Calabi quasimorphism and quantum homology

By Michael Entov, L. Polterovich. 2002

261 citations recorded by the source corpus.

We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small open subset of the sphere equals to the Calabi invariant of the diffeomorphism. This result extends to more general symplectic manifolds: If the symplectic manifold is monotone and its quantum homology algebra is semi-simple we construct a similar quasimorphism on the universal cover of the group of Hamiltonian diffeomorphisms.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1f60fbb3fc98e28d39f502e765ddf9c6d29ca4eb

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Citations in the corpus

Conjugation-invariant norms on groups of geometric origin (2007, background): “The commutator length on Ham( M, ω ) is known to be stably unbounded for various closed symplectic manifolds (see [3, 10, 11, 13, 22, 23]) including all surfaces and complex projective spaces of arbitrary dimension.”
Persistent homology and Floer-Novikov theory (2015, background): “ther restricted circumstances. While the ﬁltered Floer groupshave been studied since the early 1990’s and have been a signiﬁcant tool in symplectic topology since that time (see e.g. [FlH94], [Sc00], [EP03], [Oh05], [U13], [HLS15]), it is only very recently that they have been considered from a pers”
Spectral invariants, analysis of the Floer moduli spaces and geometry of the Hamiltonian diffeomorphism group (2004, background): “quiet time for writing and an excellent atmosphere for research. Thanks to M. Entov and L. Polterovich for enlightening discussions on spectral invariants and for explaining their applications [En2], [EnP] of the spectral invariants to the study of the Hamiltonian diﬀeomorphism group, and Y. Ostrove”
Quasi-states and symplectic intersections (2004, background): “[19], [20]) yields that for a certain E ∈ R, depending on φ, c(f φ) = c(f )+ E, c(f φ−1) = c(f )− E.”
Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms (2010, methodology): “Beginning with [12], work of Entov and Polterovich and others has shown that, on some symplectic manifolds, the asymptotic spectral invariant z H 7! limk!1 .”
CALABI QUASIMORPHISMS FOR THE SYMPLECTIC BALL (2003, background): “[8]Entov,M .,Polterovich,L.,Calabiquasim orphism and quantum hom ol- ogy, Intern.”
Rigid subsets of symplectic manifolds (2007, background): “Seidel we described in [22] the restriction of μ (in fact, for any spherically monotone M) on π1(Ham(M)) ⊂ H̃am (M) in terms of the Seidel homomorphism π1(Ham(M)) → QH inv ∗ (M), where QH inv ∗ (M) denotes the group of invertible elements in the ring QH∗(M).”
Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds (2004, background): “would like to emphasize that (5.2) is well-deﬁned because of the choice of directions of the Novikov rings Λ↑ and Λ↓. In general, the map (5.2) is injective but not an isomorphism. Polterovich [Po4], [EnP] observed that this point is closely related to certain failure of “Poincar´e duality” of the F”
Groups acting on manifolds: around the Zimmer program (2008, background): “e proof that lattices have no C1 actions on S1 via bounded cohomology [21, 22, 98]. That Diﬀ(S,ω)0 admits many interesting quasi-morphisms, follows from work of Entov-Polterovich, GambaudoGhys and Py [48, 95, 192, 193, 192]. By results of Burger and Monod, for any higher rank lattice and any homomor”
Spectral invariants for monotone Lagrangians (2015, unknown): “t descends to a well-deﬁned function ν: QH∗(L) → R∪{−∞}. Remark 20. When NL= ∞, it follows from the deﬁnitions that ν(α) = 0 for any nonzero α. Lemma 21. If α6= 0 , then ν(α) ∈ R. Proof. Just like in [EP03], we can show that ν(α) is the maximum of ν(αk) where αk ∈ QHk(D : L) is the degree-k componen”
Maximal Representations of Surface Groups: Symplectic Anosov Structures (2005, background): “• G : Hcb(G,R) → H • (5.1) c(G,R) which encodes subtle properties of Gof geometric and algebraic nature. See [1], [32], [54], [17], [18, § V.13], [13], and also [8], [9], [37], [59], [2], [27], [30], [26], [3], [48], [47] in relation with the existence of quasimorphisms. If now G is a connected semi”
On the classification of rank-two representations of quasiprojective fundamental groups (2007, background): “It has been cited recently in the papers [21], [25], and [45].”
Action and index spectra and periodic orbits in Hamiltonian dynamics (2008, background): “Our proof of the theorem relies on the technique of action selectors, developed by Hofer and Zehnder [34], Oh [44], Schwarz [52] and Viterbo [56], and on the Hamiltonian version of the Ljusternik–Schnirelman theory (rather than on an explicit use of holomorphic spheres) and bears a resemblance to ma”
Bounded differential forms, generalized Milnor–Wood inequality and an application to deformation rigidity (2007, background): “which encodes subtle properties of G of algebraic and geometric nature, see [1], [28], [43], [44], [18], [19, x V.13], [12], (see also [7], [8], [34], [48], [2], [26], [27], [ 25 ], [3], [39], [38] in relation with the existence of quasimorphisms).”
Bounds on spectral norms and barcodes (2018, methodology): “ve S ;˙([pt]) = q n[M]: Proof. The suitable loop is (t) : [z 0 : z 1 : :::: z n] 7![e 2ˇitz 0 : z 1 : :::: z n] for t2[0;1]:This follows by the original computation of Seidel element [99] (see also [39, 76], and note that this action is semi-free). This statement implies an analogous one for L= CPn ”
Quasi-morphisms and the Poisson bracket (2006, methodology): “These quasi-morphisms, which were found in [5], are rooted in Floer homology.”