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 analytic 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 categories (groupoidal categories) with smooth structures and to develop a theory of Fredholm functors. An application of such ideas to Gromov–Witten theory is outlined 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, allowing 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 associated 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]. We shall also describe a so-called pre-Fredholm theory, a new concept, which will simplify in applications the verification that a given sc-smooth section is in fact an sc-Fredholm section. Several parts of [20] have been posted on the arXiv, see [21, 22]. x Preface Background

The theory developed in the current volume has its roots in the attempt to develop an appropriate framework for the construction of Symplectic Field Theory (SFT), a general theory of symplectic invariants, outlined in [15]. SFT constructs invariants of symplectic cobordisms by analyzing the structure of solutions of (nonlinear) elliptic partial differential equations of Cauchy–Riemann type for maps from Rie- mann surfaces into compact symplectic manifolds. These Riemann surfaces are not fixed and occur in families, and in fact, the partial differential equations are de- fined for maps on varying domains and often even map into varying targets. The treatment of the nonlinear partial differential equations is technically challenging due to bubbling-off phenomena. The arising issues lead to serious compactness and transversality problems, which have to be dealt with in the presence of symmetries. Yet, these phenomena are needed to derive the underlying rich algebraic structure of SFT. Therefore they have to be ‘embraced’ and accurately accounted for. The analytical difficulties in dealing with SFT become apparent in the series of papers [46, 51, 52, 59] culminating in the SFT-compactness paper [6]. Even if the polyfold theory and the associated sc-Fredholm theory were initially prompted by the desire to construct SFT, they have many potential applications in other areas of analysis and geometry. In the upcoming project [20] we shall illustrate many of the construction tools and ideas with examples useful for symplectic geometry. Recall that the various symplectic Floer theories, the Gromov–Witten theory, the theory of Fukaya categories, and SFT are concerned with moduli spaces. These come from solution spaces of nonlinear elliptic partial differential equations. The polyfold theory can be used to address issues which arise in their study. The series of videos [47], [1], [106] illustrate the motivation behind the polyfold theory. We also recommend [17] for the intuitive ideas involved in the polyfold theory.

Historical Context

In order to deal with the problems of moduli spaces other methods have also been proposed. The earlier papers are [26, 71, 99, 88]. The papers [26, 27] prompted the research [76, 77, 78, 79] and [107, 108]. Part of the work of Yang is concerned with studying the relationship between the polyfold and Kuranishi type approaches. The papers [28, 29, 30, 31, 32, 33, 34] are further developments. Some of the work of Fukaya, Ono, Ohta and Oh, as well as the work of McDuff and Wehrheim is the starting point for Pardon in [85, 86]. These approaches rely on finite-dimensional reductions. Besides [86], which is concerned with contact homology, there is another approach based on Kuranishi structures given in [4, 5]. There is also the recent paper by Ishikawa, [64], giving a construction of SFT us- Preface xi ing Kuranishi structures. The approach in [9, 10] is different and also based on an infinite-dimensional approach. We would also like to mention the construction of cylindrical contact homology in dimension three, see [63, 83]. The construction of SFT in the context of polyfold theory will be given by Joel Fish and the first author in [23]. It is being written as a graduate text. This work will provide a short introduction to abstract polyfold theory, i.e. some of the results in the current volume, and to some of the construction methods for polyfolds contained in [20]. The polyfold theory is very effective in ‘black-boxing’ the results which one needs from the current volume and [23]. Relying on some of the results in the current volume and [20], the construction of SFT will be carried out in [23] and will be of a reasonable length. Finally, we would like to point out that some of the ideas of Joyce are potentially useful in the polyfold context, particularly [66, 67]. The project [50] will describe a perturbation theory which is fine-tuned to deal with inductive constructions.

Outlook

The present book, referred to in the following as “Book A”, is a reference volume and it is the basis for the following planned projects. In fact, results from Book A will provide important input in the further developments. The polyfold language itself is easy to learn, which allows one to at least understand the statements of many of the results in Book A, even if at the beginning the reader will not able to provide proofs. However, what helps is that the familiar classical differential geometry can be alternatively formulated in a way which is close to polyfold theory. This will be part of some of the future projects. Polyfold Constructions: Tools, Techniques, and Functors,[20], by J. Fish and H. Hofer. This project is concerned with novel methods for constructions and is illustrated by examples needed for concrete problems using the polyfold technology. Many examples are specifically focused on applications in Symplectic Geometry/Topology. The references [21] and [22] are part of this project and will be integrated in upcoming publications. This project will emphasize modularity of geometric and analytical constructions. For example it will build some kind of ‘LEGO system’ where the pieces are analytic constructions and the theory says that if the ‘lego pieces’ are built according to certain rules one can plug them together to obtain new constructions with a certain functionality. One of the highlights is a pre-Fredholm theory, which considerably simplifies the Fredholm theory in applications and makes it possible to recycle established analytical work in this context. Another feature, which will xii Preface be highlighted, is that the polyfold language is very useful and efficient in ‘black- boxing’ results and describing the property of a black-box by a universal property. Inductive Perturbation Theory by H. Hofer and J. Solomon, [50]. This will gently introduce the reader to an abstract multivalued perturbation theory, which in some sense is a bread and butter issue dealing with moduli spaces in symplectic geometry. The theory is easy to explain but relies quite often on rather nontrivial results due to serious intrinsic issues. Construction of Symplectic Field Theory by J. Fish and H. Hofer, [23]. This is a planned graduate text which will construct Symplectic Field Theory using the polyfold language. It will introduce the polyfold language and it will use and explain the ideas of ‘black-boxed’ results from the previously mentioned projects, without requiring familiarity with the latter. It will therefore focus on the underlying geometry without giving up rigor. For example, it will explain in detail the perturbation schemes underlying the algebraic treatment.

Foundations

The set-theoretic and category-theoretic assumptions are standard and we refer the reader to the Stacks Project, [102]. In standard set theory the collection of all sets is a class and certain constructions are forbidden in order not to produce paradoxes. For that reason a possible way out is to introduce the notion of a universe. In our constructions we stay at a very benign level with no possibility of running afoul with the above mentioned issue. We follow Serge Gelfand and Yu Manin’s lead, [35] (see page 58), and make sure that “whenever necessary, we always assume that all the required hygiene regulations are obeyed”. Occasionally we shall use the axiom of choice. Recall that it says that a family of non-empty sets parameterized by sets has a section. Within the Zermelo–Fraenkel set theory with the axiom of choice (ZFC) this can be generalized by allowing a family of nonempty classes parameterized by a set. ZFC guarantees that this also has a section. Hence, given (Cλ )λ∈Λ , where Λ is a set and Cλ a nonempty class, a choice function is guaranteed. The intuitive idea behind this is that Cλ is deﬁned by a formula and that it is possible to add a formula to deﬁne a nonempty set contained in Cλ reducing the question to the standard case. Further, we shall sometimes use the axiom of global choice, i.e. the framework is nonempty sets parameterized by a class. Using global choice extends ZFC and is known as a ‘conservative extension’. This means that every provable statement in this larger theory, that can be stated in ZFC, can also be proved in ZFC, but the proof might be more involved. We shall also use some category theory. When we talk about functors we mean covariant functors unless otherwise stated. Preface xiii Remarks About the Index

Finally we would like to make a remark about the indexing. 1) We give a list of notations and frequently used symbols broken down with respect to the four parts of the book. 2) The index starts with a comprehensive list of symbols and gives the first page where a symbol occurs. 3) We provide a quite comprehensive list of the Corollaries, Lemmata, Propositions, Remarks, Theorems, and Definitions (in this order) and the page numbers where they occur. For each group, keywords, which start with a capital letter, are listed. 4) The final part is a general purpose index which overlaps with the other listings. The keywords occurring in the general purpose index are not capitalized, with the exception of names. Usually using the definition listing as well as the general purpose index relevant topics can be efficiently found.

Princeton and Zurich,¨ Helmut Hofer May 2021 Eduard Zehnder Acknowledgements

The initial research introducing polyfold theory, published in journals during the period 2007–2010, was partially supported by the NSF grant DMS-0603957 (first author) and NSF grant DMS-0906280 (second author). The second and the third author would like to thank the Institute for Advanced Study (IAS) in Princeton for its support and hospitality, and the second author would like to thank the Forschungsin- stitut fur¨ Mathematik (FIM) in Zurich for its support and hospitality. The authors would like to thank Joel Fish, Dusa McDuff, and Katrin Wehrheim for many useful and enlightening discussions. The first author would also like to thank Peter Albers, Kai Cieliebak, Ben Filippenko, Urs Frauenfelder, Michael Jemison, Jo Nel- son, Matthias Schwarz, Jake Solomon, Joa Weber, Dingyu Yang, Kai Zehmisch, Zhengyi Zhou and the participants of the workshop on polyfolds at Pajaro Dunes in August 2012 for their valuable feedback. We would especially like to thank Zhengyi Zhou for his big help during the revision of the manuscript. The comprehensive feedback based on his ‘heroic act’ of reading the whole manuscript in depth led to many valuable improvements. Many thanks also to the (many) referees of this manuscript. The first author also would like to thank the Courant Institute, where some of the original ideas were conceived and the work started. He also wants to specifically thank Jalal Shatah for bringing him to the Courant Institute. It could not have been done without the great scientific atmosphere and the opportunity to test ‘unbaked ideas’ on unsuspecting graduate students* of the ‘Courant caliber’ and receiving their feedback. The first author would like to thank his colleagues and the IAS for their full-hearted support. Many thanks go specifically to Robbert Dijkgraaf and Peter Sarnak.

* A ‘sample’ can be found here: https://www.genealogy.math.ndsu.nodak.edu/id.php?id=33913

xv xvi Acknowledgements

Finally we thank our “fellow travelers” Ivar Ekeland, Yasha Eliashberg, Dusa Mc- Duff, Dietmar Salamon and Claude Viterbo for many interesting discussions and comments and most of all for their friendship. We also remember fondly our friends Charles Conley, Andreas Floer and Jurgen¨ Moser, who have inﬂuenced our work and viewpoints. Contents

Part I Basic Theory in M-Polyfolds

1 Sc-Calculus ...... 5 1.1 Sc-Structures and Differentiability ...... 5 1.2 Properties of Sc-Differentiability ...... 11 1.3 The Chain Rule and Boundary Recognition ...... 13 1.4 Appendix ...... 15 1.4.1 Proof of the sc-Fredholm Stability Result ...... 15 1.4.2 Proof of the Chain Rule ...... 15 1.4.3 Proof Lemma 1.3.4 ...... 17 1.4.4 A Useful Example ...... 17

2 Retracts ...... 21 2.1 Retractions and Retracts ...... 21 2.2 Some Basic Properties of Sc-Smooth Retracts ...... 30 2.3 M-Polyfolds and Sub-M-Polyfolds ...... 33 2.4 The Degeneracy Index and Boundary Geometry ...... 44 2.5 Tame M-polyfolds ...... 52 2.6 Strong Bundles ...... 65 2.7 Appendix ...... 69 2.7.1 Proof of Proposition 2.1.2 ...... 69 2.7.2 Proof of Theorem 2.3.10 ...... 71 2.7.3 Proof of Proposition 2.3.15 ...... 76 2.7.4 Formalism Associated to a Boundary with Corners ...... 78

3 Basic Sc-Fredholm Theory ...... 89 3.1 Sc-Fredholm Sections ...... 89 3.2 Subsets with Tangent Structure ...... 104 3.3 Contraction Germs ...... 109 3.4 Stability of Basic Germs ...... 113 3.5 The Geometry of Basic Germs...... 117

xvii xviii Contents

3.6 Implicit Function Theorems ...... 134 3.7 Conjugation to a Basic Germ ...... 140 3.8 Appendix ...... 144 3.8.1 Proof of Proposition 3.1.25 ...... 144 3.8.2 Proof of Theorem 3.3.3 ...... 155 3.8.3 Proof of Lemma 3.5.10 ...... 158 3.8.4 Proof of Lemma 3.6.9 ...... 161 3.8.5 Diffeomorphisms Between Partial Quadrants ...... 163 3.8.6 An Implicit Function Theorem in Partial Quadrants ...... 169

4 Manifolds and Strong Retracts ...... 175 4.1 Characterization...... 175 4.2 Smooth Finite-Dimensional Submanifolds ...... 184 4.3 Families and an Application of Sard’s Theorem ...... 193 4.4 Sc-Differential Forms ...... 195 4.5 Appendix ...... 199 4.5.1 Deﬁnition of the Lie Bracket ...... 200 4.5.2 Proof of Proposition 4.4.5 ...... 201 4.5.3 Proof of the Poincare´ Lemma ...... 202

5 The Fredholm Package for M-Polyfolds ...... 205 5.1 Auxiliary Norms ...... 205 5.2 Compactness Results ...... 208 5.3 Perturbation Theory and Transversality ...... 216 5.4 Remark on Extensions of Sc+-Sections...... 228 5.5 Notes on Partitions of Unity and Bump Functions ...... 229

6 Orientations ...... 237 6.1 An Overview ...... 237 6.2 Linearizations of Sc-Fredholm Sections ...... 239 6.3 Linear Algebra and Conventions ...... 241 6.4 The Determinant of a Fredholm Operator...... 246 6.5 Classical Local Determinant Bundles ...... 258 6.6 Local Orientation Propagation ...... 262 6.7 Invariants ...... 276 6.8 Appendix ...... 279 6.8.1 Proof of Lemma 6.3.3 ...... 279 6.8.2 Proof of Proposition 6.4.11 ...... 281

Part II Ep-Groupoids

7 Ep-Groupoids ...... 287 7.1 Ep-Groupoids and Basic Properties ...... 287 7.2 Effective and Reduced Ep-Groupoids ...... 307 7.3 Topological Properties of Ep-Groupoids ...... 323 7.4 Regularity Assumptions and the Zhou Condition ...... 328 Contents xix

7.5 Paracompact Orbit Spaces ...... 340 7.6 Appendix ...... 344 7.6.1 The Natural Representation ...... 344 7.6.2 Sc-Smooth Partitions of Unity ...... 346 7.6.3 On the metrizability of TR ...... 348 7.6.4 Reduced Ep-Groupoids and Raising the Index ...... 348 7.6.5 Boundary Structure of Tame Ep-Groupoids ...... 351

8 Bundles and Covering Functors...... 357 8.1 The Tangent of an Ep-Groupoid ...... 357 8.2 Sc-Differential Forms on Ep-Groupoids ...... 372 8.3 Strong Bundles over Ep-Groupoids ...... 374 8.4 Topological and Regularity Properties of Strong Bundles ...... 379 8.5 Proper Covering Functors ...... 383 8.6 Appendix ...... 396 8.6.1 Local Structure of Proper Coverings ...... 396 8.6.2 The Structure of Strong Bundle Coverings ...... 399

9 Branched Ep+-Subgroupoids ...... 401 9.1 Basic Deﬁnitions ...... 401 9.2 The Tangent and Boundary of Θ ...... 408 9.3 Orientations ...... 422 9.4 The Geometry of Local Branching Structures ...... 429 9.5 Integration and Stokes ...... 436 9.6 Appendix ...... 452 9.6.1 Proof of Proposition 9.1.12 ...... 452 9.6.2 Questions about M+-Polyfolds ...... 457 9.6.3 Questions about Branched Objects ...... 458

10 Equivalences and Localization ...... 461 10.1 Equivalences ...... 461 10.2 The Weak Fibered Product ...... 477 10.3 Localization at the System of Equivalences ...... 483 10.4 Strong Bundles and Equivalences ...... 495 10.5 Localization in the Strong Bundle Case ...... 504 10.6 Appendix ...... 509 10.6.1 Proof of Theorem 10.3.8 ...... 509 10.6.2 Proof of Theorem 10.3.10 ...... 515 10.6.3 Another Useful Example ...... 519

11 Geometry up to Equivalences ...... 521 11.1 Ep-Groupoids and Equivalences ...... 521 11.2 Sc-Differential Forms and Equivalences ...... 526 11.3 Branched Ep+-Subgroupoids and Equivalences ...... 535 11.4 Equivalences and Integration ...... 542 11.5 Strong Bundles up to Equivalence ...... 546 xx Contents

11.6 Coverings and Equivalences ...... 549

Part III Fredholm Theory in Ep-Groupoids

12 Sc-Fredholm Sections ...... 569 12.1 Introduction and Basic Deﬁnition ...... 569 12.2 Auxiliary Norms ...... 570 12.3 Sc+-Section Functors ...... 579 12.4 Compactness Properties ...... 587 12.5 Orientation Bundles ...... 592

13 Sc+-Multisections ...... 599 13.1 Structure Result ...... 599 13.2 General Sc+-Multisections ...... 607 13.3 Structurable Sc+-Multisections ...... 613 13.4 Equivalences, Coverings and Structurability ...... 621 13.5 Constructions of Sc+-Multisections ...... 629

14 Extension of Sc+-Multisections ...... 637 14.1 Deﬁnitions and Main Result ...... 637 14.2 A Good Structured Version of Λ ...... 646 14.3 Extension of Correspondences ...... 649 14.4 Implicit Structures and Local Extension ...... 657 14.5 Extension of the Sc+-Multisection ...... 670 14.6 Remarks on Inductive Constructions ...... 680

15 Transversality and Invariants ...... 683 15.1 Natural Constructions ...... 683 15.2 Transversality and Local Solution Sets ...... 688 15.3 Perturbations ...... 691 15.4 Orientations and Invariants ...... 700

16 Polyfolds ...... 705 16.1 Polyfold Structures ...... 705 16.2 Tangent of a Polyfold ...... 715 16.3 Strong Polyfold Bundles ...... 721 16.4 Branched Finite-Dimensional Orbifolds ...... 729 16.5 Sc+-Multisections ...... 732 16.6 Fredholm Theory ...... 735

Part IV Fredholm Theory in Groupoidal Categories

17 Polyfold Theory for Categories ...... 743 17.1 Polyfold Structures and Categories ...... 744 17.2 Tangent Construction ...... 767 17.3 Subpolyfolds ...... 784 Contents xxi

17.4 Boundary Formalism for Tame Polyfolds ...... 787 17.5 Branched Ep+-Subcategories ...... 812 17.6 Sc-Differential Forms and Stokes ...... 817 17.7 Strong Bundle Structures ...... 824 17.8 Proper Covering Constructions ...... 828

18 Fredholm Theory in Polyfolds ...... 843 18.1 Basic Concepts ...... 844 18.2 Compactness Properties ...... 849 18.3 Sc+-Multisection Functors ...... 851 18.4 Constructions and Extensions ...... 853 18.5 Orientations ...... 856 18.6 Perturbation Theory ...... 860

19 General Constructions ...... 863 19.1 The Basic Constructions ...... 863 19.2 The Natural Topology T for |S | ...... 869 19.3 The Natural Topology for M(Ψ,Ψ 0) ...... 872 19.4 Metrizability Criteria ...... 879 19.5 The Polyfold Structure for (S ,T ) ...... 886 19.6 A Strong Bundle Version ...... 890 19.7 Covering Constructions ...... 905 19.8 Covering Constructions for Strong Bundles ...... 930

A Construction Cheatsheet ...... 959 A.1 Groupoidal Categories ...... 959 A.1.1 Uniformizer Construction for S ...... 959 A.1.2 Basic Construction ...... 960 A.1.3 Summary ...... 962 A.2 Strong Bundle Structures ...... 963 A.2.1 Uniformizer Construction for PS : E → S ...... 964 A.2.2 Basic Construction ...... 966 A.2.3 Summary ...... 967 A.3 Finite-to-One Covering Functors ...... 968 A.3.1 Uniformizer Construction for P : A → B ...... 968 A.3.2 Basic Construction ...... 969 A.3.3 Summary ...... 969 A.4 Coverings of Strong Bundles ...... 970 A.4.1 Uniformizers for Covering Squares E ...... 971 A.4.2 Basic Construction ...... 971 A.4.3 Summary ...... 972

References ...... 973

Notation and List of Frequently Occurring Symbols ...... 979 xxii Contents

Index ...... 987