We prove the bounded isometry conjecture of F. Lalonde and L. Polterovich for a special class of closed symplectic manifolds. As a byproduct, it is shown that the flux group of a product of these special symplectic manifold is isomorphic to the direct sum of the flux group of each symplectic manifold.2010 Mathematics Subject Classification. Primary: 53D35 57R17.

Abstract. We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. In particular, we show that the unimodularity of the transverse Poisson structure of the leaf is a necessary condition for the semilocal unimodular property. Our main tool is an explicit formula for a bigraded decomposition of modular vector fields of a coupling Poisson structure on a foliated manifold.

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We lift a loop of Hamiltonian diffeomorphisms on a symplectic manifold to loop of Hamiltonian diffeomorphisms on the symplectic onepoint blow up of the symplectic manifold. Then we use Weinstein's morphism to show that the lifted loop of Hamiltonian diffeomorphisms has infinite order on the fundamental group of the group of Hamiltonian diffeomorphisms of the blown up manifold. IntroductionThe rational homotopy type of the group of Hamiltonian diffeomorphisms of the symplectic one-point blow up ( M , ω ρ ) of weight ρ is known for only a special class of symplectic manifolds. In [1], M. Abreu and D. McDuff computed the rational homotopy type of the group of symplectic diffeomorphisms of the symplectic one-point blow up of (CP 2 , ω FS ) In [5], F. Lalonde and M. Pinsonnault computed the rational homotopy type of the above group for the one-point blow up of (S 2 × S 2 , ω ⊕ µω) for 1 ≤ µ ≤ 2; and in [11] M. Pinsonnault worked out the case of the one-point blow up of rational ruled symplectic 4-manifolds; see also [2]. The case of multiple points blown up simultaneously has also been considered. J.D. Evans [3] considered this case for (CP 2 , 3ω FS ) blown up at 3, 4 and 5 generic points. The reason that all the above examples are in dimension 4, has to do with the special behavior of holomorphic curves in 4 dimensional symplectic manifolds. Apart from these cases, only partial information is known about the homotopy type of Ham( M , ω ρ ). For example in [7], D. McDuff showed that if the Hurewicz morphism π 2 (M) − → H 2 (M; Q) is non trivial then there exists a non trivial morphism π 2 (M) − → π 1 (Ham( M , ω ρ )).In this paper we will focus on determining that π 1 (Ham( M , ω ρ )) is non trivial for some particular class of symplectic manifolds (M, ω). Moreover, the way that we show that π 1 (Ham( M , ω ρ )) is non trivial is by considering a particular class of loops of Hamiltonian diffeomorphisms in Ham(M, ω), lift it Key words and phrases. Symplectic one-pint blow up, Hamiltonian diffeomorphism group, Weinstein's morphism.The author was supported by a CONACYT grant CB 2010/151846.

We give a generalization of the Atiyah-Bott-Berline-Vergne localization theorem for the equivariant cohomology of a torus action. We replace the manifold having a torus action by an equivariant map of manifolds having a compact connected Lie group action. This provides a systematic method for calculating the Gysin homomorphism in ordinary cohomology of an equivariant map. As an example, we recover a formula of Akyildiz-Carrell for the Gysin homomorphism of flag manifolds.Suppose M is a compact oriented manifold on which a torus T acts. The Atiyah-Bott-Berline-Vergne localization formula calculates the integral of an equivariant cohomology class on M in terms of an integral over the fixed point set M T . This formula has found many applications, for example, in analysis, topology, symplectic geometry, and algebraic geometry (see [2,6,8,12]). Similar, but not entirely analogous, formulas exist in K-theory [3], cobordism theory [11], and algebraic geometry [7].Taking cues from the work of Atiyah and Segal in K-theory [3], we state and prove a localization formula for a compact connected Lie group action in terms of the fixed point set of a conjugacy class in the group. As an application, the formula can be used to calculate the Gysin homomorphism in ordinary cohomology of an equivariant map. For a compact connected Lie group G with maximal torus T and a closed subgroup H containing T , we work out as an example the Gysin homomorphism of the canonical projection f : G/T → G/H , a formula first obtained by Akyildiz and Carrell [1].The application to the Gysin map in this article complements that of [12]. The previous article [12] shows how to use the ABBV localization formula to calculate the Gysin map of a fiber bundle. This article shows how to use the relative localization formula to calculate the Gysin map of an equivariant map.