D. Auroux, S. K. Donaldson, L. Katzarkov
We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon.
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We introduce a surgery operation on symplectic manifolds called coisotropic Luttinger surgery, which generalizes Luttinger surgery on Lagrangian tori in symplectic 4-manifolds. We use it to produce infinitely many distinct symplectic non-Kahler 6-manifolds $X$ with $c_1(X)=0$ which are not of the form $M\times F$ for $M$ a symplectic 4-manifold and $F$ a closed surface.
In this note we show that the Lagrangian Luttinger surgery preserves the symplectic Kodaira dimension. Some constraints on Lagrangian tori in symplectic four manifolds with non-positive Kodaira dimension are also derived.
In this paper we will show that two surfaces of the same genus and homology class in a simply connected 4-manifold are concordant. We will show they are often topologically isotopic when their complements have cyclic fundamental group. Finally, we will show that if they are 0-concordant, then surgery on one is equivalent to surgery on the other.
On Auroux's Luttinger-surgery uniqueness question
构造奇异辛四流形对，其拓扑不变量相同但辛小平维数不同，因辛小平维数在Luttinger手术下不变，从而否定Auroux的唯一性问题。
In this article we use the technique of Luttinger surgery to produce small examples of simply connected and non-simply connected minimal symplectic 4-manifolds. In particular, we construct: (1) An example of a minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to CP^2#3(-CP^2) which contains a symplectic surface of genus 2, trivial normal bundle, and simply connected complement and a disjoint nullhomologous Lagrangian torus with the fundamental group of the complement generated by one of the loops on the torus. (2) A minimal symplectic 4-manifold that is homeomorphic but …
A Dehn surgery on a knot $K$ in $S^3$ is exceptional if it produces a reducible, toroidal or Seifert fibred manifold. It is known that a large arborescent knot admits no such surgery unless it is a type II arborescent knot. The main theorem of this paper shows that up to isotopy there are exactly three large arborescent knots admitting exceptional surgery, each of which admits exactly one exceptional surgery, producing a toroidal manifold.
In [2], the first author constructed the first known examples of exotic minimal symplectic $\CP#5\CPb$ and minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to $3\CP#7\CPb$. The construction in [2] uses Y. Matsumoto's genus two Lefschetz fibrations on $M = \mathbb{T}^{2}\times \mathbb{S}^{2} #4\CPb$ over $\mathbb{S}^2$ along with the fake symplectic $\mathbb{S}^{2} \times \mathbb{S}^{2}$ construction given in [1]. The main goal in this paper is to generalize the construction in [2] using the higher genus versions of Matsumoto's fibration constructed by Mustafa Korkmaz an…
We prove that any symplectic 4-manifold which is not a rational or ruled surface, after sufficiently many blow-ups, admits an arbitrary number of nonisomorphic Lefschetz fibrations of the same genus which cannot be obtained from one another via Luttinger surgeries. This generalizes results of Park and Yun who constructed pairs of nonisomorphic Lefschetz fibrations on knot surgered elliptic surfaces. In turn, we prove that there are monodromy factorizations of Lefschetz pencils which have the same characteristic numbers but cannot be obtained from each other via partial conjugations by Dehn twi…
We study the cobordism of manifolds with boundary, and its applications to codimension 2 embeddings $M^m\subset N^{m+2}$, using the method of the algebraic theory of surgery. The first main result is a splitting theorem for cobordisms of algebraic Poincaré pairs, which is then applied to describe the behaviour on the chain level of Seifert surfaces of embeddings $M^{2n-1} \subset S^{2n+1}$ under isotopy and cobordism. The second main result (update: which is false) is that the $S$-equivalence class of a Seifert form is an isotopy invariant of the embedding, generalizing the Murasugi--Levine re…
For the purposes of this paper, Dehn surgery along a curve K in a 3-manifold M with slope r is `exceptional' if the resulting 3-manifold M_K(r) is reducible or a solid torus, or the core of the surgery solid torus has finite order in the fundamental group of M_K(r). We show that, providing the exterior of K is irreducible and atoroidal, and the distance between r and the meridian slope is more than one, and a homology condition is satisfied, then there is (up to ambient isotopy) only a finite number of such exceptional surgery curves in a given compact orientable 3-manifold M, with the boundar…
Characterizing slopes for satellite knots
本文扩展前人工作研究卫星结，证明当|q|足够大时任何斜率p/q对该结是刻画的，并确立每个非整数斜率对复合结均刻画的结论。
Lagrangian Klein bottles in $S^2 \times S^2$
本文利用 Luttinger 手术证明，当 $S^2$ 因子的辛面积至少为另一因子的两倍时，$S^2 \times S^2$ 中不存在属于该因子 $\mathbb{Z}_2$-同调类的 Lagrangian Klein 瓶。
In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along $T^4$. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group $\Z$. In this paper, we also produce the first examples of simply connected symplectic Calabi-Yau and non-Calabi-Yau 6-manifolds via coisotropic Luttinger surgery.
In this article we apply the technique of Luttinger surgery to study the complexity of the fundamental group of symplectic $4$-manifolds with holomorphic Euler number $χ_h=1$. We discuss the topology of symplectic $4$-manifolds with $b^+=1$ and provide various constructions of symplectic $4$-manifolds with $b^+=1$ and prescribed $c_1^2$.
We work in the smooth category. Let N be a closed connected n-manifold and assume that m>n+2. Denote by E^m(N) the set of embeddings N -> R^m up to isotopy. The group E^m(S^n) acts on E^m(N) by embedded connected sum of a manifold and a sphere. If E^m(S^n) is non-zero (which often happens for 2m<3n+4) then no results on this action and no complete description of E^m(N) were known. Our main results are examples of the triviality and the effectiveness of this action, and a complete isotopy classification of embeddings into R^7 for certain 4-manifolds N. The proofs are based on the Kreck modifica…
Classification of knotted tori in the 2-metastable dimension
本文研究扭结环面嵌入球面的同位素分类问题，在2-亚稳维数下利用Koschorke正合序列的类比和新β-不变量，给出了同位素类集合无限性的显式判据。
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