Holomorphic triangle invariants and the topology of symplectic four-manifolds

P. S. Ozsvath; Z. Szabo

Home/Archive/arXiv

arXiv · math/0201049

Holomorphic triangle invariants and the topology of symplectic four-manifolds

P. S. Ozsvath·Z. Szabo

View original source ↗Read source document ↗Export metadata ↓

Abstract

This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing result for the invariants of symplectic four-manifolds, which leads to new proofs of the indecomposability theorem for symplectic four-manifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of four-manifolds along a certain class of three-manifolds obtained by plumbings of spheres. This leads to restrictions on the topology of Stein fillings of such three-manifolds.

Explore related subjects

math.SG math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

P. S. Ozsvath, Z. Szabo. 2002-01-08. Holomorphic triangle invariants and the topology of symplectic four-manifolds. https://arxiv.org/abs/math/0201049

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Multiplicity Bounds for Contractible Closed Reeb Orbits on Prequantization Bundles

We study the multiplicity of contractible closed Reeb orbits on prequantization circle bundles $(M^{2n+1},ξ)$ over closed symplectic manifolds $(B^{2n},ω)$. For even $n$, under suitable geometric hypotheses, we prove that every non-degenerate contact form supporting $ξ$ whose contractible closed orbits have nonzero mean index and whose good contractible closed orbits have nonzero Conley--Zehnder index carries at least $\frac{1}{2}χ(B)+\frac{1}{2}\dim H_n(B;\mathbb{Q})+1$ geometrically distinct contractible closed Reeb orbits. In dimension five, under the corresponding geometric hypotheses and assuming $H_1(B;\mathbb{Q})=0$, we prove the conjectural lower bound $\dim H_*(B;\mathbb{Q})$ for every non-degenerate supporting contact form whose simple contractible closed orbits have nonzero Conley--Zehnder index.

math.SG↗

arXiv2026

Categorical and Floer-theoretic entropies: comparison and counting conjugacy classes in symplectic mapping class groups

Let $W$ be a Weinstein manifold and let $f: W \to W$ be a compactly supported symplectic automorphism. Using Floer theories, one can define two different entropies of $f$, called "Floer-theoretic" entropy and "categorical" entropy. The current article compares these two entropies and show that the former bounds the latter from below. As an application of the comparison, using that these entropies are invariants up to the conjugate relation, we count conjugacy classes in symplectic mapping classes. Especially, we show an exponential abundance of conjugacy classes that are smoothly isotopic to the identity, when $W$ is a Weinstein manifold of dimension $4n \geq 6$, satisfying a local condition.

math.SG↗

arXiv2026

Mapping-space Gromov--Witten classes and the Fano variety of lines

Let $Y$ be a smooth cubic 3-fold. We show that $S^2 \times Y$ admits an infinite sequence of cohomologous symplectic forms which are all deformation equivalent but pairwise non-isotopic; McDuff showed the corresponding result on $(S^2)^2 \times T^2$, but these appear to be the first simply-connected and monotone examples. The results are obtained from McDuff's bordism-flavoured Gromov--Witten invariant, re-interpreted cohomologically in terms of an evaluation map valued in the space of maps from the 2-sphere to a symplectic target. The application then relies on classical properties of the Fano space of lines on $Y$. We also discuss higher-dimensional generalisations.

math.SG↗