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The main topic is the development of a Fredholm theory in a new class ofspaces called M-polyfolds. In the subsequent Volume II the theory will begeneralized to an even larger class of spaces called polyfolds, which can alsoincorporate local symmetries. The whole package provides a functional analyticframework to deal with compactness and transversality issues as they occur inmoduli problems of symplectic geometry. Applications of the theory coverFloer-type theories as they occur in symplectic geometry. M-polyfolds and themore general polyfolds are smooth spaces which can be finite-dimensional aswell as infinite-dimensional. In applications of interest they in general havelocally varying dimensions. Despite the fact that the spaces are much moregeneral than Banach manifolds a nonlinear Fredholm theory with the usualfeatures is possible (Sard-Smale type perturbation theory). This generalizedFredholm theory can be applied to classes of nonlinear elliptic problems whichshow bubbling-off phenomena but allow for certain kind of compactifications.

Polyfold and Fredholm Theory I: Basic Theory in M-Polyfolds