arXiv research

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 $S^3$. We prove that there exists a spectral sequence of $\mathbb{F}[Q]/Q^2$-modules (where $Q$ has degree $-1$) which converges to $\widetilde{\m…

Researchers extend microlocal analysis across event horizons of rotating black holes.

This paper describes an equivalence of the canonical category of $\mathbb N$-manifolds of degree $2$ with a category of involutive double vector bundles. More precisely, we show how involutive double vector bundles are in duality with double vector bundles endowed with a linear metric. We describe then how special sect…

Real Heegaard Floer Homology extends Li's real monopole Floer homology.

Study fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.

In this study, we generalize double tangent bundles to double jet bundles. We present a secondary vector bundle structure on a 1-jet of a vector bundle. We show that 1-jet of a vector bundle carries two vector bundle structures, namely primary and secondary structures. We also show that the manifold charts induced by p…

Study proves naturality and functoriality in a type of Heegaard Floer homology.

Extends Khovanov homology spectral sequence using Heegaard Floer homology.

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…

New real invariants for 3-manifolds and links.

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 $g\geq 3$ and extend these involutions to the four-manifolds obtained by blowing up the …

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.

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 generalize the notion of involutivity to systems of differential equations of different orders and show that the classical results due to Guillemin and Quillen relating involutivity, restrictions, characteristics and characteristicity, known for first order systems, extend to the general context, though in a modifie…

The paper geometrizes N-manifolds using symmetric vector bundles.

This paper solves the generalized Kähler problem by linking it to symplectic geometry.

Study exact surgery formula in involutive Heegaard Floer homology.

Proves spectral sequence for real Heegaard Floer homology.

Scroll structures on solutions of 4D integrable equations are involutive and governed by a dispersionless hierarchy.

New exotic 4-manifolds with even $b_2^+$ and $\mathbb{Z}/2\mathbb{Z}$ fundamental group.

Using the covering involution on the double branched cover of the three-sphere branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots.

This expository monograph cuts a short path from the common, elementary background in geometry (linear algebra, vector bundles, and algebraic ideals) to the most advanced theorems about involutive exterior differential systems: (1) The incidence correspondence of the characteristic variety, (2) Guillemin normal form an…

In this work, we studied the properties of the spherical indicatrices of involute curve of a space curve and presented some characteristic properties in the cases that involute curve and evolute curve are slant helices and helices, spherical indicatrices are slant helices and helices and we introduced new representatio…

A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinator…

The canonical involution of a double (=iterated) tangent bundle may be dualized in different ways to yield relations between the Tulczyjew diffeomorphism, the Poisson anchor associated with the standard symplectic structure on the cotangent space,and the reversal diffeomorphism. We show that the constructions which yie…

We construct a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=2$ 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 $p_g=0$ and $K^2=1$. In order to construct the example, we combin…

Real Heegaard Floer homology gets a new grading for certain 3-manifolds.

New examples of manifolds with similar homotopy but different simple homotopy types.

Study $d$-invariants of L-space double branched covers of arborescent links.

Classifies involutions on spherical 3-manifolds.

We develop a theory of chain complex double-cobordism for chain complexes equipped with Poincaré duality. The resulting double-cobordism groups are a refinement of Ranicki's torsion algebraic $L$-groups for localisations of a commutative ring with involution. The refinement is analogous to the difference between metabo…

Study involutions on 3D small covers, proving quotient spaces are linked 2-spheres.

Revisits the Gauss-Bonnet formula using double forms.

The paper develops a new theory for knots and 3-manifolds with involutions.

Paper defines and proves geometric uniqueness of Einstein field equations.

New examples show high twisting doesn't guarantee open book maximality.

Classifies Real line bundles with Real connections on manifolds with involution.

The pants graph of a non-orientable surface is quasi-isometric to its Teichmüller space.

Involutive Hopf monoids yield surface invariants.

Study new involutivity theorems for Poisson quasi-Nijenhuis manifolds.

The Finsleroid-Finsler space is constructed over an underlying Riemannian space by the help of a scalar $g(x)$ and an input 1-form $b$ of unit length. Explicit form of the entailed tensors, as well as the respective spray coefficients, is evaluated. The involutive case means the framework in which the characteristic sc…

Floer homology vanishes on certain 3-manifolds formed by knots and their mirrors.

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $S^3.$ Such involutions are called hyperelliptic as the manifolds admitting such an action. We consider finite groups acting on 3-manifolds and containing hyperelliptic involutions whose fixed-point set has $r…

It is known that every nonorientable surface $Σ$ has an orientable double cover $\tildeΣ$. The covering map induces an involution on the moduli space $\tilde{\M}$ of gauge equivalence classes of flat $G$-connections on $\tildeΣ$. We identify the relation between the moduli space $\M$ and the fixed point set of the modu…

Develops Floer cohomology for 4-manifolds with involutions and links.

A new polynomial invariant for strongly involutive links.

The paper develops a new Floer theory for 3-manifolds with involutions.