arXiv research

Proves a formula for a special invariant of 4-manifolds.

Authors prove existence of exotic surfaces and invariants not detecting self-diffeomorphisms.

Proved boundary Dehn twist is exotic for Milnor fibers with specific conditions.

Novel construction of Bauer--Furuta invariant using sheaves of spectra.

Study cohomotopy classes for 4-manifolds using complex spin structures.

Study of Bauer-Furuta invariants under Lie group actions and Galois coverings.

We show how the families Seiberg-Witten invariants of a family of smooth $4$-manifolds can be recovered from the families Bauer-Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families Seiberg-Witten invariants. We give a formula for the Steenrod squares of the fami…

This paper constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.

Study BF invariants using simple type concepts.

New inequality for links in 3D using 4D invariants.

Proves a general connected sum formula for families Seiberg-Witten invariants.

Established a stable cohomotopy refinement for Pin(2) monopole invariants.

New invariant shows Dehn twist on connected sum of homology tori is not isotopic to identity.

In this paper, we study Manolescu's construction of the relative Bauer-Furuta invariants arising from the Seiberg-Witten equations on 4-manifolds with boundary. The main goal is to introduce a new gauge fixing condition in order to apply the finite dimensional approximation technique. We also hope to provide a framewor…

S.Bauer and M.Furuta defined a stable cohomotopy refinement of the Seiberg-Witten invariants. In this paper, we prove a vanishing theorem of Bauer-Furuta invariants for 4-manifolds with smooth Z/2-actions. As an application, we give a constraint on smooth Z/2-actions on homotopy K3#K3, and construct a nonsmoothable loc…

The study explores stable diffeomorphism groups in 4-manifolds using localisation and invariants.

Study Lefschetz fibrations with 4D fibers using Seiberg-Witten theory.

New methods distinguish exotic 4-manifolds using Heegaard Floer homology.

We obtain constraints on the topology of families of smooth $4$-manifolds arising from a finite dimensional approximation of the families Seiberg-Witten monopole map. Amongst other results these constraints include a families generalisation of Donaldson's diagonalisation theorem and Furuta's $10/8$ theorem. As an appli…

In a previous paper we have constructed an invariant of four-dimensional manifolds with boundary in the form of an element in the stable homotopy group of the Seiberg-Witten Floer spectrum of the boundary. Here we prove that when one glues two four-manifolds along their boundaries, the Bauer-Furuta invariant of the res…

The paper provides a new inequality for 4-manifolds and uses it to study knot sliceness and symplectic embeddings.

We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positi…

Kronheimer and Mrowka constructed a variant of Seiberg-Witten invariants for a 4-manifold $X$ with contact boundary in 1997. Using Furuta's finite dimensional approximation, we refine this invariant in the case $H^1(X, \partial X; \mathbb{R})=0$.

We use the construction of unfolded Seiberg-Witten Floer spectra of general 3-manifolds defined in our previous paper to extend the notion of relative Bauer-Furuta invariants to general 4-manifolds with boundary. One of the main purposes of this paper is to give a detailed proof of the gluing theorem for the relative i…

New invariant connects symplectic fillings and contact structures.

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 …

New spectral construction for 3-manifolds with non-zero first Betti number.

Nontrivial boundary Dehn twist found on K3#K3 manifold.

Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of $S^1$-equivariant non-linear maps between Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of cohomotopy Seiberg-Witten invariants which hav…

We show that there exist smooth, simply connected, four-dimensional spin manifolds which do not admit Einstein metrics, but nonetheless satisfy the strict Hitchin-Thorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy refinement of the Seiberg-Witten invariant, in conjunction with curvature estima…

Study determines scalar curvature invariants for 3-spheres embedded in 4-manifolds.

Diagonalizes tautological classes of definite 4-manifolds.

For closed manifolds endowed with a Riemannian foliation of codimension $4$, one can define a transversal Seiberg-Witten map. We show that there is a finite dimensional approximation for such a map. By such a method and under the condition that $H^1_b(M)\cap H^1(M,\mathbb Z)$ is a lattice of $H^1_b(M)$, we can define a…

New example of non-smooth isotopy after stabilization in 4-manifolds.

On a smooth closed oriented $4$-manifold $M$ with a smooth action by a compact Lie group $G$, we define a $G$-monopole class as an element of $H^2(M;\Bbb Z)$ which is the first Chern class of a $G$-equivariant Spin$^c$ structure which has a solution of the Seiberg-Witten equations for any $G$-invariant Riemannian metri…

Study algebraic relations of Vassiliev invariants for families of knots.

In the paper we formulate and derive the family blowup formula of family Seiberg-Witten invariants. The formula has been used in the enumerative application of counting singular curves on algebraic surfaces. We first give a topological derivation of the formula by using family index theorem. Then we define the algebrai…

We prove a gluing formula for the families Seiberg-Witten invariants of families of $4$-manifolds obtained by fibrewise connected sum. Our formula expresses the families Seiberg-Witten invariants of such a connected sum family in terms of the ordinary Seiberg-Witten invariants of one of the summands, under certain assu…

Computes Seiberg-Witten invariants for Kähler families of 4-manifolds.

Method calculates $d$-invariants for a family of Brieskorn spheres.

We explore a family of invariants obtained from linking numbers. This is a family of Kauffman finite type invariants.

The paper constructs a family of SKT metrics on the exceptional Lie group G2.

We introduce an invariant of tuples of commutative diffeomorphisms on a 4-manifold using families of Seiberg-Witten equations. This is a generalization of Ruberman's invariant of diffeomorphisms defined using 1-parameter families of Seiberg-Witten equations. Our invariant yields an application to the homotopy groups of…

We obtain new families of (1,2)-symplectic invariant metrics on the full complex flag manifolds F(n). For n > 4, we characterize n-3 different n-dimensional families of (1,2)-symplectic invariant metrics on F(n). Any of these families corresponds to a different class of non-integrable invariant almost complex structure…

New polynomials defined for quandle structures, enhancing graph invariants.

The paper calculates Seiberg-Witten invariants and finds non-symplectic diffeomorphisms.

Infinite families of quantum modular invariants for 3-manifolds are discovered.

Researchers compute mod 2 Seiberg-Witten invariants for spin structures and families.