36 research outputs found
Global Fukaya category and quantum Novikov conjecture I
Conceptually, the goal here is a construction which functorially translates a
Hamiltonian fibre bundle to a certain ``derived vector bundle'' over the same
space, with fiber an category. This ``derived vector bundle''
must remember the continuity of the original bundle. Concretely, using
Floer-Fukaya theory for a monotone we construct a natural
continuous map \begin{equation*}
BHam (M, \omega) \to (\mathcal{S}, NFuk (M)), \end{equation*} with
denoting the component of the ``space'' of
-categories, where is the -nerve of the Fukaya
category . This construction is very closely related to the theory of
the Seidel homomorphism and the quantum Chern classes of the author, and this
map is intended to be the deepest expression of their underlying geometric
theory. In part II the above map is shown to be non trivial by an explicit
calculation. In particular we arrive at a new non-trivial ``quantum'' invariant
of any smooth manifold and a ``quantum'' Novikov conjecture.Comment: v5, 41 pages. This adds significant detail and fixes some language
issue