A General Fredholm Theory I: A Splicing-Based Differential Geometry

1 Introduction In a series of papers we develop a generalized Fredholm theory and demonstrate its applicability to a variety of problems including Floer theory, Gromov-Witten theory, contact homology, and symplectic field theory. Here are some of the basic common features: • The moduli spaces are solutions of elliptic PDE's showing serious non-compactness phenomena having well-known names like bubbling-off, stretching the neck, blow-up, breaking of trajectories. These drastic names are a manifestation of the fact that one is confronted with analytical limiting phenomena where the classical analytical descriptions break down. • When the moduli spaces are not compact, they admit nontrivial compactifi-cations like the Gromov compactification, [6], of the space of pseudoholomor-phic curves in Gromov-Witten theory or the compactification of the moduli spaces in symplectic field theory (SFT) as described in [2]. • In many problems like in Floer theory, contact homology or symplectic field theory the algebraic structures of interest are precisely those created by the " violent analytical behavior " and its " taming " by suitable compactifica-tions. In fact, the algebra is created by the complicated interactions of many different moduli spaces. In the abstract theory we shall introduce a new class of spaces called polyfolds which in applications are the ambient spaces of the compactified moduli spaces. We introduce bundles p : Y → X over polyfolds which, as well as the underlying polyfolds, can have varying dimensions. We define the notion of a Fredholm section η of the bundles p whose zero sets η −1 (0) ⊂ X are in our applications precisely the compactfied moduli spaces one is interested in. The normal " Fredholm package " will be constructed consisting of an abstract perturbation and transversality theory. In the case of transver-sality the solution spaces are smooth manifolds, smooth orbifolds, or smooth weighted branched manifolds (in the sense of McDuff, [18]), depending on the generality of the situation. The usefulness of this theory will be illustrated by our 'Application Series'. The applications include Gromov-Witten theory, Floer theory and SFT, see [13, 14]. It is, however, clear that the theory applies to many more nonlinear problems showing a lack of compactness. The current paper is the first in the 'Theory Series' and deals with a generalization of differential geometry which is based on new local models. These local models are open sets in splicing cores. Splicing cores are smooth spaces with tangent spaces having in …