Sheaves on graphs, their homological invariants, and a proof of the Hanna Neumann conjecture

124 Pages·2014·0.867 MB·English

by  Joel Friedman, Warren Dicks

MEMOIRS
of the
American Mathematical Society
Volume 233 • Number 1100 (sixth of 6 numbers) • January 2015
Sheaves on Graphs,
Their Homological Invariants,
and a Proof of the
Hanna Neumann Conjecture:
with an Appendix by Warren Dicks
Joel Friedman
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 233 • Number 1100 (sixth of 6 numbers) • January 2015
Sheaves on Graphs,
Their Homological Invariants,
and a Proof of the
Hanna Neumann Conjecture:
with an Appendix by Warren Dicks
Joel Friedman
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Friedman,Joel,1962-
Sheavesongraphs,theirhomologicalinvariants,andaproofoftheHannaNeumannconjecture
/JoelFriedman;withanAppendixbyWarrenDicks.
pages cm. – (Memoirs of the AmericanMathematicalSociety, ISSN 0065-9266; volume 233,
number1100)
Includesbibliographicalreferences.
ISBN978-1-4704-0988-3(alk. paper)
1. Sheaftheory. 2. Vectorspaces. 3. Vectoranalysis. I.Dicks,Warren,1947- II.Title.
QA612.36.F75 2015
514(cid:2).224–dc23 2014033278
DOI:http://dx.doi.org/10.1090/memo/1100
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Contents
Preface vii
Introduction ix
Chapter 1. Foundations of Sheaves on Graphs and Their Homological
Invariants 1
1.1. Introduction 1
1.2. Basic Definitions and Main Results 5
1.3. Galois and Covering Theory 14
1.4. Sheaf Theory and Homology 18
1.5. Twisted Cohomology 26
1.6. Maximum Excess and Supermodularity 32
1.7. htwist and the Universal Abelian Covering 36
1
1.8. Proof of Theorem 1.10 39
1.9. Concluding Remarks 54
Chapter 2. The Hanna Neumann Conjecture 57
2.1. Introduction 57
2.2. The Strengthened Hanna Neumann Conjecture 59
2.3. Graph Theoretic Formulation of the SHNC 61
2.4. Galois and Covering Theory in the SHNC 62
2.5. ρ-kernels 65
2.6. Symmetry and Algebra of the Excess 70
2.7. Variability of k-th Power Kernels 76
2.8. Proof of the SHNC 81
2.9. Concluding Remarks 83
Appendix A. A Direct View of ρ-Kernels 85
Appendix B. Joel Friedman’s Proof of the strengthened Hanna Neumann
conjecture by Warren Dicks 91
B.1. Sheaves on graphs 91
B.2. Free groups and graphs 95
B.3. The strengthened Hanna Neumann conjecture 98
Bibliography 103
iii

Abstract
Abstract for Chapters 1,2, and Appendix A. Inthis paperwe establish
somefoundationsregardingsheavesofvectorspacesongraphsandtheirinvariants,
such as homology groups and their limits. We then use these ideas to prove the
Hanna Neumann Conjecture of the 1950’s; in fact, we prove a strengthened form
of the conjecture.
We introduce a notion of a sheaf of vector spaces on a graph, and develop the
foundations of homology theories for such sheaves. One sheaf invariant, its “max-
imum excess,” has a number of remarkable properties. It has a simple definition,
with no reference to homology theory, that resembles graph expansion. Yet it is a
“limit”ofBettinumbers, andhencehasashort/longexactsequencetheoryandre-
semblestheL2 BettinumbersofAtiyah. Also,themaximumexcessisdefinedviaa
supermodular function, which gives the maximum excess much stronger properties
than one has of a typical Betti number.
Our sheaf theory can be viewed as a vast generalization of algebraic graph
theory: each sheaf has invariants associated to it—such as Betti numbers and
Laplacian matrices—that generalize those in classical graph theory.
We shall use “Galois graph theory” to reduce the Strengthened Hanna Neu-
mann Conjecture to showing that certain sheaves, that we call ρ-kernels, have zero
maximum excess. We use the symmetry in Galois theory to argue that if the
Strengthened Hanna Neumann Conjecture is false, then the maximum excess of
“most of” these ρ-kernels must be large. We then give an inductive argument to
show that this is impossible.
ReceivedbytheeditorMay30,2011,and,inrevisedform,March23,2013.
ArticleelectronicallypublishedonMay19,2014.
DOI:http://dx.doi.org/10.1090/memo/1100
2010 Mathematics Subject Classification. Primary20E05,05C10,55N30,18F20;Secondary
20E08,05C05,05C50,18F10,18F20,14F20.
Key words and phrases. Intersection of free subgroups, graphs, skew group ring, sheaves,
HannaNeumannconjecture,homology,Galoistheory.
JoelFriedman’s researchpartially supportedby anNSERC grant andpartially done atthe
Centre Bernoulli, funded by the Swiss National Science Foundation. Warren Dicks’ research
partially supported by Spain’s Ministerio de Ciencia e Innovaci´on through Project MTM2011-
25955.
Affiliationsattimeofpublication: JoelFriedman,DepartmentofComputerScience,Univer-
sityofBritishColumbia,Vancouver,BC,V6T1Z4,CANADA,andDepartmentofMathematics,
University of British Columbia, Vancouver, BC, V6T 1Z2, CANADA, email: [email protected] and
[email protected],http://www.math.ubc.ca/˜jf;andWarrenDicks,DepartamentdeMatem`atiques,
UniversitatAut`onomadeBarcelona,E–08193Bellaterra(Barcelona),email: [email protected],
http://mat.uab.cat/˜dicks/.
(cid:3)c2014 Warren Dicks and Joel Friedman
v
vi ABSTRACT
Abstract for Appendix B. For a finite graph Z, r(Z) := e−v+t, where e, v,
andtdenotethenumber ofedges, vertices, andtreecomponentsofZ, respectively.
Let G be a finite group, Z be a finite G-free G-graph, and X and Y be subgraphs
ofZ. Usinglinearalgeb(cid:2)raandalgebraicgeometryoverasufficientlylargefield,Joel
Friedman proved that r(X∩ gY)(cid:2) r(X)r(Y). He showed that this inequality
g∈G
impliesthestrengthenedHannaNeumannconjecture. WesimplifyFriedman’sproof
of the foregoing inequality by replacing the sufficiently large field with a field F on
which G acts faithfully and then replacing all the arguments involving algebraic
geometry with shorter arguments about the left ideals of the skew group ring FG.
Preface
This memoir contains materials from three preprints:
(i) two articles of Joel Friedman, [Fri11b,Fri11a], of April 2011; and
(ii) an article of Warren Dicks, [Dic12a], of October 2012, based on Dicks’
notes of a seminar given in June 2012.
Themaingoalof[Fri11b]istodevelopfoundationalmaterialonthehomology
ofsheavesofgraphs,especiallyconceringthe“maximumexcess,”whichisa“scaled
limit Betti number” or “L2 Betti number” of a sheaf on a graph. This material
comprises Chapter 1 of this memoir.
The main goal of [Fri11a] is to prove the Hanna Neumann conjecture of the
1950’s. This material comprises Chapter 2 and Appendix A of this manuscript,
and uses results of Chapter 1.
The maingoal of [Dic12a], by WarrenDicks, istosimplify andstrengthenthe
results of Chapter 2 using skew group rings. This material comprises Appendix B
of this manuscript.
The articles [Fri11b,Fri11a] were merged into a single article for the orig-
inal submission to the Memoirs of the AMS, [Fri11c]; Dicks’ article, [Dic12a],
was incorporated into this manuscript. Some superficial aspects of [Fri11c] and
[Dic12a] were merged, such as the bibliography, keywords, table of contents, and
AMS Subject Classification; most everything else about these two articles is left in
their original form. In particular, the notation between Chapters 1,2 and Appen-
dixAdiffersfromthatofAppendixB,andthereareseparate abstractsforthetwo
parts.
* * *
We wish to thank a referee for this manuscript, who provided us with the
following eloquent historical context of the Hanna Neumann conjecture.
For a finitely generated free group, H, let rank(H) denote the rank of H, i.e.,
the size of a free-generating set of H and let r(H) denote the reduced rank, i.e.,
max(0,rank(H)−1).
In 1921, J. Nielsen proved in [Nie21] that every fintely generated subgroup
of a free group is free. In 1954, Alan Howson’s Ph.D. thesis, [How54], written
under the supervision of Graham Higman, gave a proof that the set of finitely
generatedsubgroupsofagivenfreegroupisclosedundertheintersectionoperation.
Howson’s proof entails a rough upper bound for rank(H∩K) in terms of rank(H)
and rank(K). In 1956, Hanna Neumann, in [Neu56], gave a better upper bound.
In 1957, in a short addendum [Neu57], she said that Reinhold Baer had pointed
out to her that her proof showed that r(H ∩K) (cid:2)2r(H)r(K). She asked if the 2
wasnecessaryandafewlineslatercalledr(H)r(K)“theconjecturedbound.” Thus
was born the Hanna Neumann Conjecture, abbreviated the HNC. For 54 years the
vii
viii PREFACE
conjectureresistedattemptstoproveit. In1990,WalterNeumann,oneofhersons,
made a stronger conjecture, abbreviated the SHNC.
In May1 2011, Joel Friedman proved the SHNC within 100 pages. A few days
later, Igor Mineyev gave another proof within 30 pages, spread over two articles2.
A week later, Warren Dicks reduced Mineyev’s proof to one or two pages3. A year
later, Dicks reduced Friedman’s proof to 13 pages4.
* * *
To this historical account we wish to add that work on the Hanna Neumann
conjecture goes beyond improving an inequality by a factor of two. The conjecture
suggests that there is new mathematics to understand that may manifest itself in
the conjecture’s resolution. And, indeed, our study of this conjecture has lead us
to new mathematics regarding sheaves on graphs and their symmetries; we believe
that such ideas are likely to have applications in other aspects of combinatorics
andlinear algebra. Furthermore, Dicks’ stronger5 methodof skew group rings may
renderfeasiblecertainaspectsofsheaftheoryongraphsthataredifficulttodirectly
apply.
1[Fri11a]waspostedtoourwebsiteonApril30,2011,butisgiventhedateMay1,2011on
thearXivposting.
2[Min11]and[Min12a].
3[Dic11]and[Dic12b].
4[Dic12a]
5Dicks’ method regarding the Hanna Neumann conjecture is stronger than ours; for one
thing, Dicks’ methods give new “ρ-kernels” that are, in a sense, simpler and, curiously, involve
fewertranscendentalsthanwerequired.
Introduction
Introduction for Chapters 1,2, and Appendix A. Thismemoirhastwomain
goals. First,wedevelopsomefoundationsonwhatwecall“sheavesongraphs”and
their invariants. Second, using these foundations, we resolve the Hanna Neumann
Conjecture of the 1950’s.
Althoughourfoundationsofsheavesongraphsseemlikelytoimpactanumber
of areas of graph theory, the theme that is common to most of this memoir is the
HannaNeumannConjecture(orHNC).Boththisconjectureandastrengtheningof
it, known as the Strengthened Hanna Neumann Conjecture (or SHNC) have been
extensively studied (see [Bur71,Imr77b,Imr77a,Ser83,Ger83,Sta83,Neu90,
Tar92,Dic94,Tar96,DF01,Kha02,MW02,JKM03,Neu07,Eve08,Min11];
also see related work [Iva99,Arz00,Iva01]). These conjectures are usually stated
as an inequality involving free groups, although both conjectures have well-known
reformulations in term of finite graphs. In this memoir we prove both conjectures,
usingthefinitegraphreformulations, reducingbothtothevanishingofahomology
group of certain sheaves on graphs.
This work was originally written and posted to arxiv.com as two separate
articles. The first article, [Fri11b], contains the foundational material on sheaves
ofgraphs,andcomprisesChapter1ofthismanuscript. Thesecondarticle,[Fri11a],
resolves the SHNC (and HNC), and Chapter 2 of this manuscript consists of this
material. This manuscript is easier to read than both articles separately, in that
redundant definitions have beendiscarded, and referencesin Chapter 2 tomaterial
in Chapter 1 are now more specific. Yet, as we now explain, Chapters 1 and 2 are
largelyindependentlyofoneanother, andChapter1, thefoundationsofsheaveson
graphs, is of interest beyond the HNC and SHNC. To explain this interest, let us
recall a bit about sheaf theory and its connection to discrete mathematics.
Among many (co)homology theories of topological spaces, the sheaf approach
hasmanyadvantages. Forone,itworkswithnon-Hausdorffspaces, asdonefirstby
SerreinalgebraicgeometrywiththeZariskitopology(see[Har77]). Grothendieck’s
sheaf theory of [sga72a,sga72b,sga73,sga77] defined a notion of a sheaf on very
general spaces now called “Grothendieck topologies.” While Grothendieck’s work
has had remarkable success for cohomology theories in algebraic geometry, we be-
lieve that graph theory and combinatorics may greatly benefit by studying very
special Grothendieck topologies formulated from finite, discrete structures. In par-
ticular, we will resolve the SHNC using a simple, finite Grothendieck topology
associated to any finite graph.
Anotheraspectofsheafcohomologyisthatitvastlygeneralizesthecohomology
ofaspace. Eachsheafhasinjectiveresolutionsthatgivecohomologygroups. When
thesheafistaketobethe“structuresheaf”ofthespace,werecoverthecohomology
groups of the space. However, there are many sheaves apart from the structure
ix