New knot invariants from covering involutions help deduce linear independence results.
problem Understanding the smooth concordance group of knots.
method Covering involutions and adapted ideas from previous research.
result Novel linear independence results in the smooth concordance group of knots.
Study involutions on 3D small covers, proving quotient spaces are linked 2-spheres.
problem Understanding quotient spaces of 3D small covers by involutions.
method Analyzing quotient spaces and using topological properties.
result Quotient spaces of 3D small covers by involutions are linked 2-spheres.
Extends Khovanov homology spectral sequence using Heegaard Floer homology.
problem Relating Khovanov homology and Heegaard Floer homology of branched double covers.
method Involutive Heegaard Floer homology, bordered Floer homology, surgery exact triangle.
result Establishes spectral sequence connecting Khovanov homology and Heegaard Floer homology.
Study fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
problem Characterize fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
method Construct a class of involutions, extend product involutions, and use double covering.
result Any fiber-preserving, orientation-reversing involution factors as a product of an orientation-preserving and a specific class of involutions.
We give a bordered extension of involutive HF-hat and use it to give an algorithm to compute involutive HF-hat for general 3-manifolds. We also explain how the mapping class group action on HF-hat can be computed using bordered Floer homology. As applications, we prove that involutive HF-hat satisfies a surgery exact t…
We prove that the canonical 4-dimensional surgery problems can be solved after passing to a double cover. This contrasts the long-standing conjecture about the validity of the topological surgery theorem for arbitrary fundamental groups (without passing to a cover). As a corollary, the surgery conjecture is reformulate…
We study the conjugation involution in Seiberg-Witten theory in the context of the Ozsváth-Szabó and Bloom's spectral sequence for the branched double cover of a link L in S3. We prove that there exists a spectral sequence of F[Q]/Q2-modules (where Q has degree −1) which converges to $\widetilde{\m…
In this paper, we classify smooth 5-manifolds with fundamental group isomorphic to $\z/2$ and universal cover diffeomorphic to S2×S3. This gives a classification of smooth free involutions on S2×S3 up to conjugation.
New real invariants for 3-manifolds and links.
problem Defining real structures in Floer homology.
method Introducing real spin-c structures and applying monopole Floer homology.
result Invariants for links via double branched covers.
We introduce hyperelliptic simplified (more generally, directed) broken Lefschetz fibrations, which is a generalization of hyperelliptic Lefschetz fibrations. We construct involutions on the total spaces of such fibrations of genus g≥3 and extend these involutions to the four-manifolds obtained by blowing up the …
It is known that every nonorientable surface Σ has an orientable double cover Σ~. The covering map induces an involution on the moduli space $\tilde{\M}$ of gauge equivalence classes of flat G-connections on Σ~. We identify the relation between the moduli space $\M$ and the fixed point set of the modu…
Let M be a 3-manifold admitting a strongly irreducible Heegaard surface S and f:M \to M an involution. We construct an invariant Heegaard surface for M of genus at most 8 g(S) - 7. As a consequence, given a (possibly branched) double cover π:M \to N we obtain the following bound on the Heegaard genus of N: g(N) \leq 4g…
New invariants from Seiberg-Witten theory for 3-spheres with involution.
problem Equivariant Seiberg-Witten Floer theory of rational homology 3-spheres.
method Coupling involution to Seiberg-Witten theory, constructing delta-invariants.
result New Floer-theoretic invariants with properties and applications.
Study d-invariants of L-space double branched covers of arborescent links.
problem Determine d-invariants of double branched covers of arborescent links. method Use immersed curves description of bordered Floer homology and involution of curves.
result Spin d-invariants of Σ2(L) are determined by the signatures of L. Invariant structures link to algebraic curves with specific properties.
problem Linking invariant hypercomplex structures to algebraic curves.
method Mapping invariant structures to algebraic curves with specific properties.
result Invariant hypercomplex structures correspond to algebraic curves with a flat projection and antiholomorphic involution.
In this paper, we investigate existence of inequivalent smooth structures on closed smooth non-orientable 4-manifolds building upon results of Akbulut, Cappell-Shaneson, Fintushel-Stern, Gompf, and Stolz. We add to the number of known constructions and provide new examples of exotic manifolds that are obtained as an ap…
Proves spectral sequence for real Heegaard Floer homology.
problem Real Heegaard Floer homology and its localization.
method Proves existence of a localization spectral sequence.
result Existence of spectral sequence for real Heegaard Floer hat variant.
Study of holomorphic correspondences combining entire maps and Fuchsian groups.
problem Understanding dynamics of entire maps and their interactions with Fuchsian groups.
method Systematic study of (∞:∞) holomorphic correspondences arising from conformal combinations of transcendental entire maps and Fuchsian groups. result The resulting correspondence is the composition of a Möbius involution and the deleted covering correspondence of a meromorphic function with a simple pole.
New non-cobordant hyperbolic manifolds found in certain dimensions.
problem Identifying closed hyperbolic manifolds that are not cobordant.
method Using the cobordism class and fixed point set of an involution, combined with a geodesic embedding.
result Existence of non-cobordant closed hyperbolic manifolds in dimensions not of the form 4m+3. Real Heegaard Floer Homology extends Li's real monopole Floer homology.
problem Computing invariants of three-manifolds with symmetries.
method Real version of Lagrangian Floer homology, combinatorial computation.
result Euler characteristic of real Heegaard Floer homology is a knot invariant.
We are interested in finite groups acting orientation-preservingly on 3-manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain an involution with nonempty connected fixed point set. This condition is satisfied by the isometry group of any hyperbolic cyclic …
In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of fixed points of involutions are also proved.
We construct a simply connected minimal complex surface of general type with pg=0 and K2=2 which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with pg=0 and K2=1. In order to construct the example, we combin…
The pants graph of a non-orientable surface is quasi-isometric to its Teichmüller space.
problem Understanding the relationship between pants graphs and Teichmüller spaces of non-orientable surfaces.
method Constructing a map between pants graphs induced by lifting pants decompositions and proving quasi-isometric embeddings.
result The pants graph of a non-orientable surface is quasi-isometric to its Teichmüller space.
The paper studies mapping class groups of cyclic covers and their liftable counterparts.
problem Understanding the structure of liftable mapping class groups for cyclic covers.
method Analyzes the liftable mapping class groups of regular cyclic covers and derives explicit generating sets.
result The family of self-normalizing subgroups {LModpk(Sg)}k≥2 forms an infinite family in Mod(Sg). Many three dimensional manifolds are two-fold branched covers of the three dimensional sphere. However, there are some that are not. This paper includes exposition about two-fold branched covers and many examples. It shows that there are three dimensional homology spheres that do not two-fold branched cover any manifol…
Study path spaces and their homology, extending loop products and coproducts.
problem Understanding the homology of path spaces in closed manifolds.
method Morse-Bott theory and homology operations.
result Complete computation of extended loop product and coproduct on spheres.
We consider pairs (X,Y) where X is a compact, locally CAT(-1) space, and Y is a totally geodesic subspace. The inclusion induces an embedding of the boundaries at infinity of the universal covers; we focus on the case where these are spheres whose dimensions differ by 2. We show that if the embedding is tame, then it i…
This article announces joint work with Frank Connolly and Jim Davis. We generalize our classification of pseudo-free involutions on the n-torus, by studying the action of the associated infinite group with torsion in the universal cover. Included is a non-Riemannian example obtained from the restriction of the action o…
Study of knot Floer homology and its relation to Heegaard Floer homology via equivariant surgery.
problem Understanding the action of symmetries on knot Floer homology.
method Relating knot Floer homology to Heegaard Floer homology via equivariant surgeries.
result Identify the action of the involution on Heegaard Floer homology with an action on knot Floer homology.
Following I. S. Krasilshchik and A. M. Vinogradov, we regard PDEs as infinite-dimensional manifolds with involutive distributions and consider their special morphisms called differential coverings, which include constructions like Lax pairs and Backlund transformations. We show that, similarly to usual coverings in top…
Defines character varieties for real forms of complex classical groups and proves their properties.
problem Characterizing character varieties for real forms of complex classical groups.
method Proposes and proves properties of GR-character varieties as subsets of GC-character varieties. result Character varieties cover the set of irreducible GC-characters fixed by an anti-holomorphic involution. Real Heegaard Floer homology gets a new grading for certain 3-manifolds.
problem Real Heegaard Floer homology groups get an absolute Z/2 grading under specific conditions.
method Analyzes real Heegaard Floer homology groups with an involution and nullhomologous fixed points.
result Defines a new invariant of knots equal to the Alexander polynomial evaluated at i.
Novikov initiated the study of the algebraic properties of quadratic forms over polynomial extensions by a far-reaching analogue of the Pontrjagin-Thom transversality construction of a Seifert surface of a knot and the infinite cyclic cover of the knot exterior. In this paper the analogy is applied to explain the relat…
The study classifies manifolds realized as orbit spaces of non-free Z2^k actions.
problem Classifying manifolds realized as orbit spaces of non-free Z2^k actions.
method Examining actions of subgroups H on real moment-angle manifolds and analyzing orbit spaces.
result Constructs series of manifolds homeomorphic to S^n and manifolds admitting hyperelliptic involutions.
New symplectic groups defined for Lie subgroups of algebras.
problem Defining new symplectic groups for Lie subgroups of algebras.
method Introducing symplectic group Sp2(G,σ) for Lie subgroups G of associative algebras A with anti-involution σ. result New realizations of spin groups as Sp2(G,σ) for suitable subgroups G. Classifies involutions on spherical 3-manifolds.
problem Classifying involutions on spherical 3-manifolds.
method Geometric approach to conjugacy classification.
result Insights into topological properties of involutions.
Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.
problem Understanding non-Gorenstein involutions on Calabi-Yau threefolds.
method Classification of Calabi-Yau threefolds with specific properties.
result Classification of Calabi-Yau threefolds with Picard rank one and non-Gorenstein involutions.
We study real lines on certain Moishezon threefolds which are potentially twistor spaces of 3CP^2. Here, line means a smooth rational curve whose normal bundle is O(1)^2 and the reality implies the invariance under an anti-holomorphic involution on the threefolds. Our threefolds are birational to double coverings of CP…
The paper studies circular evolutes and involutes of framed curves in Euclidean space.
problem Investigating properties of framed curves and their evolutes and involutes.
method Definition and analysis of circular evolutes and involutes of framed curves, properties of normal surfaces, and their relations.
result Circular evolutes and involutes of framed curves are opposite operations under suitable assumptions, similar to fronts in the Euclidean plane.
New formula for dual knots using involutions.
problem Understanding dual knots and their transformations.
method Involutive analog of knot surgery formula.
result Computed local equivalence class for involutive dual knots.
The study classifies involutions on del Pezzo surfaces.
problem Classifying involutions on del Pezzo surfaces.
method Mapping class group theory and hyperbolic reflection groups.
result A complete classification of involutions on del Pezzo surfaces.
Characterizes the Legendre involution on generic frontals.
problem Identifying the Legendre involution on a specific class of frontals.
method Analyzes generic frontals under mild assumptions and uses complexification.
result Any involution with the same fixed points as the Legendre involution is the Legendre involution.
The paper defines conditions for good involutions in generalized Alexander quandles.
problem Determining conditions for good involutions in generalized Alexander quandles.
method Analyzing the structure of generalized Alexander quandles and their involutions.
result Classification of all good involutions in connected generalized Alexander quandles.
The study examines conditions for symmetric and alternating subgroups in mapping class groups of surfaces.
problem Conditions for torsion elements to generate symmetric or alternating subgroups.
method Analyzes mapping class groups of surfaces, derives necessary and sufficient conditions for conjugates of torsion elements to generate symmetric or alternating subgroups.
result Symmetric or alternating subgroups cannot contain irreducible mapping classes and hyperelliptic involutions.
Study proves naturality and functoriality in a type of Heegaard Floer homology.
problem Proving naturality and functoriality in a specific type of Heegaard Floer homology.
method Used the doubling model for the involution and variations to prove results.
result First-order naturality of involutive Heegaard Floer homology proved.
The paper develops a new theory for knots and 3-manifolds with involutions.
problem Developing a new theory for knots and 3-manifolds with involutions.
method Establishing a version of Seiberg-Witten Floer K-theory for knots and 3-manifolds with involutions.
result 10/8-type inequalities for knots and involutions, yielding lower bounds on stabilizing numbers and relative genera.
The study proves symplectic quandles cannot have good involutions.
problem Existence of good involutions in symplectic quandles.
method Investigation of necessary and sufficient conditions for good involutions.
result Nonexistence of good involutions in symplectic quandles.