Association for Women in Mathematics Series

Gail Letzter · Kristin LauterErin Chambers · Nancy FlournoyJulia Elisenda Grigsby · Carla Martin Kathleen Ryan · Konstantina Trivisa Editors

Advances in the Mathematical SciencesResearch from the 2015 Association for Women in Mathematics Symposium

Association for Women in Mathematics Series

Volume 6

Series editor

Kristin Lauter, Redmond, WA, USA

Focusing on the groundbreaking work of women in mathematics past, present, andfuture, Springer’s Association for Women in Mathematics Series presents the latestresearch and proceedings of conferences worldwide organized by the Associationfor Women in Mathematics (AWM). All works are peer-reviewed to meet thehighest standards of scientific literature, while presenting topics at the cutting edgeof pure and applied mathematics. Since its inception in 1971, The Association forWomen in Mathematics has been a non-profit organization designed to helpencourage women and girls to study and pursue active careers in mathematics andthe mathematical sciences and to promote equal opportunity and equal treatment ofwomen and girls in the mathematical sciences. Currently, the organizationrepresents more than 3000 members and 200 institutions constituting a broadspectrum of the mathematical community in the United States and around theworld.

More information about this series at http://www.springer.com/series/13764

http://www.springer.com/series/13764

Gail LetzterEditor-in-Chief

Kristin Lauter • Erin ChambersNancy Flournoy • Julia Elisenda GrigsbyCarla Martin • Kathleen RyanKonstantina TrivisaEditors

Advances in theMathematical SciencesResearch from the 2015 Associationfor Women in Mathematics Symposium

123

Editor-in-ChiefGail LetzterNational Security AgencyFort Meade, MDUSA

EditorsKristin LauterMicrosoft ResearchRedmond, WAUSA

Erin ChambersDepartment of Mathematics and ComputerScience

Saint Louis UniversitySt. Louis, MOUSA

Nancy FlournoyDepartment of StatisticsUniversity of Missouri-ColumbiaColumbia, MOUSA

Julia Elisenda GrigsbyDepartment of MathematicsBoston CollegeChestnut Hill, MAUSA

Carla MartinNational Security AgencyFort Meade, MDUSA

Kathleen RyanDepartment of Mathematics and ComputerScience

DeSales UniversityCenter Valley, PAUSA

Konstantina TrivisaDepartment of MathematicsUniversity of MarylandCollege Park, MDUSA

ISSN 2364-5733 ISSN 2364-5741 (electronic)Association for Women in Mathematics SeriesISBN 978-3-319-34137-8 ISBN 978-3-319-34139-2 (eBook)DOI 10.1007/978-3-319-34139-2

Library of Congress Control Number: 2016940319

© Springer International Publishing Switzerland 2016This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof 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 transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe 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 thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made.

Gail Letzter is the Editor-in-Chief for this volume.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AG Switzerland

Dedicated to our fellow editorCarla Dee Martin1972–2015

Preface

In 2011, the Association for Women in Mathematics marked its 40th anniversarywith the organization’s first major research symposium “40 Years and Counting:AWM’s Celebration of Women in Mathematics.” The extensive scientific programwas a tribute to the depth and breadth of the technical contributions made by today'swomen mathematicians. Inspired by its success, AWM’s leadership decided tomake major research gatherings a regular event, thus beginning a new tradition ofAWM biennial research symposia. The second conference was held at Santa ClaraUniversity in 2013. This volume is based on the third of the series, the 2015 AWMResearch Symposium held at the University of Maryland.

Participants of the 2015 AWM research symposium

vii

Table 1 2015 AWM research symposium: the plenary lectures

Ingrid Daubechies Applied Mathematics helping Art Historians and Conservators: DigitalCradle Removal

Maria Chudnovsky Coloring square-free perfect graphs

Jill Pipher Dyadic Analysis: From Fourier to Haar to Wavelets, and back

Katrin Wehrheim String diagrams in Topology, Geometry and Analysis

The 2015 AWM Research Symposium attracted over 330 participants andshowcased research of women mathematicians from academia, industry, and gov-ernment. With four exciting plenary talks (Table 1) and fourteen special sessions(Table 2) on topics from across the mathematical spectrum, this conference hadsomething for everyone. There was an air of excitement throughout the meeting asparticipants learned about all sorts of new advances and novel applications to otherfields such as art history, biology, and computer science. Plenary lectures andspecial sessions had packed audiences; attendees wandered through the exhibit hallor spent time at poster sessions featuring results of recent PhDs in between talks.This volume of the new AWM Springer series commemorates the conferencethrough a set of invited peer-reviewed contributions from special session speakersand special session organizers as well as one of the plenary speakers (KatrinWehrheim). The volume, which includes papers based on talks from eight out of thefourteen special sessions (those starred in Table 2), reflects the broad range ofmathematics presented at the conference.

Table 2 2015 AWM research symposium: the special sessions

Research from the “Cutting EDGE”*

Many facets of Probability*

Topics in Computational Topology and Geometry

Low-dimensional Topology*

Number Theory

Mathematics at Government Labs and Centers*

Symplectic Topology/Geometry

Recent mathematical advancements empowering signal/image processing

Algebraic Geometry

Statistics*

PDEs in Continuum Mechanics

Discrete Math (and Theoretical Computer Science)*

Mathematical Biology*

Sharing the Joy: Engaging Undergraduate Students in Mathematics*

viii Preface

AWM presidential award winners: Rhonda Hughes (left), Sylvia Bozeman (standing up) with firstAWM president Mary Gray (right)

Speakers and organizers for the special session “Research from the ‘Cutting EDGE’”, from left toright: Kathleen Ryan, Carmen Wright, Ami Radunskaya, Candice Price, Sarah Bryant, RhondaHughes, Amy Buchmann, and Ulrica Wilson

Preface ix

A highlight of the symposium was the presentation of the first AWMPresidential Award to the co-founders Sylvia Bozeman and Rhonda Hughes of theEnhancing Diversity in Graduate Education (EDGE) program. This highly suc-cessful venture, established in 1998, was designed to increase the number ofwomen and minorities who complete graduate school in the mathematical sciences.Participants attend an initial intensive summer program and receive extensivementoring that continues through their first year of graduate school and beyond. Todate, a total of 67 EDGE participants have obtained their doctorates in the math-ematical sciences. The AWM was also honored to host a special session “Researchfrom the ‘Cutting EDGE’” at the 2015 conference; all eight speakers in this sessionattended the EDGE program and have finished or are in the process of finishingtheir PhDs. We are pleased to have a significant EDGE representation in thisvolume with three papers based on talks by EDGE special session speakers CandicePrice, Sarah Bryant, and Raegan Higgins, and a fourth paper whose authors includethe EDGE session speaker Amy Buchmann and the current EDGE co-director AmiRadunskaya. Our editorial group benefited greatly by having EDGE graduate andspecial session presenter Kathleen Ryan, as part of the team.

In addition to the special session and plenary talks, the AWM symposium fea-tured keynote speaker Shirley Malcolm, Head of Education and Human ResourcesPrograms at the American Association for the Advancement of Science and authorof “The Double Bind: The Price of Being a Minority Women in Science.” Along-time advocate for underrepresented groups in STEM, Dr. Malcolm spokeeloquently about the challenges faced by women and minorities in the mathematicalsciences. Her moving speech affirmed the importance of the AWM as well asinitiatives such as the EDGE program in providing the support women need as theypursue mathematical careers. More information about her presentation as well asother symposium events can be found on the symposium blog:

https://sites.google.com/site/awmmath/home/announcements/awmsymposiumblog

For the full symposium schedule with a list of all the talks, poster sessions,presenters, and other activities, please follow the program link on the conference’swebsite:

https://sites.google.com/site/awmmath/home/awm-research-symposium-2015

x Preface

https://sites.google.com/site/awmmath/home/announcements/awmsymposiumbloghttps://sites.google.com/site/awmmath/home/awm-research-symposium-2015

This volume opens with a part entitled From the Plenary Talks and consists of asurvey of Floer field theory written by Katrin Wehrheim, one of the symposiumplenary speakers. After that, the papers are grouped together in parts based onsubject areas. For the most part, the parts correspond to special sessions with somemodifications. Each paper from the special session “Research from the ‘CuttingEDGE’”, as well as a contribution from the “Mathematics at Government Labs andCenters” and the contributions by special session organizers have been placed in apart based on its content.

Part II, Low-Dimensional Topology, consists of three papers based on talks in thespecial session of the same name and a contribution by the special session organizerElisenda Grigsby. Part III, Mathematical Biology, contains three papers corre-sponding to presentations in the “Mathematical Biology” special session and twopapers written by participants from “Research from the ‘Cutting EDGE’” (one byCandice Price and the other by Amy Buchmann, Ami Radunskaya and others).Part IV, Probability and Stochastic Processes, is a combination of a paperco-written by Kavita Ramanan, a speaker from the “Many facets of probability”special session and a paper by “Research from the ‘Cutting EDGE’” speaker SarahBryant. Part V, Statistics, consists of three papers based on talks from the“Statistics” special session. This is followed by a Differential Equations part

Organizer Talitha Washington (bottom left), keynote speaker Shirley Malcolm (bottom middle) andAWM presidential award winner Sylvia Bozeman (top left) with AWM presidents Kristin Lauter(top middle), Jill Pipher (top right), Ruth Charney (bottom right)

Preface xi

(Part VI) that has been created for the paper by Raegan Higgins, a speaker from the“Research from the ‘Cutting EDGE’” session. The volume includes two papersfrom the special session “Sharing the Joy: Engaging Undergraduate Students inMathematics” in a part of the same name (Part VII). The last part (Part VIII),Discrete Mathematics and Computer Science, contains three papers, each related toa different special session: a paper by Shari Wiley who talked in the special sessionof the same name, a paper co-authored by Carol Woodward who spoke in the“Mathematics at Government Labs and Centers” special session, and a paperco-authored by Erin Chambers, who organized the special session “Topics inComputational Topology and Geometry”.

The editors would like to express their gratitude to the National ScienceFoundation and the National Security Agency for funding the symposium throughtheir respective grant programs and to the AWM Research Symposium's corporatesponsors Microsoft Research, Google, Springer, Elsevier, Wolfram, and INTECHfor their generous financial support. We also give special thanks to the University ofMaryland, College Park for hosting the event and providing us with appropriaterooms and other infrastructure necessary to make the meeting go smoothly. Finally,we would like to acknowledge the organizers of the conference (Ruth Charney,Shelly Harvey, Kristin Lauter, Gail Letzter, Magnhild Lien, Konstantina Trivisa,

Symposium organizers with plenary speaker and past AWM president Jill Pipher from left to right:Konstantina Trivisa, Magnhild Lien, Shelly Harvey, Talitha Washington, Kristin Lauter, JillPipher, Ruth Charney (missing: Gail Letzter)

xii Preface

Talitha Washington), Jay Popham from the Springer staff, the referees whoreviewed the papers, and the AWM staff including AWM Managing DirectorJennifer Lewis whose hard work made both the symposium and this volumepossible.

January 2016 Gail LetzterKristin LauterErin Chambers

Nancy FlournoyJulia Elisenda Grigsby

Carla MartinKathleen Ryan

Konstantina Trivisa

Preface xiii

Contents

Part I From the Plenary Talks

Floer Field Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Katrin Wehrheim

Part II Low Dimensional Topology

An Elementary Fact About Unlinked Braid Closures . . . . . . . . . . . . . . 93J. Elisenda Grigsby and Stephan M. Wehrli

Symmetric Unions Without Cosmetic Crossing Changes . . . . . . . . . . . . 103Allison H. Moore

The Total Thurston–Bennequin Number of Completeand Complete Bipartite Legendrian Graphs . . . . . . . . . . . . . . . . . . . . . 117Danielle O’Donnol and Elena Pavelescu

Coverings of Open Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Tetsuya Ito and Keiko Kawamuro

Part III Mathematical Biology

Understanding Locomotor Rhythm in the LampreyCentral Pattern Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Nicole Massarelli, Allan Yau, Kathleen Hoffman, Tim Kiemeland Eric Tytell

Applications of Knot Theory: Using Knot Theory to UnravelBiochemistry Mysteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173Candice Reneé Price

Metapopulation and Non-proportional VaccinationModels Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Mayteé Cruz-Aponte

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Controlling a Cockroach Infestation . . . . . . . . . . . . . . . . . . . . . . . . . . 209Hannah Albert, Amy Buchmann, Laurel Ohm, Ami Radunskayaand Ellen Swanson

The Impact of Violence Interruption on the Diffusionof Violence: A Mathematical Modeling Approach. . . . . . . . . . . . . . . . . 225Shari A. Wiley, Michael Z. Levy and Charles C. Branas

Part IV Probability and Stochastic Processes

Cramér’s Theorem is Atypical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253Nina Gantert, Steven Soojin Kim and Kavita Ramanan

Counting and Partition Function Asymptotics for SubordinateKilled Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271Sarah Bryant

Part V Statistics

A Statistical Change-Point Analysis Approach for Modelingthe Ratio of Next Generation Sequencing Reads . . . . . . . . . . . . . . . . . . 283Jie Chen and Hua Li

A Center-Level Approach to Estimating the Effect of CenterCharacteristics on Center Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 301Jennifer Le-Rademacher

False Discovery Rate Based on Extreme Valuesin High Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323Junyong Park, DoHwan Park and J. Wade Davis

Part VI Differential Equations

Asymptotic and Oscillatory Behavior of Dynamic Equationson Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341Raegan Higgins

Part VII Sharing the Joy: Engaging Undergraduate Studentsin Mathematics

Using Applications to Motivate the Learningof Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359Karen M. Bliss and Jessica M. Libertini

What Is a Good Question? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371Brigitte Servatius

xvi Contents

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Part VIII Discrete Math and Theoretical Computer Science

Information Measures of Frequency Distributionswith an Application to Labeled Graphs . . . . . . . . . . . . . . . . . . . . . . . . 379Cliff Joslyn and Emilie Purvine

Integrating and Sampling Cuts in Bounded Treewidth Graphs . . . . . . . 401Ivona Bezáková, Erin W. Chambers and Kyle Fox

Considerations on the Implementation and Use of AndersonAcceleration on Distributed Memory and GPU-based ParallelComputers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417John Loffeld and Carol S. Woodward

Contents xvii

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Part IFrom the Plenary Talks

Floer Field Philosophy

Katrin Wehrheim

Abstract Floer field theory is a construction principle for example, 3-manifoldinvariants via decomposition in a bordism category and a functor to the symplec-tic category, and is conjectured to have natural four-dimensional extensions. Thissurvey provides an introduction to the categorical language for the construction andextension principles and provides the basic intuition for two gauge theoretic exam-ples which conceptually frame Atiyah–Floer type conjectures in Donaldson theoryas well as the relations of Heegaard Floer homology to Seiberg–Witten theory.

Keywords Bordism bicategories · Floer theory · Topological field theory ·Quilted2-categories · Quilted Atiyah–Floer conjecturesMathematics Subject Classification 57R56 · 53R57 · 58D29 · 81T45

1 Introduction

In the 1980s, the areas of low dimensional topology and symplectic geometry bothsaw important progress arise from the studyofmoduli spaces of solutions of nonlinearelliptic PDEs. In the study of smooth 4-manifolds, Donaldson [14] introduced theuse of ASD Yang–Mills instantons,1 which were soon followed by Seiberg–Witten

1A smooth four manifold can be thought of as a curved four-dimensional space-time. ASD(anti-self-dual) instantons in this space-time satisfy a reduction of Maxwell’s equations for theelectromagnetic potential in vacuum, which has an infinite dimensional gauge symmetry.

K. Wehrheim (B)Department of Mathematics, University of California, Berkeley, CA 94720, USAe-mail: katrin@math.berkeley.edu

© Springer International Publishing Switzerland 2016G. Letzter et al. (eds.), Advances in the Mathematical Sciences, Associationfor Women in Mathematics Series 6, DOI 10.1007/978-3-319-34139-2_1

3

4 K. Wehrheim

equations [51]—another gauge theoretic2 PDE. In the study of symplectic manifolds,Gromov [28] introduced pseudoholomorphic curves3 In both subjects Floer [22, 23]then introduced a new approach to infinite dimensional Morse theory4 based on therespective PDEs. This sparked the construction of various algebraic structures—such as the Fukaya A∞-category of a symplectic manifold [63], a Chern–Simonsfield theory for 3-manifolds and 4-cobordisms [16], and analogous Seiberg–Witten3-manifold invariants [33]—from these and related PDEs, which encode significanttopological information on the underlying manifolds. Chern–Simons field theory inparticular comprises theDonaldson invariants of 4-manifolds, togetherwith algebraictools to calculate these by decomposing a closed 4-manifold into 4-manifolds whosecommon boundary is given by a three-dimensional submanifold. This strategy ofdecomposition into simpler pieces inspired the new topic of “topological (quantum)field theory” [2, 42, 62, 85], in which the properties of such theories are describedand studied.

In trying to extend the field-theoretic strategy to the decomposition of 3-manifoldsalong two-dimensional submanifolds, Floer and Atiyah [3] realized a connectionto symplectic geometry: A degeneration of the ASD Yang–Mills equation on a 4-manifold with two-dimensional fibers Σ yields the Cauchy–Riemann equation on a(singular) symplectic manifoldMΣ given by the flat connections onΣ modulo gaugesymmetries. Alongwith this, 3-dimensional handlebodiesH with boundary ∂H = Σinduce Lagrangian submanifolds LH ⊂ MΣ given by the boundary restrictions offlat connections on H. Now Lagrangians5 are the most fundamental topologicalobject studied in symplectic geometry. They are often studied by means of the Floerhomology HF(L0,L1) of pairs of Lagrangians, which arises from a complex that isgenerated by the intersection points L0 ∩ L1 and whose homology is invariant underHamiltonian deformations of the Lagrangians. For the pair LH0 ,LH1 ⊂ MΣ arisingfrom the splitting Y = H0 ∪Σ H1 of a 3-manifold into two handlebodies H0,H1,these generators are naturally identified with the generators of the instanton Floerhomology HFinst(Y), given by flat connections on Y modulo gauge symmetries.(Indeed, restricting the latter to Σ ⊂ Y yields a flat connection on Σ that extends toboth H0 and H1—in other words, an intersection point of LH0 with LH1 .)

2In mathematics, “gauge theory” refers to the study of connections on principal bundles, where“gauge symmetries” arise from the pullback action by bundle isomorphisms; see, e.g., [67, Appen-dix A].3Symplectic manifolds can be thought of as the configuration spaces of classical mechanical sys-tems, with the position-momentum pairing providing the symplectic structure as well as a class ofalmost complex structures J . Pseudoholomorphic curves can then be thought of as two-dimensionalsurfaces in a 2n-dimensional symplectic ambient space, which can be locally described as theimage of 2n real-valued functions u of a complex variable z = x + iy that satisfy a generalizedCauchy–Riemann equation ∂xu = J(u) ∂yu.4Morse theory captures the topological shape of a space by studying critical points of a functionand flow lines of its gradient vector field. In finite dimensions it yields a complex whose homologyis independent of choices (e.g., of function) and in fact equals the singular homology of the space.5Throughout this paper, the term “Lagrangian” refers to a half-dimensional isotropic submanifoldof a symplectic manifold—corresponding to fixing the integrals of motion, e.g. the momentums.

Floer Field Philosophy 5

These observations inspired the Atiyah–Floer conjecture

HFinst(H0 ∪Σ H1) � HF(LH0 ,LH1),

which asserts an equivalence between the differentials on the Floer complexes—arising from ASD instantons on R× Y and pseudoholomorphic maps R× [0, 1] →MΣ with boundary values on LH0 ,LH1 , respectively. While this conjecture is not welldefined due to singularities in the symplectic manifoldsMΣ , and the proof of a well-defined version by Dostoglou–Salamon [18] required hard adiabatic limit analysis,the underlying ideas sparked inquiry into relationships between low-dimensionaltopology and symplectic geometry. At this point, the two fields are at least astightly intertwined as algebraic and symplectic geometry (via mirror symmetry),most notably through the Heegaard–Floer invariants for 3- and 4-manifolds (aswell as knots and links), which were discovered by Ozsvath–Szabo [52] by followingthe line of argument of Atiyah and Floer in the case of Seiberg–Witten theory. Inboth cases, the concept for the construction of an invariant of 3-manifolds Y is thesame:

1. Split Y = H0 ∪Σ H1 along a surface Σ into two handlebodies Hi with ∂Hi = Σ .2. Represent the dividing surface Σ by a symplectic manifold MΣ and the two

handle bodies by Lagrangians LHi ⊂ MΣ arising from dimensional reductions ofa gauge theory which is known to yield topological invariants.

3. Take the Lagrangian Floer homology HF(LH0 ,LH1) of the pair of Lagrangians.4. Argue that different splittings yield isomorphic Floer homology groups—due to

an isomorphism to a gauge theoretic invariant of Y or by direct symplectic isomor-phisms HF(LH0 ,LH1) � HF(L˜H0 ,L˜H1) for different splittings Y = ˜H0 ∪˜Σ ˜H1.

Floer field theory is an extension of this approach to more general decompositionsof 3-manifolds, by phrasing Step 4 above as the existence of a functor betweentopological and symplectic categories that extends the association

Σ �→ MΣ, H �→ LH , ∂H = Σ ⇒ LH ⊂ MΣ.

It gives a conceptual explanation for Step 4 invariance proofs such as [52] whichbypass a comparison to the gauge theory by directly relating the Floer homologies ofLagrangians LH0 ,LH1 ⊂ MΣ and L˜H0 ,L˜H1 ⊂ M˜Σ . Since these can arise from surfacesΣ � ˜Σ of different genus, the comparison between pseudoholomorphic curves insymplectic manifolds MΣ � M˜Σ of different dimension must crucially use the factthat the Lagrangian boundary conditions encode different splittings of the same3-manifold. Floer field theory encodes this as an isomorphism between algebraiccompositions of the Lagrangians, which in turn yields isomorphic Floer homologies(a strategy that we elaborate on in Sects. 2.4 and 3.5),

H0 ∪Σ H1 � ˜H0 ∪˜Σ ˜H1 =⇒ LH0#LH1 ∼ L˜H0#L˜H1=⇒ HF(LH0 ,LH1) � HF(L˜H0 ,L˜H1).

6 K. Wehrheim

Floer field theory, in particular its key isomorphism of Floer homologies [78] hintedat above, was discovered by the author and Woodward [80, 81] when attempting toformulate well-defined versions of the Atiyah–Floer conjecture. While the isomor-phism of Floer homologies in [78] is usually formulated in terms of strip-shrinkingin a new notion of quilted Floer homology [75], it can be expressed purely in termsof Floer homologies of pairs of Lagrangians, which lie in different products of sym-plectic manifolds. In this language, strip shrinking then is a degeneration of theCauchy–Riemann operator to a limit in which the curves in one factor of the prod-uct of symplectic manifolds become trivial. (For more details, see Sect. 3.5.) Thisrelation between pseudoholomorphic curves in different symplectic manifolds thenprovides a purely symplectic analogue of the adiabatic limit in [18], which relatesASD instantons to pseudoholomorphic curves.

The value of Floer field theory to 3-manifold topology is mostly of philosoph-ical nature—giving a conceptual understanding for invariance proofs and a generalconstruction principle for 3-manifold invariants (and similarly for knots and links),which has since been applied in a variety of contexts [5, 36, 44, 56, 80, 81]. Onemain purpose of this paper and the content of Sect. 2 is to explain this philosophy andcast the construction principle into rigorous mathematical terms. For that purposeSect. 2.1 gives brief expositions of the notions of categories and functors, the cate-gory Cat, and bordism categories Bord+1. After introducing the symplectic categorySymp in Sect. 2.2, the categorical structure in symplectic geometry that can be relatedto low dimensional topology, in Sect. 2.3 we cast the concept of Cerf decompositions(cutting manifolds into simple cobordisms) into abstract categorical terms that applyequally to bordism categories and our construction of the symplectic category. Wethen exploit the existence of Cerf decompositions in Bord+1 and Symp together witha Yoneda functor Symp→ Cat (see Lemma 3.5.6) to formulate a general construc-tion principle for Floer field theories. This notion of Floer field theory is defined inSect. 2.4 as a functor Bord+1 → Cat that factors through Symp. This construction isexemplified in Sect. 2.5 by naive versions of two gauge theoretic examples related toYang–Mills–Donaldson resp. Seiberg–Witten theory in dimensions 2+ 1. Finally,Sect. 2.6 explains how this yields conjectural symplectic versions of the gauge the-oretic 3-manifold invariants, as predicted by Atiyah and Floer.

The second purpose of this paper and content of Sect. 3 is to lay some foundationsfor an extension of Floer field theories to dimension 4. Our goal here is to provide arigorous exposition of the algebraic language in which this extension principle can beformulated—at a level of sophistication that is easily accessible to geometers whilesufficient for applications. Thus, we review in detail the notions of 2-categories andbicategories in Sect. 3.1, including the 2-categoryCat of (categories, functors, naturaltransformations), explicitly construct a bordism bicategory Bor2+1+1 in Sect. 3.2, andsummarize notions and Yoneda constructions of 2-functors between these highercategories in Sect. 3.3. Moreover, Sect. 3.5 outlines the construction of symplectic2-categories, based on abstract categorical notions of adjoints and quilt diagrams thatwe develop in Sect. 3.4. The latter transfers notions of adjunction and spherical stringdiagrams from monoidal categories into settings without natural monoidal structure.

Floer Field Philosophy 7

This provides sufficient language to at least advertise an extension principle whichwe further discuss in [74]:

Any Floer field theory Bor2+1 → Symp→ Cat which satisfies a quilted naturalityaxiom has a natural extension to a 2-functor Bor2+1+1 → Symp→ Cat.

This says in particular that any 3-manifold invariant which is constructed along thelines of theAtiyah–Floer conjecture naturally induces a 4-manifold invariant.While itdoes seem surprising, such a result could bemotivated from the point of viewof gaugetheory, since the Atiyah–Floer conjecture and Heegaard–Floer theory were inspiredby dimensional reductions of 3+ 1 field theories Bor3+1 → C . It also can be viewedas a pedestrian version of the cobordism hypothesis [42],6 saying that a functorBor2+1+ε → Symp (where the ε stands for compatibility with diffeomorphisms of3-manifolds) has a canonical extension Bor2+1+1 → Symp.

Finally, the extensions of Floer field theory to dimension 4 are again expected tobe isomorphic to the associated gauge theoretic 4-manifold invariants, in a way thatis compatible with decomposition into 3- and 2-manifolds. We phrase these expecta-tions in Sect. 3.6 as quilted Atiyah–Floer conjectures, which identify field theoriesBor2+1+1 → C . The last section Sect. 3.6 also demonstrates the construction princi-ple for 2-categories via associating elliptic PDEs to quilt diagrams in several moregauge theoretic examples, which provide not only the proper context for stating allthe generalized Atiyah–Floer conjectures, but also yield conceptually clear contextsfor the various approaches to their proofs.

While this 2+ 1 field theoretic circle of ideas has been and used in various pub-lications, its rigorous abstract formulation in terms of a notion of “category withCerf decompositions” is new to the best of the author’s knowledge. Similarly, thenotions of bordism bicategories, the symplectic 2-category, generalized string dia-grams, and field theoretic proofs of Floer homology isomorphisms have been knownand (at least implicitly) used in similar contexts, but are here cast into a new conceptof “quilted bicategories” which will be central to the extension principle—both ofwhich seem significantly beyond the known circle of ideas. Finally, note that Floerfield theory should not be confused with the symplectic field theory (SFT) introducedby [21], in which another symplectic category—given by contact-typemanifolds andsymplectic cobordisms—is the domain, not the target of a functor.

We end this introduction by a more detailed explanation of the notion of an“invariant” as it applies to the study of topological or smooth compact manifolds,and a very brief introduction to the resulting classification of manifolds.

6Lurie’s constructions involve the canonical extension of a functor Bor0+1+...+ε → C toBor0+1+...+1 → C . However, this requires an extension of the field theory to dimensions 1 and0 (which we do not even have ideas for) as well as a monoidal structure on the target category C(which is lacking at present because the gauge theoretic functors are well defined only on the con-nected bordism category). On the other hand, we have other categorical structures at our disposal,which we formalize in Sect. 3.4 as the notion of a quilted 2-category (akin to a spherical 2-categoryas described in [43]). In that language, the diagram of a Morse 2-function as in [26] expresses a4-manifold as a quilt diagram in the bordism bicategory Bor2+1+1. Now the key idea for [74] isthat a functor Bor2+1 → Symp translates the diagram of a 4-manifold into a quilt diagram in thesymplectic 2-category in Sect. 3.5, where it is reinterpreted in terms of pseudoholomorphic curves.

8 K. Wehrheim

1.1 A Brief Introduction to Invariants of Manifolds

In order to classifymanifolds of a fixed dimension n up to diffeomorphism, onewouldideally like to have a complete invariant I : Mann → C . Here Mann is the categoryof n-manifolds and diffeomorphisms between them (see Example 2.1.4), and C is acategory such asC = Zwith trivialmorphisms or the categoryC = Gr of groups andhomomorphisms. Such I is an invariant if it is a functor (see Definition 2.1.3), sincethis guarantees that diffeomorphic manifolds are mapped to isomorphic objects of C(e.g., the same integer or isomorphic groups). In otherwords, functoriality guaranteesthat I induces a well-defined map |I| : |Mann| → |C | from diffeomorphism classesof manifolds to, e.g., Z or isomorphism classes of groups. Such an invariant letsus distinguish manifolds: If I(X), I(Y) are not isomorphic (i.e. |I|([X]) = |I|([Y ]))then X and Y cannot be diffeomorphic. Moreover, an invariant is called “complete”if an isomorphism I(X) � I(Y) implies the existence of a diffeomorphism X � Y ,i.e., |I|([X]) = |I|([Y ])⇒ [X] = [Y ].

Simple examples of invariants—when restricting Mann to compact orientedmanifolds—are the homology groupsHk : Mann → Groups for fixed k ∈ N0 or theirrank, i.e., the Betti numbers βk : Mann → Z. These are in fact topological—ratherthan smooth—invariants since homeomorphic—rather than just diffeomorphic—manifolds have isomorphic homology groups. The 0th Betti number β0 is completefor n = 0, 1 since it determines the number of connected components, and thereis only one compact, connected manifold of dimension 0 (the point) or 1 (the cir-cle). The first more nontrivial complete invariant—now also restricting to connectedmanifolds—is the first Betti number β1 : Man2 → Z, since compact, connected, ori-ented 2-manifolds are determined by their genus g = 12β1.

The fundamental group π1 : Mann → Gr is not strictly well-defined since itrequires the choice of a base point and thus is a functor on the category of man-ifolds with a marked point. However, for connected manifolds it still induces awell defined map |π1| : |Mann| → |Gr| from manifolds modulo diffeomorphism togroups modulo isomorphism, since change of base point induces an isomorphismof fundamental groups. Viewing this as an invariant, it is complete for n = 0, 1, 2.In dimension n = 3, completeness would mean that the isomorphism type of thefundamental group of a (compact, connected) 3-manifold determines the 3-manifoldup to diffeomorphism. This is true in the case of the trivial fundamental group: Bythe Poincaré conjecture, any simply connected 3-manifold “is the 3-sphere,” i.e., isdiffeomorphic to S3. It is also true for a large class (irreducible, nonspherical) of3-manifolds, but there are plenty of groups that can be represented by many non-diffeomorphic 3-manifolds, e.g., lens spaces and connected sums with them (see[1, 30] for surveys). Thus |π1| is a useful but incomplete invariant of closed, con-nected 3-manifolds. In dimension n ≥ 4 however, the classification question shouldbe posed for fixed |π1| since on the one hand any finitely presented group appears as

Floer Field Philosophy 9

the fundamental group of a closed, connected n-manifold, and on the other hand theclassification of finitely presented groups is a wide open problem itself.7

Moreover,while in dimensionn ≤ 3, the classifications up tohomeomorphismandup to diffeomorphism coincide (i.e., topological n-manifolds can be equipped with aunique smooth structure), these differ in dimensions n ≥ 4. In dimension n ≥ 5, bothclassifications can be undertakenwith the help of surgery theory introduced byMilnor[49]. In dimension 4, the classification of smooth 4-manifolds differs drastically fromthat of topological manifolds (see [61] for a survey). Here, gauge theory—startingwith thework ofDonaldson, and continuingwith Seiberg–Witten theory—is themainsource of invariants which can differentiate between different smooth structures onthe same topological manifold. In particular, Donaldson’s first results using ASDYang–Mills instantons [14] showed that a large number of topological manifolds(those with nondiagonalizable definite intersection form H2(X;Z)× H2(X;Z) →Z) in fact do not support any smooth structure.

2 Floer Field Theory

2.1 Categories and Functors

Definition 2.1.1 A category8 C consists of

• a set ObjC of objects,• for each pair x1, x2 ∈ ObjC a set of morphisms MorC (x1, x2),• for each triple x1, x2, x3 ∈ ObjC a composition map

MorC (x1, x2)×MorC (x2, x3)→ MorC (x1, x3), (f12, f23) �→ f12 ◦ f23,

such that

• composition is associative, i.e., we have (f12 ◦ f23) ◦ f34 = f12 ◦

(

f23 ◦ f34)

for anytriple of composable morphisms f12, f23, f34,

• composition has identities, i.e., for each x ∈ ObjC there exist a unique9 mor-phism idx ∈ MorC (x, x) such that idx ◦ f = f and g ◦ idx = g hold for any f ∈MorC (x, y) and g ∈ MorC (y, x).

7As a matter of curiosity: The group isomorphism problem—determining whether different finitegroup presentations define isomorphic groups—is undecidable, i.e., cannot be solved for all generalpresentations by an algorithm; see, e.g., [32].8Throughout, all categories are meant to be small, i.e., consist of sets of objects and morphisms.However, we will usually neglect to specify constructions in sufficient detail—e.g., require mani-folds to be submanifolds of some RN—in order to obtain sets.9Note that uniqueness follows immediately from the defining properties: If id′x is another identitymorphism then we have id′x = id′x ◦ idx = idx .

10 K. Wehrheim

The very first example of a category consists of objects which are sets (possiblywith extra structure such as a linear structure, metric, or smooth manifold structure),morphisms that are maps (preserving the extra structure), composition given by com-position of maps, and identities given by the identity maps. The following bordismcategories contain more general morphisms, which are more rigorously constructedin Remark 3.2.2.

Example 2.1.2 ThebordismcategoryBord+1 in dimensiond ≥ 0 is roughlydefinedas follows; see Fig. 1 for illustration.

• Objects are the closed, oriented, d-dimensional manifolds Σ .• Morphisms inMor(Σ1,Σ2) are the compact, oriented, (d + 1)-dimensional cobor-disms Y with identification of the boundary ∂Y � Σ−1 �Σ2, modulo diffeomor-phisms relative to the boundary.

• Composition of morphisms [Y12] ∈ Mor(Σ1,Σ2) and [Y23] ∈ Mor(Σ2,Σ3) isgiven by gluing [Y12] ◦ [Y23] := [Y12 ∪Σ2 Y23] ∈ Mor(Σ1,Σ3) along the commonboundary.

Here one needs to be careful to include the choice of boundary identifications inthe notion of morphism. Thus, a diffeomorphism φ : Σ0 → Σ1 can be cast as amorphism

Zφ :=[([0, 1] ×Σ1, {0} × φ, {1} × idΣ1

)] ∈ MorBord+1(Σ0,Σ1) (2.1.1)

Fig. 1 The bordism category Bor2+1

Floer Field Philosophy 11

given by the cobordism [0, 1] ×Σ1 with boundary identifications {0} × φ : Σ0 →{0} ×Σ1 and {1} × idΣ1 : Σ1 → {1} ×Σ1, as illustrated in Fig. 2. In that sense, theidentity morphisms idΣ = ZidΣ are given by the identity maps idΣ : Σ → Σ .

Equipping the composed morphism [Y12] ◦ [Y23] with a smooth structure more-over requires a choice of tubular neighborhoods of Σ2 in the gluing operation. Thegoodnews is that gluingwith respect to different choices yields diffeomorphic results,so that composition is well defined. The interesting news is that this ambiguity in thecomposition precludes the extension to a 2-category; see Example 3.2.3.

The notion of categories becomes most useful in the notion of a functor relatingtwo categories, since preservation of various structures (composition and identities)can be expressed efficiently as “functoriality.”

Definition 2.1.3 A functor F : C → D between two categories C ,D consists of• a map F : ObjC → ObjD between the sets of objects,• for each pair x1, x2 ∈ ObjC a map Fx1,x2 : MorC (x1, x2)→ MorD(F (x1),F (x2)),

that are compatible with identities and composition in the sense that

idF (x) = Fx,x(idx), Fx1,x3(f12 ◦ f23) = Fx1,x2(f12) ◦Fx2,x3(f23).

For example, the inclusion of diffeomorphisms into the bordismcategory inExam-ple 2.1.2 can be phrased as a functor as follows.

Example 2.1.4 Let Mand be the category consisting of the same objects as Bord+1,morphisms given by diffeomorphisms, and composition given by composition ofmaps. Then, there is a functor Mand → Bord+1 given by• the identity map between the sets of objects,• for each pair Σ0,Σ1 of diffeomorphic d-manifolds the map MorMand (Σ0,Σ1) →MorBord+1(Σ0,Σ1) that associates to a diffeomorphismφ the cobordismZφ definedin (2.1.1).

A more algebraic example of a category is given by categories and functors.

Example 2.1.5 The category of categories Cat consists of

• objects given by categories C ,• morphisms in MorCat(C1,C2) given by functors F12 : C1 → C2,• composition of morphisms given by composition of functors—i.e. composition ofthe maps on both object and morphism level.

12 K. Wehrheim

2.2 The Symplectic Category

The vision of Alan Weinstein [82] was to construct a symplectic category along thefollowing lines. (See [11, 47] for introductions to symplectic topology.)

• Objects are the symplectic manifolds M := (M, ω).• Morphisms are the Lagrangian submanifolds10 L ⊂ M−1 ×M2, where we denoteby M−1 := (M1,−ω1) the same manifold with reversed symplectic structure.• Composition of morphisms L12 ⊂ M−1 ×M2 and L23 ⊂ M−2 ×M3 is defined bythe geometric composition (where ΔM ⊂ M ×M− denotes the diagonal)

L12 ◦ L23 := prM−1 ×M3(

L12 × L23 ∩ M−1 ×ΔM2 ×M3) ⊂ M−1 ×M3.

This notion includes symplectomorphisms φ : M1 → M2, φ∗ω2 = ω1 as morphismsgiven by their graph gr(φ) = {(x, φ(x)) | x ∈ M1} ⊂ M−1 ×M2. Also, geometriccomposition is defined exactly so as to generalize the composition of maps. Thatis, we have gr(φ) ◦ gr(ψ) = gr(ψ ◦ φ). On the other hand, this more generalizednotion allows one to view pretty much all constructions in symplectic topology asmorphisms—for example, symplectic reduction from CP2 to CP1 is described bya Lagrangian 3-sphere Λ ⊂ (CP2)− × CP1; see [29, 75, 82] for details and moreexamples.

Unfortunately, geometric composition generally—even after allowing for pertur-bations (e.g., isotopy through Lagrangians)—at best yields immersed or multiplycovered Lagrangians.11 However, Floer homology12 is at most expected to be invari-ant under embedded geometric composition, i.e., when the intersection in

prM−1 ×M3 : L12 ×M2 L23 := L12 × L23 ∩ M−1 ×ΔM2 ×M3 −→ M−1 ×M3(2.2.1)

is transverse, and the projection is an embedding. In the linear case—for symplec-tic vector spaces and linear Lagrangian subspaces—this issue was resolved in [29]by observing that linear composition, even if not transverse, always yields anotherLagrangian subspace. In higher generality, and compatible with Floer homology, asymplectic category Symp = Symp#/∼ was constructed in [76] by the followinggeneral algebraic completion construction for a partially defined composition.

10Other terms for a Lagrangian, viewed as a morphism M1 → M2, are “Lagrangian relation” or“Lagrangian correspondence,” but we will largely avoid such distinctions in this paper.11Even the question of finding a Lagrangian L ⊂ CP2 with embedded composition L ◦Λ ⊂ CP1was open until the recent construction of a new Lagrangian embedding RP2 ↪→ L ⊂ CP2 in [12].12Floer homology is a central tool in symplectic topology introduced by Floer [23] in the 1980s,inspired by Gromov [28] and Witten [84]. It has been extended to a wealth of algebraic structuressuch as Fukaya categories; see, e.g., [63]. It can be thought of as theMorse homology of a symplecticaction functional on the space of paths connecting twoLagrangians, and recasts the ill posed gradientflow ODE as a Cauchy–Riemann PDE (whose solutions are pseudoholomorphic curves).

Floer Field Philosophy 13

Definition 2.2.1 The extended symplectic category Symp# is defined as follows.

• Objects are the symplectic manifolds (M, ω).• Simple morphisms L12 ∈ SMor(M1,M2) are the Lagrangian submanifolds L12 ⊂M−1 ×M2.• General morphisms L = (L01, . . . ,L(k−1)k) ∈ MorSymp# (M,N) are the compos-able chains of simple morphisms Lij ∈ SMor(Mi,Mj) between symplectic mani-folds M = M0,M1, . . . ,Mk = N .

• Composition of morphisms L = (L01, . . . ,L(k−1)k) ∈ MorSymp# (M,N) and L′ =(L′01, . . . ,L′(k′−1)k′) ∈ MorSymp# (N,P) is given by algebraic concatenationL#L′ :=(L01, . . . ,L(k−1)k,L′01, . . . ,L′(k′−1)k′).

For this to form a strict category, we include trivial chains ( ) ∈ Mor(M,M) of lengthk = 0 as identity morphisms.

While this is a welldefined category, its composition notion is not related togeometric composition yet. However, the following quotient construction ensuresthat composition is given by geometric composition when the result is embedded.

Definition 2.2.2 The symplectic category Symp is defined as follows.

• Objects are the symplectic manifolds (M, ω).• Morphisms are the equivalence classes inMorSymp(M,N) := MorSymp# (M,N)/∼.• Composition [L] ◦ [L′] := [L#L′] is induced by the composition in Symp#.Here, the composition-compatible equivalence relation ∼ on the morphism spacesof Symp# is obtained as follows.

• The subset of geometric compositionmovesComp ⊂ MorSymp# ×MorSymp# con-sists of all pairs

(

(L12,L23),L12 ◦ L23)

and(

L12 ◦ L23, (L12,L23))

for which thegeometric composition L12 ◦ L23 is embedded as in (2.2.1).

• The equivalence relation ∼ on MorSymp# is defined by L ∼ L̃ if there is a finitesequence of moves L � L′ � L′′ . . . � L(N) = L̃ in which each move replacesone subchain of simple morphisms by another,

L(k) = (. . . ,Lij,Ljl, . . .)

� L(k+1) = (. . . ,Lij ◦ Ljl, . . .)

resp. L(k) = (. . . ,Lij ◦ Ljl, . . .)

� L(k+1) = (. . . ,Lij,Ljl, . . .)

according to a geometric composition move(

(Lij,Ljl),Lij ◦ Ljl) ∈ Comp resp.

(

Lij ◦ Ljl, (Lij,Ljl)) ∈ Comp.

The result of this quotient construction is that the composition of morphisms isgiven by geometric composition [L12] ◦ [L23] = [L12 ◦ L23] if the latter is embedded.We will later recast this construction in terms of an extension of the symplecticcategory Symp# to a 2-category in which the equivalence relation∼ is obtained from2-isomorphisms; see Example 3.1.6 and Sect. 3.5.

14 K. Wehrheim

Remark 2.2.3 The present equivalence relation does not identify a LagrangianL ⊂ M− × N with its image φH(L) ⊂ M− × N under a Hamiltonian symplectomor-phism φH . Indeed, any morphism L inMorSymp# (M,N) induces a (Lagrangian whereimmersed) subset of M− × N by complete geometric composition, and this subsetis invariant under geometric composition moves. However, such equivalences underHamiltonian deformation can also be cast as 2-isomorphisms; see Example 3.5.1.

2.3 Categories with Cerf Decompositions

The basic idea of Cerf decompositions is to decompose a (d + 1)-manifold Y =Y01 ∪Σ1 Y12 . . . ∪Σk−1 Y(k−1)k into simpler pieces Yij = f −1([bi, bj]) by cutting at reg-ular level sets Σi = f −1(bi) of a Morse function Y → R as illustrated in Fig. 4below. By viewing Y as a cobordism between empty sets, i.e., as a morphismin MorBord+1(∅,∅), this can be seen as a factorization [Y ] = [Y01] ◦ [Y12] ◦ . . . ◦[Y(k−1)k] in Bord+1. Here, the Morse function f and regular levels bi can be chosensuch that each piece Yi(i+1) contains either none or one critical point, and thus is eithera cylindrical cobordism—diffeomorphic to the product cobordism Zφ = [0, 1] ×Σjas in (2.1.1)—or a handle attachment as in the following remark. These “simplecobordisms” are illustrated in Figs. 2 and 3.

Remark 2.3.1 A k-handle attachment Yα of index 0 ≤ k ≤ d + 1 is a (d + 1)-dimensional cobordism, which is obtained by attaching to a cylinder [0, 1] ×Σ ahandle Bk × Bd+1−k along an attaching cycle Sk−1 ↪→ α ⊂ {1} ×Σ , as illustrated inFig. 3. Here, Bk denotes a k-dimensional ball with boundary ∂Bk = Sk−1.

By reversing the orientation and boundary identifications of any k-handle attach-ment Yα from Σ to Σ ′, we obtain a cobordism Y−α from Σ ′ to Σ . This reversedcobordism is also a d + 1− k-handle attachment Y−α = Yα∗ for an attaching cycleSd−k ↪→ α∗ ⊂ {1} ×Σ ′. It moreover is the adjoint of Yα in the sense of Remark 2.4.3and will become useful in the formulation of Cerf moves below.

Specifying to dimension d = 2 and the connected bordism category, it will suf-fice to consider 2-handle attachments (and their adjoints) with attaching circles

Fig. 2 A cylindrical cobordism supports a Morse function without critical points, whose gradientflow induces a diffeomorphism to the product cobordism [0, 1] ×Σj with natural identificationΣj ∼= {1} ×Σj and boundary identification φ : Σi→̃{0} ×Σj arising from the flow as well

Floer Field Philosophy 15

Fig. 3 Handle attachments in dimension d = 2 are “simple cobordisms” which support a Morsefunction f with a single critical point of index 0 ≤ k ≤ 3. The attaching cycles are given by intersec-tion of the unstable manifold (in red) with the boundary. Index k = 0 and k = 3 handle attachmentsare adjoint via orientation reversal and only appear between the empty set and sphere S2. Indexk = 1 and k = 2 handle attachments are adjoint via orientation reversal (which interchanges unsta-ble and stable manifolds) and appear between surfaces Σg,Σg+1 of adjacent genus g, g+ 1 ∈ N0

that are homologically nontrivial and thus do not disconnect the surface. Moreprecisely, any attaching circle S1 � α ⊂ Σ in a closed surface Σ determines a2-handle attachment as follows: Replacing an annulus neighborhood of α by twodisks specifies a lower genus surface Σ ′ = Σα together with a diffeomorphismπα : Σ�α → Σ ′�{2 points}. Given this construction, the 2-handle attaching cobor-dism Yα from Σ to Σ ′ is unique up to diffeomorphism fixing the boundary.

More detailed introductions to Cerf theory can be found in e.g. [13, 27, 50].Here, we concentrate on the algebraic structure that it equips the bordism categorieswith. To describe this structure, we may think of Cerf decompositions as a primedecomposition of (d + 1)-manifolds, and more generally of (d + 1)-cobordisms: Adecomposition into simple cobordisms (cylindrical cobordisms and handle attach-ments) always exist and simple cobordisms have no further simplifying decompo-sition. And while these Cerf decompositions are not unique, any two choices ofdecomposition are related via just a few moves, some of which are shown in Fig.4.These moves reflect changes in the Morse function (critical point cancelations andcritical point switches), cutting levels (cylinder cancellation), and the ways in whichpieces are glued together (diffeomorphism equivalences which in particular encodehandle slides). All of these Cerf moves are local in the sense13 that they replace only

13While a diffeomorphism equivalence is not local, it decomposes into a sequence of local moves.