Stefano Vidussi
In this note we prove that, for any integer n, there exist a smooth 4-manifold, homotopic to a K3 surface, defined by applying the link surgery method of Fintushel-Stern to a certain 2-component graph link, which admits n inequivalent symplectic structures Keywords: Symplectic topology of 4-manifolds; Seiberg-Witten theory
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A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is shown that an effective action by various maximal symplectic K3 groups forces the corresponding homotopy K3 surface to be minimally exotic with respect to our measure. (However, the standard K3 is the only known example of such minimally exotic homotopy K3 surfaces.) The possible str…
We answer a question of Oprea-Tralle on the realizability of symplectic algebras by symplectic manifolds in dimensions divisible by four, along with a question of Lupton-Oprea in all even dimensions. This will also allow us to address, in all even dimensions six and higher, another question of Oprea-Tralle on the possibility of algebraic conditions on the rational homotopy minimal model of a closed smooth manifold implying the existence of a symplectic structure on the manifold.
This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic 4-manifold E(1)_K, obtained by knot surgery on the rational elliptic surface E(1) with the left-handed trefoil knot K. E(1)_K has the simplest homotopy type among simply-connected symplectic 4-manifolds known to exhibit such property.
Let E(1)_K denote the closed 4-manifold that is homotopy equivalent (hence homeomorphic) to the rational elliptic surface E(1) and is obtained by performing Fintushel-Stern knot surgery on E(1) using a knot K in S^3. We construct an infinite family of homologous non-isotopic symplectic tori representing a primitive homology class in E(1)_K when K is any nontrivial fibred knot in S^3. We also show how these tori can be non-isotopically embedded as homologous symplectic submanifolds in other symplectic 4-manifolds.
We prove that any two irreducible cuspidal Hurwitz curves $C_0$ and $C_1$ (or more generally, curves with A-type singularities) in the Hirzebruch surface $F_N$ with coinciding homology classes and sets of singularities are regular homotopic; and symplectically regular homotopic if $C_0$ and $C_1$ are symplectic with respect to a compatible symplectic form.
We show that a stabilized convex symplectic (also called Liouville) manifold with the homotopy type of a half dimensional CW-complex is symplectomorphic to a flexible Weinstein manifold.
A contact homotopy type
本文将global Kuranishi charts调整至切触设定，基于Reeb orbits和伪全纯建筑模空间为非退化contact manifold构建流范畴，该构造提升了contact homology并具有自然性。
On totally c-symplectically aspherical manifolds
该研究在≥6的偶数维构造了具非平凡第二同伦群的全辛非球面凯勒流形例子，在四维构造了具非平凡第二同伦群的全c-辛非球面近辛流形例子，拓展了相关流形的存在性结果。
Let L be an exact Lagrangian submanifold inside the cotangent bundle of a closed manifold N. We prove that if N satisfies a mild homotopy assumption then the image of π_2(L) in π_2(N) has finite index. We make no assumption on the Maslov class of L, and we make no orientability assumptions. The homotopy assumption is either that N is simply connected, or more generally that π_m(N) is finitely generated for each m \geq 2. The result is proved by constructing the Novikov homology theory for symplectic cohomology and generalizing Viterbo's construction of a transfer map between the homologies of …
In this short article we give a criterion whether a given minimal symplectic 4-manifold with $b_{2}^{+}=1$ having a torsion-free canonical class is rational or ruled. As a corollary, we confirm that most of homotopy elliptic surfaces $E(1}_{K}$, K is a fibered knot in $S^3$, constructed by R. Fintushel and R. Stern are minimal symplectic 4-manifolds with $b_{2}^{+}=1$ which do not admit a complex structure.
We introduce the secondary Stiefel-Whitney class $\tilde w_2$ of homotopically trivial diffeomorphisms and show that a homotopically trivial symplectomorphism of a ruled 4-manifold is isotopic to identity if and only if the class $\tilde w_2$ vanishes. Using this, we give a detailed description of the combinatorial structure of the diffeotopy group of ruled symplectic 4-manifolds $X$, either minimal or blown-up, and its action on the homology and homotopy groups $H_2(X,\zz)$, $π_1(X)$, and $π_2(X)$.
Symplectic submanifolds in dimension $6$ from hyperelliptic Lefschetz fibrations
本研究利用4-流形上的超椭圆Lefschetz纤维化，构造出含无穷多同调等价但同伦不等价且无复结构的余维2辛子流形的闭单连通辛6-流形，并推广至高维，还在附录证明了ℂℙ³中低次余维2辛子流形的微分同胚性。
We show that simply connected contact manifolds that are subcritically Stein fillable have a unique symplectically aspherical filling up to diffeomorphism. Various extensions to manifolds with non-trivial fundamental group are discussed. The proof rests on homological restrictions on symplectic fillings derived from a degree-theoretic analysis of the evaluation map on a suitable moduli space of holomorphic spheres. Applications of this homological result include a proof that compositions of right-handed Dehn twists on Liouville domains are of infinite order in the symplectomorphism group. We a…
Towards a symplectic Khovanov homology for links in fibered $3$-manifolds
本文针对纤维化闭3-流形中的横向链环定义辛Khovanov型同调，并猜想相关曲面范畴的组合dga描述，该dga是带边Heegaard Floer同调中链代数的高维类似物。
$S^1$-localisation by pseudocycles, lifts to $S^1$-localisation of moduli spaces, and application to invariants of $S^1$-equivariant symplectic cohomology
该研究提出伪循环局部化（LbP）作为Atiyah-Bott $S^1$-局部化的新诠释，将其从参数空间提升到特定全纯曲线模空间，推导了等变辛类与Gromov-Witten不变量的关系，证明了Seidel的猜想。
The rational homotopy type of a differential graded algebra (DGA) can be represented by a family of tensors on its cohomology, which constitute an $A_\infty$-minimal model of this DGA. When only the cohomology is needed to determine the rational homotopy type, then the DGA is called formal. By a theorem of Miller, a compact $k$-connected manifold is formal if its dimension is not greater than $4k+2$. We expand this theorem and a result of Crowley-Nordström to prove that if the dimension of a compact $k$-connected manifold $N\leq (l+1)k+2$, then its de Rham complex has an $A_\infty$-minimal mod…
Two constructions of contact manifolds are presented: (i) products of S^1 with manifolds admitting a suitable decomposition into two exact symplectic pieces and (ii) fibre connected sums along isotropic circles. Baykur has found a decomposition as required for (i) for all closed, oriented 4-manifolds. As a corollary, we can show that all closed, oriented 5-manifolds that are Cartesian products of lower-dimensional manifolds carry a contact structure. For symplectic 4-manifolds we exhibit an alternative construction of such a decomposition; this gives us control over the homotopy type of the co…
Let M be a 4-manifold with residually finite fundamental group G having b_1(G) > 0. Assume that M carries a symplectic structure with trivial canonical class K = 0 in H^2(M). Using a theorem of Bauer and Li, together with some classical results in 4-manifold topology, we show that for a large class of groups $ is determined up to homotopy and, in favorable circumstances, up to homeomorphism by its fundamental group. This is analogous to what was proven by Morgan-Szabo in the case of b_1 = 0 and provides further evidence to the conjectural classification of symplectic 4-manifolds with K = 0$. A…
Let $M$ be either $S^2\times S^2$ or the one point blow-up $\cp# \bcp$ of $\cp$. In both cases $M$ carries a family of symplectic forms $\om_\la$, where $\la > -1$ determines the cohomology class $[\om_\la]$. This paper calculates the rational (co)homology of the group $G_\la$ of symplectomorphisms of $(M,\om_\la)$ as well as the rational homotopy type of its classifying space $BG_\la$. It turns out that each group $G_\la$ contains a finite collection $K_k, k = 0,...,\ell = \ell(\la)$, of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups "asymptotically …
We define an simple invariant of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4-manifolds have infinitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We further show that for a large class of examples that lambda(T) is actually a C-infinity invariant. In addition, this invariant is used to show that many symplectic 4-manifolds have nontrivial homology classes which are represented by infinitely many pairwise inequivalent Lagrangian tori, a result first proved by S Vidussi for the homotopy K3-surface obtained from …
Topological line arrangements and their topological invariants
研究拓扑线排列补集拓扑，用同调方法证其补集上同调环同构于Orlik - Solomon代数，还研究补集同伦型，发现辛线排列补集有特定同伦型，且拓扑线排列组合类型的补集有多种非同伦等价情况。
In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3-manifolds are ``symplectic cobordism equivalent.''
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