Defines an equivariant cobordism category for finite groups.
problem Understanding geometric structures with group actions.
method Introduces a new category for (d−1)-dimensional closed smooth G-manifolds and their cobordisms. result Identifies the homotopy type of the classifying space of the equivariant cobordism category as fixed points of an infinite loop space.
Classifies knots that bound equivariant surfaces with free symmetries.
problem Classifying knots that bound equivariant surfaces with free symmetries.
method Homology cobordism classification of lens spaces using d-invariants.
result Numerical condition determining free periods for torus knots.
Constructs a functor for equivariant smooth h-cobordisms.
problem Defines a functor for equivariant smooth h-cobordisms.
method Constructs an (∞,1)-functor mapping smooth G-manifolds to spaces of equivariant h-cobordisms. result The functor structure is subtle and relies on new ideas.
Proves an equivariant version of Heegaard Floer link surgery formula.
problem Equivariant knot surgery in S3. method Naturality theorem for bordered modules.
result Kernel of forgetful map contains a Z∞-summand. Study of equivariant movie moves for involutive links.
problem Equivariant cobordisms between involutive links.
method Equivariant Morse theory and singularity theory.
result 39 equivariant movie moves for isotopic cobordisms.
Study of equivariant ribbon concordance using Khovanov homology.
problem Understanding equivariant ribbon concordance.
method Functoriality of equivariant Khovanov homology under equivariant cobordisms, and induced split injection.
result Equivariant ribbon concordances induce a split injection on equivariant Khovanov homology.
Research examines triangulations and homology cobordism groups.
problem Understanding the three-dimensional homology cobordism group.
method Local equivalence methods from Pin(2)-equivariant Seiberg-Witten Floer spectra and involutive Heegaard Floer homology.
result Review of known results and new insights into the group.
Fixed-point techniques compute semifree geometric circle-equivariant complex cobordism.
problem Computing the coefficient ring of semifree geometric circle-equivariant complex cobordism.
method Fixed-point techniques applied to 19th-century methods.
result Recover a 2004 result of Sinha.
New invariants refine link homology, showing large genus differences.
problem Understanding genus differences in equivariant cobordisms.
method Refined Bar-Natan homology for involutive links, constructing new numerical invariants.
result Difference between equivariant and isotopy-equivariant slice genera can be arbitrarily large.
We give a simple and explicit presentation of the Z/2-equivariant complex cobordism ring.
We compute the Pin(2)-equivariant Seiberg-Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu's conjecture that β=−μˉ for Seifert integral homology three-spheres. We show that the Manolescu invari…
We show that the theory of stable complex G-cobordisms, for a torus G, is embedded into the theory of stable complex G-cobordisms of not necessarily compact manifolds equipped with proper abstract moment maps. Thus the introduction of such non-compact cobordisms in the stable complex G-cobordism theory does not…
The paper shows how to transform certain 3D shapes into hyperbolic ones.
problem Transforming 3D shapes with group actions into hyperbolic forms.
method Using homology cobordisms and equivariant corks.
result Any 3D manifold with a finite group action is equivariantly homology cobordant to a hyperbolic manifold.
New map constructed from equivariant spectra for manifold study.
problem Understanding equivariant parametrized h-cobordism in non-manifold settings.
method Constructed a map from suspension G-spectrum to equivariant A-theory spectrum, compatible with tom Dieck splitting formulas.
result Fiber of constructed map is wedge of stable h-cobordism spectra.
Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.
problem Finite group actions on rational homology 3-spheres.
method Equivariant version of Seiberg-Witten-Floer stable homotopy type.
result Definition of d-invariants with Froyshov-type inequality. Calculates cobordism ring of stably almost complex C_p-manifolds.
problem Calculating the cobordism ring of stably almost complex C_p-manifolds.
method Complete calculation using generators and relations.
result Generators compared with geometrically-defined ones by Kosniowski.
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.
The paper studies hyperbolic three-manifolds and their geometric constraints.
problem Understanding the interaction between hyperbolic geometry and homology cobordism.
method Derived explicit bounds on relative grading and invariants in monopole Floer homology.
result Explicit bounds on numerical invariants and subgroup structure of homology cobordism.
New invariants detect corks obstructing homology ball extensions.
problem Detecting corks obstructing homology ball extensions.
method Adapting Heegaard Floer homology formalism for local equivalence, using monotonicity and surgery.
result Established new families of corks and proved various examples are strongly non-extendable.
Develops equivariant grid homology for strongly invertible knots.
problem Invariants of strongly invertible knots.
method Equivariant grid diagrams and mapping cones.
result Equivariant unknotting numbers and genus bounds.
New geometric model for equivariant cohomotopy using bordism.
problem Identifying equivariant cohomotopy classes with fixed point bordism.
method Equivariant Pontrjagin-Thom construction for global quotient orbifolds.
result Recovery of Wasserman's construction and new perspective on equivariant bordism.
Study cyclic group actions on spin 4-manifolds with boundary using Seiberg-Witten theory.
problem Understanding cyclic group actions on spin 4-manifolds with boundary.
method Using Seiberg-Witten equations and lattice constructions, define equivariant refinements of the invariant κ.
result Equivariant relative 10/8-ths type inequalities for spin equivariant cobordisms between rational homology spheres.
Paper defines half-Conway polynomial and computes it for knots up to 12 crossings.
problem Computing and characterizing half-Conway polynomials of knots.
method Normalized Conway polynomial, equivariant skein relation, diagrammatic interpretation.
result First examples of non-slice strongly negative amphichiral knots with determinant one.
Researchers compute K-theory for cohomogeneity-one actions.
problem Computing equivariant K-theory for cohomogeneity-one actions.
method Equivariant homotopy theory, representation theory, Lie theory.
result Derived generators and relations for K-theory ring.
Using Seiberg-Witten Floer spectrum and Pin(2)-equivariant KO-theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology 3-spheres. As an application, we give explicit constrains on the intersection forms of spin 4-manifolds bounded by Brieskorn spheres $\pmΣ(2,3,6k\…
Let D be a (generalized) Dirac operator on a non-compact complete Riemannian manifold M acted on by a compact Lie group G. Let v:M−−>Lie(G) be an equivariant map, such that the corresponding vector field on M does not vanish outside of a compact subset. These data define an element of K-theory of the tran…
New knot invariant from equivariant Heegaard Floer cohomology.
problem Defining a knot invariant from Heegaard Floer cohomology.
method Defined a transverse knot invariant using equivariant Heegaard Floer cohomology.
result The invariant is nonvanishing and functorial under certain cobordisms.
This work extends knot homology theory to links, proving exact triangles and categorifying link signatures.
problem Extending knot homology theory to links and proving exact triangles.
method Equivariant singular instanton Floer theory, circle-equivariant Morse-Floer theory, cobordism constructions.
result Established unoriented skein exact triangles and categorified link signatures.
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.
Witt algebra acts on Khovanov-Rozansky homology of links.
problem Understanding algebraic structures in knot theory.
method Construction of Witt algebra action on Khovanov-Rozansky homology.
result Induced maps between twists of the homology via link cobordisms.
In this paper we study the (equivariant) topological types of a class of 3-dimensional closed manifolds (i.e., 3-dimensional small covers), each of which admits a locally standard (Z2)3-action such that its orbit space is a simple convex 3-polytope. We introduce six equivariant operations on 3-dimensional …
We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there…
We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg-Witten Flo…
Paper studies spectral invariants and monopole Floer homology for rational homology three-spheres.
problem Tackles the existence of positive scalar curvature metrics on ribbon homology cobordisms.
method Defines an R-filtration on the equivariant complex of monopole Floer homology via Chern-Simons-Dirac functional, leading to a spectral invariant.
result Shows that the spectral invariant provides an obstruction to the existence of positive scalar curvature metrics on ribbon homology cobordisms.
We give a proof of the cobordism invariance of the index of elliptic pseudodifferential operators on sigma-compact manifolds, where, in the non-compact case, the operators are assumed to be multiplication outside a compact set. We show that, if the principal symbol class of such an elliptic operator on the boundary of …
New instanton homology theories for 3-manifolds and bundles.
problem Defining instanton homology for a class of 3-manifolds and bundles.
method Functorial construction for 4-manifold cobordisms, algebraic construction of homology theories for dg-modules.
result Isomorphic to Floer's instanton homology for admissible bundles, calculates I∞ in all cases it is defined. Closed oriented 4-manifolds with the same geometrically 2-dimensional fundamental group (satisfying certain properties) are classified up to s-cobordism by their w2-type, equivariant intersection form and the Kirby-Siebenmann invariant. As an application, we obtain a complete homeomorphism classification of closed…
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.
New instanton invariants for rational homology spheres defined and shown to be functorial.
problem Defining and proving invariance of instanton homology groups for rational homology spheres.
method Novel suspended flow category technique to handle obstructed cobordisms and prove wall-crossing formula.
result Instanton invariant λI(Y) conjecturally equals Casson-Walker invariant for rational homology spheres. Manolescu correction terms are numerical invariants of homology three-spheres arising from Pin(2)-equivariant Seiberg-Witten theory that contain information about homology cobordism. We discuss several constraints on these invariants for homology spheres obtained by Dehn surgery on a knot in the three-sphere…
New mappings solve long-standing problems in 3-space.
problem Longstanding problems for bounded locally homeomorphic quasiregular mappings in 3-space.
method Constructed bounded locally homeomorphic quasiregular mappings in the unit 3-ball using non-trivial hyperbolic cobordisms.
result Solved long-standing problems for quasiregular mappings in 3-space.
Using the conjugation symmetry on Heegaard Floer complexes, we define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z4-equivariant Seiberg-Witten Floer homology. Further, we obtain two new invariants of homology cobordism, d and dˉ…
We prove Furuta-type bounds for the intersection forms of spin cobordisms between homology 3-spheres. The bounds are in terms of a new numerical invariant of homology spheres, obtained from Pin(2)-equivariant Seiberg-Witten Floer K-theory. In the process we introduce the notion of a Floer K_G-split homology sphere; thi…
New conditions prevent non-trivial relations in local equivalence group.
problem Understanding non-trivial relations in homology cobordism groups.
method Filtered instanton Floer homology and rs-invariants. result Certain conditions prevent non-trivial relations in the local equivalence group.
New invariants derived from link homology for 4-manifolds.
problem Creating invariants for surfaces in smooth 4-manifolds.
method Skein lasagna modules based on equivariant and deformed glN link homology. result Non-vanishing and decomposition results for deformed glN skein lasagna modules. Let n be a positive integer, and let ℓ>1 be square-free odd. We classify the set of equivariant homeomorphism classes of free Cℓ-actions on the product S1×Sn of spheres, up to indeterminacy bounded in ℓ. The description is expressed in terms of number theory. The techniques are various appl…
Study topological 4-manifolds with specific fundamental groups.
problem Classify topological 4-manifolds with 4-dimensional fundamental group.
method Use algebraic topology, Poincaré duality, and Kirby-Siebenmann invariant.
result Two manifolds are homeomorphic if they are s-cobordant and have same Kirby-Siebenmann invariant.
Let G be a finite group. To every smooth G-action on a compact, connected and oriented surface we can associate its data of singular orbits. The set of such data becomes an Abelian group B_G under the G-equivariant connected sum. We will show that the map which sends G to B_G is functorial and carries many features of …