# Commentarii Mathematici Helvetici

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**Volume 85, Issue 1, 2010, pp. 95–133**

**DOI: 10.4171/CMH/189**

Published online: 2009-12-23

Monodromy in Hamiltonian Floer theory

Dusa McDuff^{[1]}(1) Columbia University, New York, USA

Schwarz showed that when
a closed symplectic manifold (`M`,`ω`) is symplectically aspherical (i.e. the symplectic form and the first
Chern class vanish on `π`_{2}(`M`)) then
the spectral invariants, which are initially defined on the universal cover
of the Hamiltonian group, descend to the Hamiltonian group
Ham(`M`,`ω`).
In this note we describe less stringent conditions
on the Chern class and quantum homology of `M` under which the
(asymptotic) spectral invariants descend to Ham(`M`,`ω`). For example, they
descend if the quantum multiplication of
`M` is undeformed and `H`_{2}(`M`) has rank > 1, or if
the minimal Chern number is at least `n` + 1 (where dim `M` = 2`n`) and the even cohomology
of `M` is generated by divisors.
The proofs are based on certain calculations of genus zero
Gromov–Witten invariants.
As an application, we show that the Hamiltonian group
of the one point blow up of `T`^{4}
admits a Calabi quasimorphism. Moreover, whenever the
(asymptotic) spectral invariants descend it is easy to see that
Ham(`M`,`ω`) has infinite diameter in the Hofer norm. Hence our results
establish the infinite diameter of Ham in many new cases. We also
show that the area pseudonorm – a geometric version of the Hofer norm – is nontrivial on the
(compactly supported) Hamiltonian group for all noncompact manifolds
as well as for a large class of closed manifolds.

*Keywords: *Hamiltonian group, Floer theory, Seidel representation, spectral invariant, Hofer norm, Calabi quasimorphism

McDuff Dusa: Monodromy in Hamiltonian Floer theory. *Comment. Math. Helv.* 85 (2010), 95-133. doi: 10.4171/CMH/189