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A general Fredholm theory. I: A splicing-based differential geometry. (English) Zbl 1149.53053
The paper under review is the first of a sequel of papers, and to studying the generalized Fredholm theory. The motivation of the series comes from the basic common analytic feature moduli spaces of solutions of elliptic PDEs with serious noncompact phenomena (the bubbling-tree, the neck-shrinking, blowing up and breaking of trajectories). The algebraic structure of the interesting compactifications leads to rich theories, like Floer theory, symplectic field theory and Gromov-Witten theory. The series of papers develops the ambient Banach spaces of the compactified moduli spaces in a systematic way for all these known examples. Potentially, it may lead to a new discovery (there is no example and application given in the first part).
The sc-structure (structure of a scale) on a Banach space is given by a nested sequence with compact imbedding operators from the top stratum and with the dense proper stratum in definition 2.1. A generalized Fredholm operator is an operator preserving the sc-structures with sc-sub-Banach finite dimensional kernel and cokernel. sc-Fredholm operators are preserved under the composition and addition of an sc-Fredholm operator and an s$$c^+$$-operator. In section 2.3, the authors develop the sc-smooth notion with respect to the structure of a scale, and prove the chain rule in theorem 2.16. This allows them to define sc-smooth manifolds, in short sc-manifolds. For both, gauge theory and symplectic theory the tangent splicing space is defined in section 3, and a local $$M$$-polyfold model consists of a pair of open subsets in the splicing core and the sc-smooth splicing in tangent direction.
It is certainly better to see the applications for this notion in symplectic field theory, Gromov-Witten theory and Floer theory (as sc-smooth within splicing world). Therefore the M-polyfold (generalizing manifold, orbifold) is an object with local chart having the sc-smooth splicing and local homeomorphism onto an open subset of the splicing core, the two charts are compatible if the transition maps are sc-smooth, with a maximal atlas.
The first important result on M-polyfolds is that there exists a subordinate sc-smooth partition of unity for M-polyfolds with local models on separable sc-Hilbert spaces. Another important result on the corner structure for $$M$$-polyfolds is given in subsection 3.4. The sub-M-polyfold similar to the submanifold definition is studied in section 3.5, and finite dimensional submanifolds of an M-polyfold will be given in the later work of the authors. Then, the authors continue to define M-polyfold bundles in section 4 as a generalization of the bundle over manifolds. Some new concepts, like fillability and fillers, are given due to the study of the symplectic field theory and contact homology theory. The linearization is sc-Fredholm if and only if the linearization of the filled version is sc-Fredholm (proposition 4.17). The index of the sc-Fredholm operator is not fully exploited in this current paper. The sc-index of the sc-Fredholm operator may be related to sc-characteristic classes.

##### MSC:
 53D40 Symplectic aspects of Floer homology and cohomology 58B15 Fredholm structures on infinite-dimensional manifolds 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 47A53 (Semi-) Fredholm operators; index theories
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