Given Lagrangian (real, complex) projective spaces
in a Liouville
manifold
satisfying a certain cohomological condition, we show there is a Lagrangian correspondence
(in the sense of Wehrheim and Woodward (2012)) that assigns a Lagrangian sphere
of another Liouville
manifold
to any given
projective Lagrangian
for
.
We use the Hopf correspondence to study
projective twists, a class of
symplectomorphisms akin to Dehn twists, but defined starting from Lagrangian
projective spaces. When this correspondence can be established, we show
that it intertwines the autoequivalences of the compact Fukaya category
induced by the
projective twists
with
the autoequivalences of
induced by the Dehn twists
for
.
Using the Hopf correspondence, we obtain a free generation result for
projective twists in a
clean plumbing of projective spaces and various results
about products of positive powers of Dehn/projective twists in Liouville
manifolds.
The same techniques are also used to show that the Hamiltonian
isotopy class of the projective twist (along the zero section in
) in
does depend on a
choice of framing for
.
Another application of the Hopf correspondence delivers smooth homotopy complex projective spaces
that do not admit
Lagrangian embeddings into
for
.
Keywords
symplectic topology, Dehn twists, symplectic mapping class
group, product of twists, framing of twists, nearby
Lagrangian conjecture
