We present partial results towards a classification of symplectic mapping tori using dynamical
properties of wrapped Fukaya categories. More precisely, we construct a symplectic manifold
associated to a
Weinstein domain
,
and an exact, compactly supported symplectomorphism
. The symplectic
manifold
is another Weinstein domain and its contact boundary is independent of
. We
distinguish
from
,
under certain assumptions (Theorem 1.1). As an application, we obtain pairs
of diffeomorphic Weinstein domains with the same contact boundary and
whose symplectic cohomology groups are the same, as vector spaces, but
that are different as Liouville domains. To our knowledge, this is the first
example of such pairs that can be distinguished by their wrapped Fukaya
category.
Previously, we have suggested a categorical model
for the wrapped
Fukaya category
, and
we have distinguished
from the mapping torus category of the identity. We prove
and
are
derived equivalent (Theorem 1.9); hence, deducing the promised Theorem 1.1.
Theorem 1.9 is of independent interest as it preludes an algebraic description of
wrapped Fukaya categories of locally trivial symplectic fibrations as twisted tensor
products.
