The arnold conjecture

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Fixed Points ◽  
Generating Functions ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Index Theory ◽  
Intersection Problem ◽  
Arnold Conjecture ◽  
Conley Index Theory ◽  
Cotangent Bundles ◽  
Symplectic Action

This chapter contains a proof of the Arnold conjecture for the standard torus, which is based on the discrete symplectic action. The symplectic part of this proof is very easy. However, for completeness of the exposition, one section is devoted to a fairly detailed discussion of the relevant Conley index theory and of Ljusternik–Schnirelmann theory. Closely related to the problem of finding symplectic fixed points is the Lagrangian intersection problem. The chapter outlines a proof of Arnold’s conjecture for cotangent bundles that again uses the discrete symplectic action, this time to construct generating functions for Lagrangian submanifolds. The chapter ends with a brief outline of the construction and applications of Floer homology.

Generating functions

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Discrete Time ◽  
Generating Functions ◽  
Final Section ◽  
Lagrangian Submanifolds ◽  
Variational Problems ◽  
Action Functional ◽  
Arnold Conjecture ◽  
Cotangent Bundles ◽  
Symplectic Action

This chapter discusses generating functions in more detail. It shows how generating functions give rise to discrete-time analogues of the symplectic action functional and hence lead to discrete variational problems. The results of this chapter form the basis for the proofs in Chapter 11 of the Arnold conjecture for the torus and in Chapter 12 of the existence of the Hofer–Zehnder capacity. The final section examines generating functions for exact Lagrangian submanifolds of cotangent bundles.

scholarly journals Spectral invariants for monotone Lagrangians

2018 ◽  
Vol 10 (03) ◽  
pp. 627-700 ◽  
Author(s):  
Rémi Leclercq ◽  
Frol Zapolsky
Keyword(s):  
Generating Functions ◽  
Floer Homology ◽  
Spectral Invariants ◽  
Seminal Paper ◽  
Floer Theory ◽  
Cotangent Bundles

Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper [73], they have been defined in various contexts, mainly via Floer homology theories, and then used in a great variety of applications. In this paper we extend their definition to monotone Lagrangians, which is so far the most general case for which a “classical” Floer theory has been developed. Then, we gather and prove the properties satisfied by these invariants, and which are crucial for their applications. Finally, as a demonstration, we apply these new invariants to symplectic rigidity of some specific monotone Lagrangians.

scholarly journals Mappings and homological properties in the Conley index theory

1988 ◽  
Vol 8 (8) ◽  
pp. 175-198 ◽  
Keyword(s):  
Fixed Points ◽  
Euler Characteristic ◽  
Index Theory ◽  
Conley Index ◽  
Homology Theory ◽  
Conley Index Theory ◽  
Characteristic Condition ◽  
Lefschetz Theorem ◽  
Index Level

AbstractThe role in the Conley index of mappings between flows is considered. A class of maps is introduced which induce maps on the index level. With the addition of such maps to the theory, the homology Conley index becomes a homology theory. Using this structure, an analogue of the Lefschetz theorem is proved for the Conley index. This gives a new condition for detecting fixed points of flows, extending the classical Euler characteristic condition.

LUSTERNIK–SCHNIRELMANN CATEGORY BASED ON THE DISCRETE CONLEY INDEX THEORY

2018 ◽  
Vol 61 (03) ◽  
pp. 693-704
Author(s):  
KATSUYA YOKOI
Keyword(s):  
Index Theory ◽  
Conley Index ◽  
Invariant Sets ◽  
Conley Index Theory ◽  
Lusternik Schnirelmann

AbstractWe study Lusternik–Schnirelmann type categories for isolated invariant sets by the use of the discrete Conley index.

scholarly journals Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces

2019 ◽  
Vol 6 (1) ◽  
pp. 303-319
Author(s):  
Yoshihiro Ohnita
Keyword(s):  
Symmetric Space ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Lagrangian Submanifold ◽  
Riemannian Symmetric Space ◽  
Canonical Embedding ◽  
Real Form ◽  
Isotropy Representation ◽  
Totally Geodesic

AbstractAn R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.

scholarly journals Exact Lagrangian submanifolds in simply-connected cotangent bundles

2007 ◽  
Vol 172 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Kenji Fukaya ◽  
Paul Seidel ◽  
Ivan Smith
Keyword(s):  
Lagrangian Submanifolds ◽  
Cotangent Bundles ◽  
Simply Connected
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