Cluster homology: An overview of the construction and results

By O. Cornea, Francois M. Lalonde. 2006

49 citations recorded by the source corpus.

We associate, to a Lagrangian submanifold L of a symplectic manifold, a new homology, called the cluster homology of L, which is invariant up to ambient symplectic diffeomorphisms. We discuss various applications concerning analytical, topological, and dynamical properties of Lagrangian submanifolds. We also deduce a new universal Floer homology, defined without obstruction, for pairs of Lagrangian submanifolds.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/38dd3b7660cac255801d2098b7288fc47b325ca9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. Nick Sheridan
2. P. Seidel
3. T. Ekholm
4. P. Biran
5. O. Cornea
6. Ben Webster
7. I. Smith
8. E. Witten

In Geometry and Topology

1. P. Seidel
2. M. Abouzaid
3. Nick Sheridan
4. T. Ekholm
5. O. Cornea
6. I. Smith
7. P. Biran
8. D. Auroux

In Mathematical Physics

1. P. Seidel
2. M. Abouzaid
3. Nick Sheridan
4. T. Ekholm
5. O. Cornea
6. P. Biran
7. I. Smith
8. D. Auroux

Citations in the corpus

• Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “In Section 6, we compute CF ∗(L,L) to first order in F(CP,D), using a Morse-Bott model for the relative Fukaya category, based on the ‘cluster homology’ of [9].”
• Rational symplectic field theory over Z2 for exact Lagrangian cobordisms (2006, methodology): “In this sense, the method for dealing with boundary bubbling in the present paper is related to [17] and to [6], though in [6], only one Morse function and flow lines appears, while here and in [17], there are many Morse functions and flow trees.”
• On the homological mirror symmetry conjecture for pairs of pants (2010, background): “Bott’ approach which is a modification of the approach in [48], combining it with the approach of [3]. The outline of this approach has appeared in [49, Section 7], and is related to the ‘clusters’ of [11]. However, the geometric situation we consider is simpler than that of [11], namely we work only”
• Quantum Structures for Lagrangian Submanifolds (2007, methodology): “Indeed, as introduced in [8] and in [24] there is a natural notion of Gromov width of a Lagrangian and obviously one can also consider the width of the complement of that Lagrangian.”
• Rigidity and uniruling for Lagrangian submanifolds (2008, background): “In the perspective of [24; 25] the complex C.”
• Homological mirror symmetry for the genus two curve (2008, methodology): “A point of this moduli space consists of the following data (similar to, but not quite the same as, the “clusters” from [16]):”