Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds
Abstract
In this paper, we develop a mini-max theory of the action functional over the semi-infinite cycles via the chain level Floer homology theory and construct spectral invariants of Hamiltonian diffeomorphisms on arbitrary, especially on {\it non-exact and non-rational}, compact symplectic manifold
(M,ω)
. To each given time dependent Hamiltonian function
H
and quantum cohomology class
0≠a∈Q
H
∗
(M)
, we associate an invariant
ρ(H;a)
which varies continuously over
H
in the
C
0
-topology. This is obtained as the mini-max value over the semi-infinite cycles whose homology class is `dual' to the given quantum cohomology class
a
on the covering space
Ω
˜
0
(M)
of the contractible loop space
Ω
0
(M)
. We call them the {\it Novikov Floer cycles}. We apply the spectral invariants to the study of Hamiltonian diffeomorphisms in sequels of this paper.