arXiv research

The aim of this paper is to study Seifert bundle structures on simply connected 5--manifolds. We classify all such 5--manifolds which admit a Seifert bundle structure, and in a few cases all Seifert bundle structures are also classified. These results are then used to construct positive Ricci curvature Einstein metrics…

New systolic inequality for 3D contact forms on Seifert bundles.

A Seifert manifold is a 3-dimensional manifold with a circle action. It is a circle bundle (with singularities) over a 2-dimensional orbifold. In this note, we discuss a generalized Seifert manifolds. By definition, they have bundle-like structures whose fibers are infra- homogeneous spaces; that is, the fibers are fla…

This paper shows that the Seifert volume of each closed non-trivial graph manifold is virtually positive. As a consequence, for each closed orientable prime 3-manifold $N$, the set of mapping degrees $cD(M,N)$ is finite for any 3-manifold $M$, unless $N$ is finitely covered by either a torus bundle, or a trivial circle…

It is known that there are surface bundles of arbitrarily high genus which have genus two Heegaard splittings. The simplest examples are Seifert fibered spaces with the sphere as a base space, three exceptional fibers and which allow horizontal surfaces. We characterize the monodromy maps of all surface bundles with ge…

We study totally geodesic codimension 1 smooth foliations on Lorentzian manifold. We are in particular interested by the relations between riemannian flows and geodesic foliations. We prove that, up to a 2-cover, any Seifert bundle admit such a foliation.

To each connected component in the space of semisimple representations from the orbifold fundamental group of the base orbifold of a Seifert fibered homology 3-sphere into the Lie group U(2,1), we associate a real number called the "orbifold Toledo invariant." For each such orbifold, there exists an elliptic surface ov…

New minimal hypersurfaces in 4D sphere found.

We find a geometric invariant of isotopy classes of strongly irreducible Heegaard splittings of toroidal 3-manifolds. Combining this invariant with a theorem of R Weidmann, proved here in the appendix, we show that a closed, totally orientable Seifert fibered space M has infinitely many isotopy classes of Heegaard spli…

Study harmonic measures and rigidity in Seifert 3-manifolds using $S^1$-connections.

The study finds infinitely many 7-manifolds with non-negative curvature but not homotopy equivalent to bundles.

We derive a simple closed formula for the SL(2,C) Casson invariant for Seifert fibered homology 3-spheres using the correspondence between SL(2,C) character varieties and moduli spaces of parabolic Higgs bundles of rank two. These results are then used to deduce the invariant for Dehn surgeries on twist knots by combin…

Mathematical study supports connection between 3D manifolds and modular tensor categories.

Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Kollár proved that, if W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Our Main Theorem, answering in the affirmative three questions of Kollár, gives s…

We consider circle bundles over compact three-manifolds with symplectic total spaces. We show that the base of such a space must be irreducible or the product of the two-sphere with the circle. We then deduce that such a bundle admits a symplectic form if and only if it admits one that is invariant under the circle act…

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

New geometry based on Siegel upper half-space with volume formula.

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 constructs 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.

The paper uses symplectic homology to study 3D Besse manifolds with vanishing first Chern class.

A Seifert surgery is an integral surgery on a knot in S^3 producing a Seifert fiber space which may contain an exceptional fiber of index 0. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; its edges correspond to single twistings along "seiferters" or "annular pair…

We show that a PSL(2;R)-representation of a Fuchsian group induces the asymptotics of the Reidemeister torsion for the Seifert manifold corresponding to the euler class of the PSL(2;R)-representation. We also show that the limit of leading coefficient of the Reidemeister torsion is determined by the euler class of a PS…

In this paper, we study Euler classes in groups of homeomorphisms of Seifert fibered 3-manifolds. We show that, in contrast to the familiar Euler class for $\mathrm{Homeo}_0(S^1)^δ$, these Euler classes for $\mathrm{Homeo}_0(M^3)^δ$ are unbounded classes. In fact, we give examples of flat topological M bundles over a g…

How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the ot…

The paper presents a chain complex for 3-manifold covers, including surface bundles and surgeries.

New Euclidean supersymmetric solutions found for a specific metric.

We study Chern-Simons theory on 3-manifolds M that are circle-bundles over 2-dimensional orbifolds S by the method of Abelianisation. This method, which completely sidesteps the issue of having to integrate over the moduli space of non-Abelian flat connections, reduces the complete partition function of the non-Abelian…

We compute eta invariants of various Dirac type operators on circle bundles over Riemann surfaces via two approaches: an adiabatic approach based on the results of Bismut-Cheeger-Dai and a direct elementary one. These results, coupled with some delicate spectral flow computations are then used to determine the virtual …

We prove that a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is virtually trivial. This generalizes in every dimension the case of circle bundles over hyperbolic surfaces, for which the result was known…

The aim of this paper is to construct infinitely many families of Einstein metrics on the connected sums of arbitrary number of copies of $S^2\times S^3$. We realize these 5-manifolds as total spaces of Seifert bundles over Del Pezzo orbifolds. A Kähler--Einstein metric on the Del Pezzo orbifold is then lifted to an Ei…

We find an infinite family of Seifert fibered surgeries on strongly invertible knots which do not have primitive/Seifert positions. Each member of the family is obtained from a trefoil knot after alternate twists along a pair of seiferters for a Seifert fibered surgery on a trefoil knot.

We call a pair (K, m) of a knot K in the 3-sphere S^3 and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K, m), K can be embedded in a genus 2 Heegaard surface of S^3 in a primitive/Seifert position, the concept introduced by Dean as a na…

A Seifert surgery is a pair (K, m) of a knot K in the 3-sphere and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K, m) yields Seifert surgeries. We study Seifert surgeries obtained from those o…

We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The (-3,3,5)-pretzel…

This paper constructs Seifert-fibered Dehn surgeries for hyperbolic tunnel-number-one knots.

Let E be a Real or Quaternionic Hermitian vector bundle over a Klein surface M. We study the action of the gauge group of E on the space of Galois-invariant unitary connections and we show that the closure of a semi-stable orbit contains a unique unitary orbit of projectively flat, Galois-invariant connections. We then…

Let M be a U(1) bundle over a smooth Riemann surface. I show that for Chern-Simons theory on M, with structure group G, the path integral is an integral over the space of G-connections on the Riemann surface involving characteristic classes as well as a certain 4-dimensional class that comes from a universal bundle. Wh…

For Seifert manifold $M=X({p_1}/_{\f{q_1}},{p_2}/_{\f{q_2}}, ...,{p_n}/_ {\f{q_n}}), τ^{'}_r(M)$ is calculated for all $r$ odd $\geq 3$. If $r$ is coprime to at least $n-2$ of $p_k$ (e.g. when $M$ is the Poincare homology sphere), it is proved that $(\sqrt {\dfrac{4}{r}}\sin \dfracπ{r})^ντ^{'}_r(M)$ is an algebraic int…

We classify the Seifert fibrations of any given lens space L(p,q). We give an algorithmic construction of a Seifert fibration of L(p,q) over the base orbifold S^2(m,n) with the coprime parts of m and n arbitrarily prescribed. This algorithm produces all possible Seifert fibrations, and the equivalences between the resu…

Extends Seifert algorithm to 3-manifolds via surgery.

Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a rational Seifert surface for the link. In the case when this condition is satisfied, we…

Dominant knots have isomorphic Seifert and Tait graphs.

The paper explores representations of graph manifolds to Seifert motion groups.

Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.

Study on Seifert fibered spherical 3-orbifolds, determining their unique fibrations.

Classifies generalized Seifert fiber spaces and their branched covers.

New knots found with Seifert genus not matching minimal genus Seifert surfaces.

We define the surface complex for $3$-manifolds and embark on a case study in the arena of Seifert fibered spaces. The base orbifold of a Seifert fibered space captures some of the topology of the Seifert fibered space, so, not surprisingly, the surface complex of a Seifert fibered space always contains a subcomplex is…