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Ergebnisse Der Mathematik Und Ihrer Grenzgebiete. 3. Folge / a Series of Modern Surveys in Mathematics Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics Volume 72 Series Editors L. Ambrosio, Pisa V. Baladi, Paris G.-M. Greuel, Kaiserslautern M. Gromov, Bures-sur-Yvette G. Huisken, Tübingen J. Jost, Leipzig J. Kollár, Princeton G. Laumon, Orsay U. Tillmann, Oxford J. Tits, Paris D.B. Zagier, Bonn More information about this series at http://www.springer.com/series/728 The Virtual Series on Symplectic Geometry Series Editors Alberto Abbondandolo Helmut Hofer Tara Suzanne Holm Dusa McDuff Claude Viterbo Associate Editors Dan Cristofaro-Gardiner Umberto Hryniewicz Emmy Murphy Yaron Ostrover Silvia Sabatini Sobhan Seyfaddini Jake Solomon Tony Yue Yu More information about this series at http://www.springer.com/series/16019 Helmut Hofer • Krzysztof Wysocki Eduard Zehnder Polyfold and Fredholm Theory Helmut Hofer Krzysztof Wysocki Institute for Advanced Study Department of Mathematics Princeton, NJ, USA Pennsylvania State University University Park, State College Eduard Zehnder PA , USA Department of Mathematics ETH Zurich Zürich, Switzerland ISSN 0071-1136 ISSN 2197-5655 (electronic) Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics ISBN 978-3-030-78006-7 ISBN 978-3-030-78007-4 (eBook) https://doi.org/10.1007/978-3-030-78007-4 Mathematics Subject Classification (2020): 58, 53, 46 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Two lives were tragically lost while writing this book. My coauthor and friend Kris Wysocki passed away on February 16, 2016 and my son David S. Hofer on March 1, 2016. I dedicate this book to them. Helmut Hofer I dedicate this book to my friend and coauthor Kris Wysocki who lost his long fight against cancer on February 16, 2016, and to his wife and our friend, Beata Wysocka. We all miss Kris. Eduard Zehnder Preface Goals The abstract theory developed in the current volume was initiated in a series of papers [54, 55, 56]. It generalizes differential geometry and nonlinear Fredholm theory to a class of spaces which are much more general than (Banach) manifolds. These spaces may have (locally) varying dimensions (finite or even infinite) and are described locally by retracts on scale Banach spaces, replacing the open sets of Banach spaces in the familiar local description of manifolds. A version of the theory also combines the differential geometric and functional analytic aspects with category theory. In this generality the theory also allows certain kinds of categories to be equipped with smooth structures and the development of a nonlinear Fredholm theory, i.e. Fredholm functors, in this context. The figure below illustrates some aspects of the spaces which will occur. vii viii Preface The top part shows a topological space obtained from an open three-ball and an open two-disk connected by two curves. It will turn out that it admits a new kind of smooth structure, called a (finite-dimensional) M-polyfold structure. The M-polyfold structures can have locally finite or locally infinite dimensions which change rapidly, but nevertheless they are in some sense smooth structures. In this context we can also have bundles and they can also change dimensions. These smooth bundles have smooth sections and if the dimension jumps are correlated the zero set might be something resembling a smooth classical manifold. This is shown in the bottom part of the figure, where we have a submanifold diffeomorphic to a circle as the zero set obtained from a section of the bundle over our space, which has a fiber-dimension one dimension less than the underlying local dimension. As it turns out, one can use the theory developed in this volume to give functional an- alytic treatments of nonlinear problems which show bubbling-off phenomena. The methods cover considerations about compactness, symmetry and transversality. This book is written as a reference volume for polyfold theory and the accompany- ing Fredholm theory. In Part I of this book, based on [54, 55], we develop the Fredholm theory in a class of spaces called M-polyfolds. These spaces can be viewed as a generalization of the notion of a manifold (finite or infinite-dimensional). In Part II M-polyfolds, based on [56], will be generalized to a class of spaces called ep-groupoids, which can be viewed as the polyfold generalization of etale´ proper Lie groupoids. This generalization is useful in dealing with problems having local symmetries, which is needed in more advanced applications. In Part III we develop, following [55, 56, 57, 58], the Fredholm theory in ep- groupoids. Transversality and symmetry are antagonistic concepts, since perturb- ing a problem to achieve transversality might not be possible when we also want to preserve the symmetries. Although it seems hopeless in many cases to achieve transversality, there is a version of transversality theory over the rational numbers, which is based on a perturbation theory using set-valued perturbations. Such an idea was introduced by Fukaya and Ono, [26], in their Kuranishi framework for the Arnold conjectures. The use of set-valued operators in nonlinear analysis is older, and a description can be found in [109]. It seems that the latter was never used to develop a transversality theory in the context of symmetries. We merge these ideas into a convenient formalism which was in a very special case suggested in [11]. In some sense the transversality theory in a symmetric setting locally replaces the equivariant problem by a symmetric weighted family of problems. One of the by- products of our considerations is a version of Fredholm theory, generalizing what in classical terms would be a Fredholm theory on (Banach) orbifolds with boundary with corners. Preface ix In summary, Parts I–III describe a wide array of nonlinear functional analytic tools to study perturbation and transversality questions for a large class of so-called sc- Fredholm sections and Fredholm section functors in ep-groupoids. We note that ‘sc’ is short for ‘scale’. In Part IV, the whole theory is generalized even a step further to equip certain cat- egories (groupoidal categories) with smooth structures and to develop a theory of Fredholm functors. An application of such ideas to Gromov–Witten theory is out- lined in [49]. The language and results of the abstract polyfold theory turn out to be very useful in concrete applications. In fact, right from the beginning it was a declared goal to create a theory with a rich language and many internal results, which would allow the effective study of moduli spaces associated to families (or even infinitely many interacting families) of nonlinear elliptic problems on possibly varying domains, al- lowing for symmetries, bubbling-off and transversality issues. In applications, for example in symplectic geometry/topology, these analytic problems require massive constructions usually covering hundreds of pages. The reader can imagine the asso- ciated difficulties not only for the authors, but also for the referees. A natural idea to soften the burden is to make it possible to have modular constructions. Small pieces of checkable analysis and a theory which would allow such pieces to be combined and guarantee the properties of the newly constructed ‘cluster’. This would make it possible to recycle useful constructions and reuse them in new work, and a theory would guarantee that if the different pieces live up to certain ‘specifications’ the newly created object would have the required ‘functionality’. The reader should not be deceived by the length of the book. The language is rather compact and intuitive, and the length of this volume results from the fact that there are many internal results, where in applications only a few are needed. Since this is a new theory the authors spend an inordinate amount of time pushing it to its limits, since this is a way to spot possible issues and a way to find improvements of the core ideas, besides making sure that the theory covers many of the important applications. As it stands, this is now a reference volume and the follow-up projects will provide the reader with streamlined introductions to the theory designed and explained around important applications, described in the outlook below. The forthcoming project [20] by J. Fish and the first author will illustrate the utility of the theory by developing methods and tools for the constructions of concrete polyfolds. One of the very useful ideas for concrete polyfold constructions, as they occur in applications, is the imprinting method, see [21].
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