.
result Compact subgroups of homotopy equivalences can be realized by simplicial isomorphisms of graphs.
Researchers solve Nielsen Realization problems for K3 surfaces in various categories.
problem Realizing finite groups of mapping classes as diffeomorphisms, isometries, or automorphisms in K3 surfaces.
method Introduced a computable invariant LG and constructed an S4 action by isometries. result Some finite groups are realizable while others are not, depending on preserved structures.
Study shows Dehn twists on certain 4-manifolds cannot be realized by finite order diffeomorphisms.
problem Realization of Dehn twists as finite order diffeomorphisms on spin 4-manifolds.
method Use Y. Kato's 10/8-type inequality for involutions and its refinement.
result Dehn twists about specific spheres in certain 4-manifolds are not homotopic to finite order diffeomorphisms.
Extends Nielsen realization to infinite-type surfaces, classifying torsion elements and topological groups.
problem Realizing finite subgroups of mapping class groups on infinite-type surfaces.
method Extending Kerckhoff's result to infinite-type surfaces, using hyperbolic metrics and topological group properties.
result Compact subgroups of mapping class groups are finite, and locally compact subgroups are discrete.
Solves a problem related to Nielsen realization for certain groups.
problem Whether a cocompact proper topological manifold is equivariantly homotopy equivalent to a classifying space.
method Assumes a zero-dimensional singular set and uses properties of hyperbolic groups and aspherical manifolds.
result Solves the problem for specific groups containing a normal torsion-free subgroup.
Researchers solved a complex problem for a specific type of 4-manifolds.
problem Realizing mapping classes of finite order on del Pezzo surfaces.
method Synthesized results from reflection group theory and 4-manifold topology.
result Provided both positive and negative examples of realizability.
Study on realizing subgroup twists in 3-manifolds.
problem Realizing subgroups of twist groups in 3-manifolds.
method Analyzing Nielsen realization problem for Twist(M) subgroups, applying to Burnside problem.
result Nontrivial subgroups of Twist(M) are realized by diffeomorphisms if and only if they are cyclic and M is a connected sum of lens spaces.
Solves a problem related to classifying spaces for proper actions and Nielsen Realization.
problem Whether a cocompact proper topological Γ-manifold is equivariantly homotopy equivalent to the classifying space for proper actions.
method Using Poincaré models and assuming a zero-dimensional singular set, the problem is solved in the Poincaré category.
result New results about Brown's problem are obtained under certain conditions on the underlying group.
A criterion for Whitney disks connects intersections in 3-manifold homology.
problem Existence of Whitney disks in Heegaard Floer homology.
method Use Nielsen theory to establish a criterion.
result Simple criterion for the existence of Whitney disks.
The purpose of this expository paper is to present new directions in the classical Nielsen-Reidemeister fixed point theory. We describe twisted Burnside-Frobenius theorem, groups with R∞ \emph{property} and a connection between Nielsen fixed point theory and symplectic Floer homology.
The paper solves the Nielsen realization problem for high degree del Pezzo surfaces.
problem Which finite subgroups of the mapping class group of a del Pezzo surface lift to the diffeomorphism group?
method Classification and partial answers for d≥7, equivariant connected sum for d=6. result Complete classification for d≥7, partial answer for d=6. 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…
The main goal of this paper is to give the first examples of equivariant aspherical Poincare complexes, that are not realized by group actions on closed aspherical manifolds M. These will also provide new counterexamples to the Nielsen realization problem about lifting homotopy actions of finite groups to honest grou…
Innovates rotation index for matrix pairs, solving group action problems.
problem Solving group actions problems, especially Nielsen realization and higher-rank Anosov actions.
method Rotation index and Milnor--Munkres--Novikov pairing applied to Z2 group actions. result Solved specific group action problems using new matrix pair invariant.
We study the action of the mapping class group on the real homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen-Thurston …
For a based manifold (M,*), the question of whether the surjection Diff(M,*) \rightarrow π_0 Diff(M,*) admits a section is an example of a Nielsen realization problem. This question is related to a question about flat connections on M-bundles and is meaningful for M of any dimension. In dimension 2, Bestvina-Church-Sou…
Nielsen realization problem for the mapping class group Mod(Sg) asks whether the natural projection pg:Homeo+(Sg)→Mod(Sg) has a section. While all the previous results use torsion elements in an essential way, in this paper, we focus on the much more difficult problem of realization of…
We discuss recent results and open questions on the broad theme of (Nielsen) realization problems. Beyond realizing subgroups of mapping class groups, there are many other natural instances where one can ask if a surjection from a group of diffeomorphisms of a manifold to another group admits a section over particular …
We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed homotopy class and all hyperbolic metrics on the surface. We give explicit example…
The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough then no section exists over the entire mapping class group. We prov…
We give a Dehn-Nielsen type theorem for the homology cobordism group of homology cylinders by considering its action on the acyclic closure, which was defined by Levine, of a free group. Then we construct an additive invariant of those homology cylinders which act on the acyclic closure trivially. We also describe some…
The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism f of a closed manifold is isotopic to a map realizing the Nielsen number of f, which is a lower bound for the number of fixed points among all maps homotopic to f. The main theorem of this paper proves this conjecture for all orientation pre…
Conditions for Baumslag-Solitar subgroups in mapping class groups.
problem Characterizing Baumslag-Solitar subgroups in mapping class groups.
method Analyzing necessary and sufficient conditions for subgroup embeddings.
result Conditions for embedding BS(p,q) in Mod(Sg), including ∣p∣=∣q∣ and reducibility. The pure braid group cannot be realized as area-preserving homeomorphisms.
problem Realizing the pure braid group inside area-preserving homeomorphisms.
method Using rotation numbers to show non-realizability.
result The pure braid group has no realization inside area-preserving homeomorphisms.
In this article, we propose two algorithms for determining the Nielsen-Thurston classification of a mapping class ψ on a surface S. We start with a finite generating set X for the mapping class group and a word ψ in ⟨X⟩. We show that if ψ represents a reducible mapping class in $\Mod(S)$ then …
Quantum complexity lowerbound proved using differential geometry.
problem Proving lower bounds on quantum complexity.
method Applied the Bishop-Gromov bound to Nielsen's complexity geometry.
result Lower bounds on quantum complexity are exponentially large.
Study supports conjecture about pretzel links' homology.
problem Investigate extreme Khovanov homology of pretzel links.
method Geometric realizations of the independence simplicial complex.
result Supports conjecture that extreme Khovanov homology is torsion-free.
We define generalized currents associated with immersions of abstract oriented solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geo…
Friedl and Kim show any taut sutured manifold can be realized as a twisted homology product, but their proof gives no practical description of how complicated the realizing representation needs to be. We give a number of results illustrating the relationship between the topology of a taut sutured handlebody and the com…
Let Mod(Sg) denote the mapping class group of the closed orientable surface Sg of genus g≥2, and let f∈Mod(Sg) be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on Sg that realizes f as an isometry. In other words, this procedure yield…
We consider a classical N. Steenrod's problem on realization of homology classes by images of the fundamental classes of manifolds. It is well-known that each integral homology class can be realized with some multiplicity as an image of the fundamental class of a manifold. Our main result is an explicit purely combinat…
We introduce the notion of partial presimplicial set and construct its geometric realization. We show that any semiadequate diagram yields a partial presimplicial set leading to a geometric realization of the almost-extreme Khovanov homology of the diagram. We give a concrete formula for the homotopy type of this geome…
The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with as…
This paper develops a new homology theory for biquandles and discusses geometric realizations.
problem Constructing knot invariants using set-theoretic Yang-Baxter equation.
method Developed a normalized (co)homology theory for biquandles and geometrically realized them.
result Geometric realization of biquandles has finitely generated second homotopy group for finite biquandles.
Determine lens spaces as closures of homology cobordisms over planar surfaces.
problem Identify conditions for lens spaces to be closures of homology cobordisms over planar surfaces.
method Use Chebotarev density theorem in the proof.
result Every lens space is represented as a closure of homology cobordism over a planar surface with three boundary components.
We show that every area-minimizing hypercone and every oriented Lawlor cone in [Law91] can be realized as a tangent cone at a point of some homologically area-minimizing singular compact submanifold. In particular this generalizes the result of N. Smale [Sma99].
Defines Fenchel-Nielsen coordinates for SL(3,C) representations.
problem No specific problem stated; coordinates defined for a new context.
method Introduced Fenchel-Nielsen coordinates for mSL(3,C) representations. result Relates to classical and generalized Fenchel-Nielsen coordinates.
Unified proof of Nielsen-Thurston classification via Teichmüller's theorem.
problem Classifying mapping class groups and rational maps.
method Unified proof following Bers' approach.
result Unified proof of Nielsen-Thurston classification.
We conjecture the existence of four independent gradings in the colored HOMFLY homology. We describe these gradings explicitly for the rectangular colored homology of torus knots and make qualitative predictions of various interesting structures and symmetries in the colored homology of general knots. We also give a si…
This paper models CSI 300 index volatility using machine learning and addresses jump prediction.
problem Volatility modeling and jump prediction for high-frequency CSI 300 index data.
method Generalized Barndorff-Nielsen and Shephard model with machine learning algorithms for parameter estimation and forecast evaluation.
result Deterministic component of stochastic volatility processes can be captured over short and longer-term windows.
Study strong Nielsen equivalence on punctured disc homeomorphisms.
problem Characterize strong Nielsen equivalence of periodic points in punctured disc homeomorphisms.
method Braid theory and trace formula application.
result Necessary and sufficient conditions for strong Nielsen equivalence of periodic points.