Floer-Fukaya theory and topological elliptic objects
arXiv:1202.4118
Abstract
Inspired by Segal-Stolz-Teichner project for geometric construction of elliptic (tmf) cohomology, and ideas of Floer theory and of Hopkins-Lurie on extended TFT's, we geometrically construct some -valued representable cofunctors on the homotopy category of topological spaces. Using a classical computation in Gromov-Witten theory due to Seidel we show that for one version of these cofunctors of the representing space is non trivial, provided a certain categorical extension of Kontsevich conjecture holds for the symplectic manifold , for some some . This gives further evidence for existence of generalized cohomology theories built from field theories living on a topological space.
Withdrawn as most of what I intended here now appears in complete detail in: Global Fukaya category and the space of A_\infty categories I and II, after trading complete Segal spaces for quasi-categories. There are some further algebraic topological ambitions in this withdrawn research announcement, and I do plan to get to them at some point