Breadth in Contemporary Topology

298 Pages·2019·5.77 MB·English

by  David T. Gay, Weiwei Wu

Proceedings of Symposia in
P M
URE ATHEMATICS
Volume 102
Breadth in
Contemporary Topology
2017 Georgia International
Topology Conference
May 22–June 2, 2017
University of Georgia, Athens, Georgia
David T. Gay
Weiwei Wu
Editors
Licensed to AMS.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
10.1090/pspum/102
Volume 102
Breadth in
Contemporary Topology
2017 Georgia International
Topology Conference
May 22–June 2, 2017
University of Georgia, Athens, Georgia
David T. Gay
Weiwei Wu
Editors
Licensed to AMS.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
Licensed to AMS.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
Proceedings of Symposia in
P M
URE ATHEMATICS
Volume 102
Breadth in
Contemporary Topology
2017 Georgia International
Topology Conference
May 22–June 2, 2017
University of Georgia, Athens, Georgia
David T. Gay
Weiwei Wu
Editors
Licensed to AMS.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
2010 Mathematics Subject Classification. Primary 57-06, 20F36,
53D35, 53D12, 55P48,57M25, 57M27, 57R17, 57R91.
The 2017 Georgia International Topology Conference was supported by a grant from
the National Science Foundation (grant DMS-1719320). Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the authors
and do not necessarilly reflect the views of the National Science Foundation.
Library of Congress Cataloging-in-Publication Data
Names: GeorgiaInternationalTopologyConference(2017: UniversityofGeorgia)|Gay,DavidT.,
1968–editor. |Wu,Weiwei,1983–editor.
Title: Breadth in contemporary topology: 2017 Georgia International Topology Conference,
May 22–June 2, 2017, University of Georgia, Athens, Georgia / David T. Gay, Weiwei Wu,
editors.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2019]|Series: Proceed-
ingsofsymposiainpuremathematics;volume102|Includesbibliographicalreferences.
Identifiers: LCCN2019001949|ISBN9781470442491(alk. paper)
Subjects: LCSH:Topology–Congresses. |Homologytheory–Congresses. |Manifolds(Mathemat-
ics)–Congresses. |AMS:Manifoldsandcellcomplexes–Proceedings,conferences,collections,
etc. msc | Group theory and generalizations – Special aspects of infinite or finite groups
– Braid groups; Artin groups. msc | Differential geometry – Symplectic geometry, contact
geometry – Global theory of symplectic and contact manifolds. msc | Differential geometry
– Symplectic geometry, contact geometry – Lagrangian submanifolds; Maslov index. msc |
Algebraic topology – Homotopy theory – Loop space machines, operads. msc | Manifolds
andcellcomplexes–Low-dimensionaltopology–Knotsandlinksinmsc|Manifoldsandcell
complexes–Low-dimensionaltopology–Invariantsofknotsand3-manifolds. msc|Manifolds
andcellcomplexes–Differentialtopology–Symplecticandcontacttopology. msc|Manifolds
andcellcomplexes–Differentialtopology–Equivariantalgebraictopologyofmanifolds. msc
Classification: LCCQA611.A1G4562017|DDC514/.34–dc23
LCrecordavailableathttps://lccn.loc.gov/2019001949
DOI:https://doi.org/10.1090/pspum/102
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Contents
Preface vii
Structure of the flow and Yamada polynomials of cubic graphs
Ian Agol and Vyacheslav Krushkal 1
Inequivalent Lefschetz fibrations on rational and ruled surfaces
R. I˙nanc¸ Baykur 21
Virtual and welded periods of classical knots
Hans U. Boden and Andrew J. Nicas 29
Transverse universal links
Roger Casals and John B. Etnyre 43
Groups and polytopes
Stefan Friedl, Wolfgang Lu¨ck, and Stephan Tillmann 57
Functions on surfaces and constructions of manifolds in dimensions three,
four and five
David T. Gay 79
On braided, banded surfaces and ribbon obstructions
J. Elisenda Grigsby 95
A remark on the geography problem in Heegaard Floer homology
Jonathan Hanselman, C¸ag˘atay Kutluhan, and Tye Lidman 103
A note on knot concordance and involutive knot Floer homology
Kristen Hendricks and Jennifer Hom 113
A Heegaard Floer analog of algebraic torsion
C¸ag˘atay Kutluhan, Gordana Matic´, Jeremy Van Horn-Morris,
and Andy Wand 119
Realization problems for diffeomorphism groups
Kathryn Mann and Bena Tshishiku 131
Problems, questions, and conjectures about mapping class groups
Dan Margalit 157
Totally disconnected groups (not) acting on two-manifolds
John Pardon 187
Fukaya A∞-structures associated to Lefschetz fibrations. IV
Paul Seidel 195
2017 GITC Problem Sessions 277
v
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Preface
Since1961theUniversityofGeorgiahasorganizedandhostedanannualweek-
longtopologyconferenceandamajorinternationaltopologyconferenceeveryeight
years. The tradition started in 1961 with a memorable four-week conference, the
proceedings of which, The Topology of 3-Manifolds and Related Topics, were pub-
lished by Prentice Hall. The 2017 Georgia International Topology Conference,
continuing this long tradition, was held at the University of Georgia in Athens,
Georgia, from May 22 to June 2, 2017. The conference had 41 speakers, ranging
from the leading figures to talented young mathematicians, who covered a wide
spectrum of the topics in topology and attracted a total of over 190 participants.
The main goals of our conference are to give wide exposure to new and impor-
tant results and toencourage interaction among graduate students and researchers
in different stages of their careers. Toward this end, we encouraged speakers to
aim their talks at a broad audience of topologists, and we also scheduled a series
of expository lectures that introduced graduate students to several of the themes
of the conference.
The papers in this volume were written by both speakers and participants
of the conference and cover topics ranging from symplectic topology to classical
knot theory to the topology of 3- and 4-dimensional manifolds to geometric group
theory. Several papersfocuson openproblems, especially thepaper of Margalit on
mapping class groups of surfaces. Other papers present new and insightful proofs
of classical results, such as Pardon’s paper on groups which can and cannot act on
2-manifolds. Still other papers lay out rich and technical foundational material or
survey material presented in more depth elsewhere. Taken as a whole, this volume
captures the spirit of the conference, both in terms of public lectures and informal
conversations, and presents a sampling of some of the great new ideas generated in
topology over the preceding eight years.
The conference was organized by David Gay, Jean Gutt, Will Kazez, Gordana
Matic, Juanita Pinzo´n-Caicedo, Adam Saltz, Michael Usher, and Weiwei Wu. The
Scientific Advisory Committee, consisting of Ian Agol, Yakov Eliashberg, David
Gabai, Dusa McDuff, Tomasz Mrowka, Peter Ozsvath, Peter Teichner, and Karen
Vogtmann, provided great assistance selecting speakers. The logistical support of
theconferencewasduetothehardworkofJulieMcEver,ConniePoore,GailSuggs,
Laura Ackerley, and Christy McDonald. Finally, the organizers are grateful to the
generous support of the National Science Foundation (grant DMS-1719320) and to
the University of Georgia.
D. G.
W. W.
vii
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Licensed to AMS.
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10.1090/pspum/102/01
ProceedingsofSymposiainPureMathematics
Volume102,2019
https://doi.org/10.1090/pspum/102/01806
Structure of the flow and Yamada polynomials
of cubic graphs
Ian Agol and Vyacheslav Krushkal
Abstract. We establish a quadratic identity for the Yamada polynomial of
ribboncubicgraphsinR3,extendingtheTuttegoldenidentityforplanarcubic
graphs. Anapplicationisgiventothestructureoftheflowpolynomialofcubic
graphs at zero. The golden identity for the flow polynomial is conjectured
to characterize planarity of cubic graphs, and we prove this conjecture for a
certaininfinitefamilyofnon-planargraphs.
Further,weestablishexponentialgrowthofthenumberofchromaticpoly-
nomials of planar triangulations, answering a question of D. Treumann and
E. Zaslow. The structure underlying these results is the chromatic algebra,
andmoregenerallytheSO(3)topologicalquantumfieldtheory.
1. Introduction
Using the interplay between classical and quantum polynomials of graphs and
ideas from topological quantum field theory (TQFT), we establish results on the
structure of the Yamada and flow polynomials of cubic graphs. It has bee√n known
since the work of Birkhoff and Lewis in the 1940s [3] that the values (3± 5)/2 of
the parameter play a special role in the theory of the chromatic polynomial χ of
T
planar triangulations. In a series of papers [18,19] in the 1960s, Tutte established
further remarkable properties, including the golden identity. Formulated dually in
terms of the flow polynomial F of planar cubic graphs G, it reads
G
(1.1) F (φ+2) = φEF (φ+1)2,
G G √
where E is the number of edges of G and φ denotes the golden ratio (1+ 5)/2.
The special role played by φ+1, and more generally by the Beraha numbers [2]
B = 2+2cos(2π/n), was conceptually explained in [6] where these results were
n
placed in the context of SO(3) TQFT.
We show in Theorem 3.1that Tutte’s identity (1.1) admits an extensionto the
Yamada polynomial R of ribbon cubic graphs G in R3:
G
(1.2) R (eπi/5) = (−1)V−EφER (e−2πi/5)2,
G G
2010MathematicsSubjectClassification. Primary05C31;Secondary05C10,57M27,57R56.
The first author was supported by NSF grant DMS-1406301,a Simons InvestigatorAward,
andthe Institute for Advanced Study (in particular the Simonyi Fund and the Fund for Mathe-
matics).
ThesecondauthorwassupportedinpartbyNSFgrantDMS-1612159.
(cid:2)c2019 American Mathematical Society
1
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