scholarly journals DUALITY IN FILTERED FLOER–NOVIKOV COMPLEXES

2010 ◽  
Vol 02 (02) ◽  
pp. 233-258 ◽  
Author(s):  
MICHAEL USHER
Keyword(s):  
Chain Complex ◽  
Bilinear Pairing ◽  
Symplectic Manifolds ◽  
Duality Relation ◽  
Spectral Invariants ◽  
Floer Theory ◽  
Intersection Pairing ◽  
Period Map ◽  
New Formula ◽  
Special Case

We prove that a certain bilinear pairing (analogous to the Poincaré–Lefschetz intersection pairing) between filtered sub- and quotient complexes of a Floer-type chain complex and of its "opposite complex" is always nondegenerate on homology. This implies a duality relation for the Oh–Schwarz-type spectral invariants of these complexes which (in Hamiltonian Floer theory) was established in the special case that the period map has discrete image by Entov and Polterovich. The duality relation served as a key lemma in Entov and Polterovich's construction of a Calabi quasimorphism on certain rational symplectic manifolds, and the result that we prove here implies that their construction remains valid when the rationality hypothesis is dropped. Apart from this, we also use the nondegeneracy of the pairing to establish a new formula for what we have previously called the boundary depth of a Floer chain complex; this formula shows that the boundary depth is unchanged under passing to the opposite complex.

scholarly journals The Guillemin–Sternberg conjecture for noncompact groups and spaces

2008 ◽  
Vol 1 (3) ◽  
pp. 473-533 ◽  
Author(s):  
P. Hochs ◽  
N.P. Landsman
Keyword(s):  
Normal Subgroup ◽  
Dirac Operator ◽  
Lie Groups ◽  
Group Action ◽  
Symplectic Manifolds ◽  
Lie Group Action ◽  
Compact Case ◽  
Assembly Map ◽  
High Degree ◽  
Special Case

AbstractThe Guillemin–Sternberg conjecture states that “quantisation commutes with reduction” in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved for compact Lie groups G acting on compact symplectic manifolds, and, largely due to the use of Spinc Dirac operator techniques, has reached a high degree of perfection under these compactness assumptions. In this paper we formulate an appropriate Guillemin–Sternberg conjecture in the general case, under the main assumptions that the Lie group action is proper and cocompact. This formulation is motivated by our interpretation of the “quantisation commuates with reduction” phenomenon as a special case of the functoriality of quantisation, and uses equivariant K-homology and the K-theory of the group C*-algebra C*(G) in a crucial way. For example, the equivariant index – which in the compact case takes values in the representation ring R(G) – is replaced by the analytic assembly map – which takes values in K0(C*(G)) – familiar from the Baum–Connes conjecture in noncommutative geometry. Under the usual freeness assumption on the action, we prove our conjecture for all Lie groups G having a discrete normal subgroup Γ with compact quotient G/Γ, but we believe it is valid for all unimodular Lie groups.

scholarly journals Spectral invariants in Lagrangian Floer theory

2008 ◽  
Vol 2 (2) ◽  
pp. 249-286 ◽  
Author(s):  
Rémi Leclercq ◽  
Keyword(s):  
Spectral Invariants ◽  
Floer Theory

scholarly journals Lagrangian floer Theory over integers: spherically positive symplectic manifolds

2013 ◽  
Vol 9 (2) ◽  
pp. 189-289 ◽  
Author(s):  
Kenji Fukaya ◽  
Yong-Geun Oh ◽  
Hiroshi Ohta ◽  
Kaoru Ono
Keyword(s):  
Symplectic Manifolds ◽  
Floer Theory

scholarly journals Invariance of symplectic cohomology and twisted cotangent bundles over surfaces

2020 ◽  
Vol 31 (09) ◽  
pp. 2050070
Author(s):  
Gabriele Benedetti ◽  
Alexander F. Ritter
Keyword(s):  
Symplectic Form ◽  
Cotangent Bundle ◽  
Local System ◽  
Symplectic Manifolds ◽  
Existence Results ◽  
Open Convex ◽  
Floer Theory ◽  
Cotangent Bundles ◽  
Sample Application ◽  
Symplectic Cohomology

We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be nonexact and noncompactly supported, provided one uses the correct local system of coefficients in Floer theory. As a sample application beyond the Liouville setup, we describe in detail the symplectic cohomology for disc bundles in the twisted cotangent bundle of surfaces, and we deduce existence results for periodic magnetic geodesics on surfaces. In particular, we show the existence of geometrically distinct orbits by exploiting properties of the BV-operator on symplectic cohomology.

scholarly journals DESCENT AND C0-RIGIDITY OF SPECTRAL INVARIANTS ON MONOTONE SYMPLECTIC MANIFOLDS

2012 ◽  
Vol 04 (04) ◽  
pp. 481-498 ◽  
Author(s):  
SOBHAN SEYFADDINI
Keyword(s):  
Symplectic Manifolds ◽  
Spectral Invariants ◽  
Open Set ◽  
Continuity Properties ◽  
Hofer Metric

We obtain estimates showing that on monotone symplectic manifolds (asymptotic) spectral invariants of Hamiltonians which vanish on a non-empty open set, U, descend from [Formula: see text] to Hamc(M\U). Furthermore, we show that these invariants are continuous with respect to the C0-topology on Hamc(M\U).We apply the above results to Hofer geometry and establish unboundedness of the Hofer diameter of Hamc(M\U) for stably displaceable U. We also answer a question of F. Le Roux about C0-continuity properties of the Hofer metric.

scholarly journals Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory

2019 ◽  
Vol 260 (1254) ◽  
pp. 0-0
Author(s):  
Kenji Fukaya ◽  
Yong-Geun Oh ◽  
Hiroshi Ohta ◽  
Kaoru Ono
Keyword(s):  
Spectral Invariants ◽  
Floer Theory

scholarly journals Comparison of spectral invariants in Lagrangian and Hamiltonian Floer theory

Filomat ◽  
2016 ◽  
Vol 30 (5) ◽  
pp. 1161-1174 ◽  
Author(s):  
Jovana Djuretic ◽  
Jelena Katic ◽  
Darko Milinkovic
Keyword(s):  
Periodic Orbits ◽  
Symplectic Manifold ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Spectral Invariants ◽  
Floer Theory

We compare spectral invariants in periodic orbits and Lagrangian Floer homology case, for a closed symplectic manifold P and its closed Lagrangian submanifolds L, when ?|?2(P,L)=0, and ?|?2(P,L)=0. We define a product HF*(H)?HF*(H:L) ? HF*(H:L) and prove subadditivity of invariants with respect to this product.

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.

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