Journal of the European Mathematical Society

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Volume 9, Issue 4, 2007, pp. 841–876
DOI: 10.4171/JEMS/99

A general Fredholm theory I: a splicing-based differential geometry

David Geraghty[1], Kris Wysocki[2] and Eduard Zehnder[3]

(1) School of Mathematics, Institute for Advanced Study, Einstein Drive, NJ 08540, PRINCETON, UNITED STATES
(2) Department of Mathematics, Penn State University, PA 16802, UNIVERSITY PARK, UNITED STATES
(3) Departement Mathematik, ETH Zürich, LEO D 3, Leonhardstrasse 27, 8092, ZÜRICH, SWITZERLAND

This is the first paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. These spaces, in general, are locally not homeomorphic to open sets in Banach spaces. The present paper describes some of the differential geometry of this new class of spaces. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory, and Symplectic Field Theory.

Keywords: Banach scales, sc-smoothness, M-polyfolds, splicings, splicing cores, fillers, strong bundles

Geraghty David, Wysocki Kris, Zehnder Eduard: A general Fredholm theory I: a splicing-based differential geometry. J. Eur. Math. Soc. 9 (2007), 841-876. doi: 10.4171/JEMS/99