Two Pin(2)-equivariant Floer homologies are shown to be equivalent.
problem Equivalence of Pin(2)-equivariant Floer homologies.
method Comparison of definitions by Lin and Manolescu.
result Pin(2)-equivariant monopole Floer homology is isomorphic to Seiberg-Witten Floer homology.
Computes Pin(2)-equivariant Seiberg-Witten Floer homology for Seifert fibrations.
problem Homology cobordism between Seifert fiber spaces and integral homology spheres.
method Uses Heegaard Floer homology and Pin(2)-equivariant Seiberg-Witten Floer homology.
result Proves Manolescu's conjecture and identifies new homology cobordism obstructions.
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.
Floer theory applied to manifold surgery and triangulation.
problem Understanding manifold surgeries and triangulations.
method Floer homology and Pin(2)-equivariant Seiberg-Witten Floer homology.
result Floer stable homotopy types and Pin(2)-equivariant Seiberg-Witten Floer homology provide insights into the triangulation conjecture.
Researchers compute mod 2 Seiberg-Witten invariants for spin structures and families.
problem Computing mod 2 Seiberg-Witten invariants for spin structures and families.
method Using Pin(2)-symmetry and localisation in equivariant cohomology, the researchers enhance and compute the invariants.
result The mod 2 Seiberg-Witten invariants are computed for spin structures and families, confirming the simple type conjecture mod 2.
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…
New equations use Pin(2) symmetry to study spinor and connection solutions.
problem Study of spinor and connection solutions on 4-manifolds.
method Define and analyze Seiberg-Witten-like equations with Rarita-Schwinger operators.
result Moduli space of solutions is non-compact and non-empty under certain conditions.
Exotic diffeomorphism survives stabilizations on a contractible 4-manifold.
problem Finding exotic diffeomorphisms on contractible 4-manifolds.
method Developed a Pin(2) × Z_2-equivariant refinement for computing Seiberg-Witten Floer homotopy types.
result Constructed an exotic diffeomorphism that survives two stabilizations.
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\…
Study constraints on knot surgery invariants using Seiberg-Witten theory.
problem Understanding invariants of knots in the three-sphere.
method Use Manolescu correction terms and Seiberg-Witten theory.
result Constraints on invariants for knots in the three-sphere.
Established a stable cohomotopy refinement for Pin(2) monopole invariants.
problem Refining monopole invariants for Pin(2) structures.
method Adapted Bauer-Furuta's Seiberg-Witten refinement method to Pin(2) monopole invariants.
result Connected sum formula for the refined invariants.
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.
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…
Remark on Pin(2)-equivariant Floer homology and its connections.
problem Understanding the relationship between Pin(2)-equivariant Floer homology and other invariants.
method Analyzing the monopole Frøyshov invariant and Involutive Heegaard Floer correction terms.
result Identifying the only interesting correction terms as coming from subgroups Z/4, S^1, and Pin(2).
New method computes Yamabe invariants for specific 4D manifolds.
problem Computing Yamabe invariants for new 4D manifolds.
method Using twisted Seiberg-Witten equations, Pin(2)-monopole equations.
result Computed Yamabe invariants for a new class of 4D manifolds.
In the present work we generalize the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singularities. Focusing then on the special case of a three-manifold equipped with a spinc structure isomorphic to its conjugate, we define the counterpart in this…
New methods prove weak homotopy inequivalence for manifold symmetries.
problem Understanding symmetries of smooth 4-manifolds.
method Using Pin(-2)-monopole equations.
result New examples of 4-manifolds where symmetries are not fully represented.
Proof outlines existence of non-triangulable manifolds in high dimensions.
problem Existence of non-triangulable manifolds in dimensions >4.
method Homology cobordism invariants from Pin(2) symmetry of Seiberg-Witten equations.
result Existence of non-triangulable manifolds proven in dimensions >4.
Paper solves a 4D topology problem using Pin(2)-equivariant Mahowald invariants.
problem Existence of Pin(2)-equivariant stable maps between certain representation spheres.
method Analysis of Pin(2)-equivariant Mahowald invariants and use of cell diagrams, Atiyah-Hirzebruch spectral sequence.
result Proves a '10/8+4'-Theorem and solves the 11/8-Conjecture.
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.
Computes Pin(2)-equivariant monopole Floer homology for certain plumbed 3-manifolds.
problem Computing Pin(2)-equivariant monopole Floer homology for specific 3-manifolds.
method Uses Heegaard Floer/monopole Floer lattice complex and ranks of monopole Floer homology groups.
result Pin(2)-equivariant monopole Floer homology can be determined by ranks of monopole Floer homology groups.
Study real structures and Pin−(2)-monopoles in symplectic and Kähler manifolds.
problem Investigate Pin−(2)-monopole invariants in real symplectic and Kähler manifolds. method Prove nonvanishing theorem and give Kobayashi-Hitchin type correspondence.
result Analogous nonvanishing theorem for real symplectic 4-manifolds.
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. Two Seiberg-Witten homologies are proven equivalent.
problem Equivalence of Seiberg-Witten homologies.
method Isomorphism proof between monopole Floer homology and Seiberg-Witten spectrum.
result Monopole Floer homology is isomorphic to Seiberg-Witten spectrum.
For a rational homology 3-sphere Y with a $\spinc$ structure $\s$, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology lead to a collection of variants which are topological invariants. We establish exact sequences relating them, we show that they satisfy a dua…
Study of Seiberg-Witten invariants for 4-manifolds with group actions.
problem Understanding Seiberg-Witten invariants for manifolds with group actions.
method Introduced equivariant Seiberg-Witten invariants, studied their properties and established relations.
result Established localisation formulas and gluing formulas for the invariants.
New 4-manifolds found without certain Einstein metrics, using Seiberg-Witten theory.
problem Constructing 4-manifolds without specific Einstein metrics.
method Using Seiberg-Witten theory and constructing solutions on noncompact manifolds.
result Infinitely many examples of 4-manifolds without cusped asymptotically hyperbolic Einstein metrics.
We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and give its applications to intersection forms with local coefficients of 4-manifolds. The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients. The second is a local …
We construct Bott-type and stable equivariant Seiberg-Witten Floer homology and cohomology for rational homology spheres, and prove their diffeomorphism invariance.
We use localization formulas in the theory of equivariant cohomology to rederive the wall crossing formulas of Li-Liu and Okonek-Teleman for Seiberg-Witten invariants.
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.
We construct Bott-type and equivariant Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. This paper is a revised version of math.DG/9701010. Some typos are removed.
Proves a specific knot is not smoothly slice using real invariants.
problem Determining the smooth sliceness of (2n,1)-cables of the figure-eight knot. method Used real Seiberg-Witten Frøyshov invariant and developed an equivariant lattice homotopy type.
result Proves the (2n,1)-cable of the figure-eight knot is not smoothly slice when n is odd. New invariant recovers known contact element and considers finite coverings.
problem Defining and studying new contact invariants in Seiberg-Witten Floer spectra.
method Cohomotopy set of Seiberg-Witten Floer spectrum, equivariant Borel cohomology.
result New invariant recovers known contact element and considers finite coverings.
We construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory: the singular version, the de Rham version and the Cartan version, with the first…
New constraints on embedded spheres and projective planes in 4-manifolds from Seiberg-Witten theory.
problem Constraints on configurations of embedded spheres and real projective planes in 4-manifolds.
method Equivariant Seiberg-Witten invariants and gluing formula for relative Seiberg-Witten invariants.
result Existence of certain configurations of surfaces leads to 4-manifolds of non-simple type.
The monopole map defines an element in an equivariant stable cohomotopy group refining the Seiberg-Witten invariant. This first of two articles presents the details of the definition of the stable cohomotopy invariant and discusses its relation to the integer valued Seiberg-Witten invariant.
Furuta's ``10/8-th's'' theorem gives a bound on the magnitude of the signature of a smooth spin 4-manifold in terms of the second Betti number. We show that in the presence of a Z/2^p action, his bound can be strengthened. As applications, we give new genus bounds on classes with divisibility and we give a classificati…
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 knot concordance invariants from Seiberg-Witten theory bound slice genus.
problem Bounding the slice genus of knots.
method Equivariant Seiberg-Witten-Floer cohomology applied to cyclic covers.
result Lower bounds on slice genus from knot concordance invariants.
Let M be a manifold homotopy equivalent to the complex projective space $\C P^m$. Petrie conjectured that M has standard total Pontrjagin class if M admits a non-trivial action by S1. We prove the conjecture for m<12 under the assumption that the action extends to a nice Pin(2)-action with fixed point. The…
Researchers compute Floer homotopy types and eta invariants for Seifert 3-manifolds.
problem Computing Floer homotopy types and eta invariants for Seifert 3-manifolds.
method Floer homology, Seiberg-Witten Floer homotopy type, adiabatic connections, spin^c-Dirac operators, eta invariants, orbifold pin^c-connections.
result Floer homotopy types are suspensions of S^0, and Seifert 3-manifolds are L-spaces.
We study the space of positive scalar curvature (psc) metrics on a 4-manifold, and give examples of simply connected manifolds for which it is disconnected. These examples imply that concordance of psc metrics does not imply isotopy of such metrics. This is demonstrated using a modification of the 1-parameter Seiberg-W…
Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S^1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the …
Authors prove existence of exotic surfaces and invariants not detecting self-diffeomorphisms.
problem Existence and properties of exotic surfaces and diffeomorphisms.
method Vanishing theorem of family Bauer--Furuta invariant for diffeomorphisms on spin 4-manifolds.
result Family Bauer--Furuta invariants do not detect exotic self-diffeomorphisms on S4 or S2imesS2. Study of Bauer-Furuta invariants under Lie group actions and Galois coverings.
problem Investigating invariants of 4-manifolds under group actions and Galois coverings.
method Functorial approach to equivariant invariants and study in Galois covering situations.
result Ordinary invariants of quotients are determined by equivariant invariants of the covering manifold.
New invariant shows Dehn twist on connected sum of homology tori is not isotopic to identity.
problem Determining when Dehn twists on connected sums of homology tori are isotopic to identity.
method Generalized Pin(2)-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds and constructed a refinement.
result Dehn twist on X1#X2 is not isotopic to identity if determinants r1,r2 are odd. New example of non-smooth isotopy after stabilization in 4-manifolds.
problem Non-smooth isotopy of Dehn twist after stabilization.
method Pin(2)-equivariant Bauer-Furuta invariant.
result Dehn twist is not smoothly isotopic even after stabilization.