arXiv research

Sphere bundles over 4-manifolds are trivial after looping, except for two cases.

Conditions for equivariant bundles on 4-manifolds with cyclic actions.

New exotic 4-manifolds found with fiber bundles.

Involutory Hopf group-coalgebras provide new invariants for 4-manifold bundles.

Classifies torus bundles bounding 4-manifolds with rational homology.

New definition of skein lasagna module for specific 4-manifolds.

Symplectic 4-manifolds with Kodaira dimension zero can be viewed as symplectic Calabi-Yau surfaces. We are able to completely determine their Betti numbers by proving two general results on quaternionic vector bundles.

The paper constructs symplectic forms on 4-manifolds using branched coverings and holomorphic line bundles.

This paper constructs a Weinstein trisection for a surface bundle.

We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S^4…

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

New configuration space integrals show nontrivial formal smooth structures on 4-manifold bundles.

Proves properties of 4-manifolds with scalar curvature constraints.

In this paper we first show that on projective manifolds (M, ω), there are holomorphic determinant bundles (in the sense of Knusden-Mumford used by Bismut, Gillet, Soule) which play the role of the geometric quantum bundle, namely one for each input data of a Hermitian holomorphic line bundle L of non-trivial Chern cla…

Study shows looping a 6-manifold over a 4-manifold results in a product of loops on spheres.

This work is a continuation of the former paper in which principal bundles are given by compact spin toric manifolds and compact connected semisimple Lie groups. In this paper, ambient manifolds are assumed to be compact toric manifolds and Lie groups are compact connected. The main result is that locally smooth manifo…

Non-orientable 4-manifolds are simple branched coverings of RP^4 and twisted S^3-bundles.

Every 1-connected topological 4-manifold M admits a $S^{1}$-covering by $#_{r-1}S^{2}\times S^{3}$, where $r=$rank$H^{2}(M;\QTR{Bbb}{Z})$.

We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifold…

Classification of homogeneous almost complex 4-manifolds with non-degenerate torsion bundle

A 4-manifold is parallelizable if its Stiefel-Whitney and Pontryagin classes vanish.

The paper constructs Poincaré-Einstein 4-manifolds with various cusps.

In this paper we use the G-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe the Spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtl…

The paper studies mapping class groups of nontrivial $S^2$ fiber bundles.

Study of Yang-Mills fields on 4-manifolds using modified Lévy Laplacians.

Proves surface embedding theorem for 4-manifolds with good fundamental group.

Plumbing of surfaces embeds in hyperbolic 4-manifolds.

In this paper we give the first example of a surface bundle over a surface with at least three fiberings. In fact, for each $n \ge 3$ we construct $4$-manifolds $E$ admitting at least $n$ distinct fiberings $p_i: E \to Σ_{g_i}$ as a surface bundle over a surface with base and fiber both closed surfaces of negative Eule…

This paper constructs symplectic surfaces in 4-manifolds with transversal intersections.

The paper studies $ext{Pin}^{\pm}$-structures on non-oriented 4-manifolds via Lefschetz fibrations.

We prove that a compact smooth 4-manifold admits generalized complex structures of odd type if and only if it has a transversely holomorphic 2-foliation. Consequently, there exist generalized complex structures of odd type on a circle bundle over a closed Seifert fibered 3-manifold.

The paper studies $π_1$-injective bounding of manifolds and its applications.

We prove that if a closed oriented 4-manifold X fibers over a 2- or 3-dimensional manifold, in most cases all of its virtual Betti numbers are infinite. In turn, we show that a closed oriented 4-manifold X which is not a tower of torus bundles and fibering over a 2- or 3-dimensional manifold does not admit a torsion sy…

The paper extends trisection theory to non-orientable 4-manifolds using colored triangulations.

The Vafa-Witten equations on an oriented Riemannian 4- manifold are first order, non-linear equations for a pair of connection on a principle SO(3) bundle over the 4-manifold and a self-dual 2-form with values in the associated Lie algebra bundle. The main theorem in this paper characterizes in part the behavior of seq…

The main results of this paper describes a formula for the Seiberg-Witten invariant of a 4-manifold which admits a nontrivial free S^1-action. We use this theorem to produce a nonsymplectic 4-manifold with a free circle action whose orbit space fibers over S^1. We also describe a 3-manifold which is not the orbit space…

We study the natural G_2 structure on the unit tangent sphere bundle SM of any given orientable Riemannian 4-manifold M, as it was discovered in \cite{AlbSal}. A name is proposed for the space. We work in the context of metric connections, or so called geometry with torsion, and describe the components of the torsion o…

We introduce a surgery for generalized complex manifolds whose input is a symplectic 4-manifold containing a symplectic 2-torus with trivial normal bundle and whose output is a 4-manifold endowed with a generalized complex structure exhibiting type change along a 2-torus. Performing this surgery on a K3 surface, we obt…

Study torsion parallel spinors on Lorentzian 4-manifolds and their evolution flows.

Generalized Donaldson invariants of 4-manifolds are defined, using moduli spaces of anti-self-dual connections with structure group SU(N) or PSU(N). Some values of the invariants are calculated for the case that the 4-manifold arises by the knot-complement construction of Fintushel and Stern. The results are consistent…

The paper creates non-isotopic but homotopic diffeomorphisms in 4-manifolds.

We construct characteristic classes of 4-manifold bundles using $SO(3)$-Yang-Mills theory and Seiberg-Witten theory for families.

In this paper, we first prove that any closed simply connected 4-manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4-manifolds: $\mathbb{S}^4$, $\mathbb{CP}^2$, $\mathbb{S}^2\times\mathbb{S}^2$, or $\mathbb{CP}^2\#\pm \mathbb{CP}^2$. As an…

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over $S^1$ with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over $S^1$. In a different direction, we prove that if the …

The paper examines Kodaira dimensions of specific complex 4-manifolds with torsion first Chern class.

By studying the Higgs bundle equations with the gauge group replaced by the group of symplectic diffeomorphisms of the 2-sphere we encounter the notion of a folded hyperkaehler 4-manifold and conjecture the existence of a family of such metrics parametrised by an infinite-dimensional analogue of Teichmueller space.

We show that any 4-manifold, after surgery on a curve, admits an achiral Lefschetz fibration. In particular, we show that the connected sum of any simply connected 4-manifold with a 2-sphere bundle over the 2-sphere will admit an achiral Lefschetz fibration. We also show these surgered manifolds admit near-symplectic s…

Study finite group actions on symplectic Calabi-Yau 4-manifolds with non-zero first Betti number.