Global Fukaya category I

Abstract

Let $Ham (M,\omega ) $ denote the Frechet Lie group of Hamiltonian symplectomorphisms of a monotone symplectic manifold $(M, \omega) $. Let $NFuk (M, \omega)$ be the $A _{\infty} $-nerve of the Fukaya category $Fuk (M, \omega)$, and let $(|\mathbb{S}|, NFuk (M, \omega))$ denote the $NFuk (M, \omega)$ component of the ``space of $\infty$-categories'' $|\mathbb{S}| $. Using Floer-Fukaya theory for a monotone $(M, \omega)$ we construct a natural up to homotopy classifying map \begin{equation*} BHam (M, \omega) \to (|\mathbb{S}|, NFuk (M, \omega)). \end{equation*} This verifies one sense of a conjecture of Teleman on existence of action of $Ham (M , \omega)$ on the Fukaya category of $(M, \omega ) $. This construction is very closely related to the theory of the Seidel homomorphism and the quantum characteristic 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 nontrivial by an explicit calculation. In particular, we arrive at a new non-trivial ``quantum'' invariant of any smooth manifold, which motives the statement of a kind of ``quantum'' Novikov conjecture.