A General Fredholm Theory and Applications

By Helmut Hofer. 2005

67 citations recorded by the source corpus.

The theory described here results from an attempt to find a general abstract framework in which various theories, like Gromov-Witten Theory (GW), Floer Theory (FT), Contact Homology (CH) and more generally Symplectic Field Theory (SFT) can be understood from a general point of view. Let us describe the general landscape in a somewhat oversimplified form. The common feature (with the exception of GW which has less structure) is the fact that we have infinitely many different Fredholm problems defined on spaces with boundary with corners, where the boundary strata can be explained in terms of products (or more generally fibered products) of other problems (on the list). In oversimplified form, the solution sets are zeros of a section f of some bundle τ : Y → X, where the space has a boundary ∂X, and where moreover there exists a recipe (or even many recipes) to construct from two given solutions x and x of f = 0 a new solution, say the product, x = x ◦x. The recipes for constructing new solutions are defined even for non-solutions and ∂X is precisely the space of points which are products. Hence we have

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f054c6a35506c79e0701a2636993afe18a9ac24f

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Citations in the corpus

Weak and strong fillability of higher dimensional contact manifolds (2011, unknown): “[36].”
Bordered Heegaard Floer homology (2008, background): “(3) The map u extends to a proper map ue : Se → Σe × [0, 1]× R. (4) The map ue has nite energy in the sense of Bourgeois, Eliashberg, Hofer, Wysocki and Zehnder [2].”
$S^1$-equivariant symplectic homology and linearized contact homology (2012, methodology): “In particular, we do not appeal to abstract perturbations like the ones used in the polyfold theory of Hofer, Wysocki and Zehnder [38, 39, 40, 41].”