Hamiltonian loops on the symplectic one-point blow up

Abstract

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.

On the rank of $π_1(\text{Ham})$

“…Using the embedded ball ιB r 0 ⊂ M, define ( M , ω r 0 ) to be the one-point blow up at ι(0) of (M, ω) of weight r 0 . From the above remarks on the loop ψ, it follows from [10,Sec. 3] that ψ induces a Hamiltonian loop ψ = { ψ t } 0≤t≤2 in Ham( M , ω r 0 ).…”

“…In [10] it is proved that if a closed symplectic manifold admits a Hamiltonian circle action, then after blowing up one point the fundamental group of the group of Hamiltonian diffeomorphisms has positive rank. In this section we prove that the above result always holds, namely we prove Theorem 1.2.…”

“…, ι k (p k ). Therefore, from [10,Sec. 3] it follows that ψ (j) induces a loop ψ (j) in Ham(CP 2 # k CP 2 , ω r ).…”