arXiv research

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

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

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

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…

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

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…

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

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

Study BF invariants using simple type concepts.

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

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…

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

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

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

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 $H^1(X, \partial X; \mathbb{R})=0$.

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

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…

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 cohomotopy classes for 4-manifolds using complex spin structures.

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

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

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 …

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…

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

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…

New invariant connects symplectic fillings and contact structures.

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 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…

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

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…

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

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

Nontrivial boundary Dehn twist found on K3#K3 manifold.

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…

We construct non-constructible simplicial $d$-spheres with $d+10$ vertices and non-constructible, non-realizable simplicial $d$-balls with $d+9$ vertices for $d\geq 3$.

A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system(RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal su…

Modified curve shortening flow constructs $λ$-Angenent curve.

Article constructs coassociative submanifolds in Joyce's $G_2$-manifolds.

The twist construction is a geometric model of T-duality that includes constructions of nilmanifolds from tori. This paper shows how one-dimensional foliations on manifolds may be used in a shear construction, which in algebraic form builds certain solvable Lie groups from Abelian ones. We discuss other examples of geo…

We describe various constructions in Sasakian geometry. First we generalize the join construction of the first two authors to arbitrary Sasakian manifolds. We then give several examples, including ones which prove the existence of Sasakian-Einstein metrics on manifolds homeomorphic to $S^2\times S^5.$ Then we use a gen…

Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation and functional analysis. Existing constructions include the reflexive RKBS via a bilinear form, the semi-inner-p…

We construct bi-invariant total orderings of residually torsion-free nilpotent groups by using Chen's iterated integrals. This construction can be seen as a generalization of the Magnus ordering of the free groups, and equivalent to the classical construction which uses an iteration of central extensions. Our geometric…

The paper constructs infinitely many surfaces with specific mean curvature.

Explains a 1978 construction for Yang-Mills instantons.

We construct several non-trivial examples of CAT(1) spaces by using the idea of free construction.

We study the performance of the adaptive construction scheme for a Bayesian inference on the Quadratic GARCH model which introduces the asymmetry in time series dynamics. In the adaptive construction scheme a proposal density in the Metropolis-Hastings algorithm is constructed adaptively by changing the parameters of t…

New pseudomodular groups constructed from jigsaw construction.