# An absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields

@article{Ramos2011AnAG,
title={An absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields},
author={Vinicius G. B. Ramos and Yang Huang},
journal={arXiv: Symplectic Geometry},
year={2011}
}
• Published 1 December 2011
• Mathematics
• arXiv: Symplectic Geometry
For a closed oriented 3-manifold Y, we define an absolute grading on the Heegaard Floer homology groups of Y by homotopy classes of oriented 2-plane fields. We show that this absolute grading refines the relative one and that it is compatible with the maps induced by cobordisms. We also prove that if {\xi} is a contact structure on Y, then the grading of the contact invariant c({\xi}) is the homotopy class of {\xi}.
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## References

SHOWING 1-10 OF 13 REFERENCES
On the contact class in Heegaard Floer homology
• Mathematics
• 2006
We present an alternate description of the Ozsvath-Szabo contact class in Heegaard Floer homology. Using our contact class, we prove that if a contact structure (M,\xi) has an adapted open book
Floer homology and knot complements
We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It
Holomorphic disks and genus bounds
• Mathematics
• 2004
We prove that, like the Seiberg-Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the
HF=HM, I : Heegaard Floer homology and Seiberg–Witten Floer homology
• Mathematics
Geometry & Topology
• 2020
Let M be a closed, connected and oriented 3-manifold. This article is the first of a five part series that constructs an isomorphism between the Heegaard Floer homology groups of M and the
Embedded contact homology and open book decompositions
• Mathematics
• 2010
This is the first of a series of papers devoted to proving the equivalence of Heegaard Floer homology and embedded contact homology (abbreviated ECH). In this paper we prove that, given a closed,
Embedded contact homology and its applications
Embedded contact homology (ECH) is a kind of Floer homology for contact three-manifolds. Taubes has shown that ECH is isomorphic to a version of Seiberg-Witten Floer homology (and both are
Holomorphic disks and topological invariants for closed three-manifolds
• Mathematics
• 2001
The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y, equipped with a Spiny structure. Given a Heegaard splitting of Y = U 0o U Σ U 1 , these
Handlebody construction of Stein surfaces
The topology of Stein surfaces and contact 3-manifolds is studied by means of handle decompositions. A simple characterization of homeomorphism types of Stein surfaces is obtained-they correspond to
A cylindrical reformulation of Heegaard Floer homology
We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold U a0;1c R, where U is the Heegaard surface, instead of Sym g .U/. We then show that the entire