The Reidemeister Torsion of 3-Manifolds

264 Pages·2003·1.126 MB·English

by  Liviu I. Nicolaescu

de Gruyter Studies in Mathematics 30
Editors: Carlos Kenig · Andrew Ranicki · Michael Röckner
de Gruyter Studies in Mathematics
1 Riemannian Geometry, 2nd rev. ed., Wilhelm P.A.Klingenberg
2 Semimartingales, Michel Me´tivier
3 Holomorphic Functions of Several Variables, Ludger Kaup and Burchard Kaup
4 Spaces of Measures, Corneliu Constantinescu
5 Knots, 2nd rev. and ext. ed., Gerhard Burde and Heiner Zieschang
6 Ergodic Theorems, Ulrich Krengel
7 Mathematical Theory of Statistics, Helmut Strasser
8 Transformation Groups, Tammo tom Dieck
9 Gibbs Measures and Phase Transitions, Hans-Otto Georgii
10 Analyticity in Infinite Dimensional Spaces, Michel Herve´
11 Elementary Geometry in Hyperbolic Space, Werner Fenchel
12 Transcendental Numbers, Andrei B. Shidlovskii
13 Ordinary Differential Equations, Herbert Amann
14 Dirichlet Forms and Analysis on Wiener Space, Nicolas Bouleau and
Francis Hirsch
15 Nevanlinna Theory and Complex Differential Equations, Ilpo Laine
16 Rational Iteration, Norbert Steinmetz
17 Korovkin-typeApproximationTheoryanditsApplications,FrancescoAltomare
and Michele Campiti
18 Quantum Invariants of Knots and 3-Manifolds, Vladimir G. Turaev
19 Dirichlet Forms and Symmetric Markov Processes, Masatoshi Fukushima,
Yoichi Oshima and Masayoshi Takeda
20 Harmonic Analysis of Probability Measures on Hypergroups, Walter R.Bloom
and Herbert Heyer
21 Potential Theory on Infinite-Dimensional Abelian Groups, Alexander Bendikov
22 Methods of Noncommutative Analysis, Vladimir E. Nazaikinskii,
Victor E. Shatalov and Boris Yu. Sternin
23 Probability Theory, Heinz Bauer
24 Variational Methods for Potential Operator Equations, Jan Chabrowski
25 The Structure of Compact Groups, Karl H. Hofmann and Sidney A. Morris
26 Measure and Integration Theory, Heinz Bauer
27 Stochastic Finance, Hans Föllmer and Alexander Schied
28 Painleve´ Differential Equations in the Complex Plane, Valerii I. Gromak, Ilpo
Laine and Shun Shimomura
29 Discontinuous Groups of Isometries in the Hyperbolic Plane, Werner Fenchel
and Jakob Nielsen
Liviu I. Nicolaescu
The Reidemeister Torsion
of 3-Manifolds
≥
Walter de Gruyter
Berlin · New York 2003
Author
LiviuNicolaescu
DepartmentofMathematics
UniversityofNotreDame
NotreDame,IN46556-4618
USA
SeriesEditors
CarlosE.Kenig AndrewRanicki MichaelRöckner
DepartmentofMathematics DepartmentofMathematics FakultätfürMathematik
UniversityofChicago UniversityofEdinburgh UniversitätBielefeld
5734UniversityAve MayfieldRoad Universitätsstraße25
Chicago,IL60637,USA EdinburghEH93JZ,Scotland 33615Bielefeld,Germany
Mathematics Subject Classification 2000: 57-02; 43A40; 55N25, 57M05, 57M12, 57M25, 57M27,
57Q10,57R15,57R57,58J28
Keywords:3-manifolds;Reidemeistertorsion;Turaev’srefinedtorsion;spin-cstructures;linkingforms;
harmonicanalysisondiscreteAbeliangroups;knotsandlinks;surgeryformulae;plumbings;Seiber(cid:1)
Witteninvariants;surfacesingularities
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TheReidemeistertorsionof3-manifolds/LiviuI.Nicolaescu.
p. cm.(cid:1)(DeGruyterstudiesinmathematics;30)
Includesbibliographicalreferencesandindex.
ISBN3-11-017383-2(cloth:alk.paper)
1.Three-manifolds(Topology). 2.Reidemeistertorsion. I.Title:
Reidemeistertorsionofthree-manifolds. II.Title. III.Series.
QA613.2.N53 2003
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ISBN3-11-017383-2
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To the memory of my father, Ion I. Nicolaescu,
who shared with me his passion for books.

Introduction
Thetorsionofacellular(simplicial)complexwasintroducedinthe30sbyW.Franz
[29]andK.Reidemeister[90]intheirstudyoflensspaces. ThelensspacesL(p,q)
(p fixed) have the same fundamental groups and thus the same homology groups.
However,theyarenotallhomeomorphic. Theyarenotevenhomotopicallyequivalent.
This can be observed by detecting some below the radar interactions between the
fundamental group and the simplicial structure. The torsion captures some of these
interactions. Inparticular,itisabletodistinguishlensspaceswhicharehomotopically
equivalentbutnothomeomorphic,andmoreovercompletelyclassifythesespacesup
to a homeomorphism. This suggests that this invariant is reaching deep inside the
topologicalstructure.
What is then this torsion? What does it compute? These are the kind of ques-
tionswetrytoaddressinthesenotes,throughmanyexamplesandvariousequivalent
descriptionsofthisinvariant.
From an algebraic point of view, the torsion is a generalization of the notion of
determinant. Themostnaturalandgeneralcontexttodefinethetorsionwouldinvolve
the Whitehead group and algebraic K-theory as in the very elegant and influential
Milnorsurvey[72],butwedidnotadoptthismoregeneralpointofview. Insteadwe
lookatwhatMilnordubbedR-torsion.
ThisinvariantcanbeviewedasahigherEulercharacteristictypeinvariant. Much
liketheEulercharacteristic,thetorsionsatisfiesaninclusion–exclusion(a.k.a. Mayer–
Vietoris)principlewhichcanberoughlystatedthatforanytopologicalspacesAand
B wehave
Tors(A∪B)=Tors(A)+Tors(B)−Tors(A∩B)
whichsuggeststhatthetorsioncouldbeinterpretedascountingsomething.
TheclassicalPoincaré–HopftheoremstatesthattheEuler–Poincarécharacteristic
ofasmoothmanifoldcountsthezerosofagenericvectorfield. IftheEuler-Poincaré
characteristic is zero then most vector fields have no zeroes but may have periodic
orbits. The torsion counts these closed orbits, at least for some families of vector
fields. As D. Fried put it in [34], “the Euler characteristic counts points while the
torsioncountscircles”.
One of the oldest results in algebraic topology equates the Euler-Poincaré char-
acteristic of simplicial complex, defined as the alternating sums of the numbers of
simplices,withamanifestlycombinatorialinvariant,thealternatingsumoftheBetti
numbers. Similarly,theR-torsioncanbegivenadescriptionintermsofchaincom-
plexes or, a plainly invariant description, in homological terms. Just like the Euler
characteristic, the R-torsion of a smooth manifold can be given a Hodge theoretic
description,albeitmuchmorecomplicated.
viii Introduction
Morerecently,thisinvariantturnedupin3-dimensionalSeiberg–Wittentheory,in
theworkofMeng–Taubes([68]). Thisresultgaveustheoriginalimpetustounderstand
themeaningoftorsion.
This is a semi-informal, computationally oriented little book which grew out of
oureffortstounderstandtheintricaciesoftheMeng–Taubes–Turaevresults,[68,115].
ForthisreasonalotofemphasisisplacedontheReidemeistertorsionof3-manifolds.
Thesenotestriedtoaddresstheauthor’sownstrugglewiththeoverwhelmingamount
of data involved and the conspicuously scanty supply of computational examples in
thetraditionalliteratureonthesubject. Weconsideredthatataninitialstageagood
intuitive argument or example explaining why a certain result could be true is more
helpful than a complete technical proof. The classical Milnor survey [72] and the
recent introductory book [117] byV. Turaev are excellent sources to fill in many of
ourdeliberatefoundationalomissions.
Whenthinkingoftopologicalissuesitisveryimportantnottogetdistractedbythe
uglylookingbutelementaryformalismbehindthetorsion. Forthisreasonwedevoted
theentirefirstchaptertothealgebraicfoundationsoftheconceptoftorsion. Wegive
several equivalent definitions of the torsion of an acyclic complex and in particular,
wespendagoodamountoftimeconstructingasetupwhichcoherentlydealswiththe
torturous sign problem. We achieved this using a variation of some of the ideas in
Deligne’ssurvey[18].
Thegeneralalgebraicconstructionsarepresentedinthefirsthalfofthischapter,
while in the second half we discuss Turaev’s construction of several arithmetically
defined subrings of the field of fractions of the rational group algebra of anAbelian
group. Thesesubringsprovidetheoptimalalgebraicframeworktodiscussthetorsion
ofamanifold. WeconcludethischapterbypresentingadualpictureoftheseTuraev
subringsviaFouriertransform. Theseresultsseemtobenewandsimplifysubstantially
manygluingformulæforthetorsion,tothepointthattheybecomequasi-tautological.
TheReidemeistertorsionofanarbitrarysimplicial(orCW)complexisdefinedin
the second chapter. This is simply the torsion of a simplicial complex withAbelian
localcoefficients,orequivalentlythetorsionofthesimplicialcomplexofthemaximal
Abeliancover. Wepresentthebasicpropertiesofthisinvariant: theMayer–Vietoris
principle, duality, arithmeticpropertiesandanEuler–Poincarétyperesult. Wecom-
putethetorsionofmanymostlylowdimensionalmanifoldsandinparticularweexplain
howtocomputethetorsionofany3-manifoldwithb >0usingtheMayer–Vietoris
1
principle,theFouriertransform,andtheknowledgeoftheAlexanderpolynomialsof
links in S3. Since the literature on Dehn surgery can be quite inconsistent on the
varioussignconventions,wehavedevotedquiteasubstantialappendixtothissubject
wherewekeptanwatchfuleyeontheseoftentroublesomesings.
TheapproachbasedonAlexanderpolynomialshasonemajordrawback,namelyit
requiresahugevolumeofcomputations. Wespendthewholesection§2.6explaining
how to simplify these computation for a special yet very large class of 3-manifolds,
namelythegraphmanifolds. Thelinksofisolatedsingularitiesofcomplexsurfacesare
includedinthisclassandtherecentwork[75,76]provesthattheReidemeistertorsion
Introduction ix
captures rather subtle geometrical information about such manifolds. We conclude
thischapterwithsomeofthetraditionalapplicationsofthetorsionintopology.
Chapter 3 focuses on Turaev’s ingenious idea of Euler structure and how it can
beusedtorefinetheconceptoftorsionbyremovingtheambiguitiesinchoosingthe
basesneededforcomputingthetorsion. Turaevlaterobservedthatfora3-manifold
a choice of an Euler structure is equivalent to a choice of spinc-structure. After we
review a few fundamental properties of this refined torsion for 3-manifolds we then
goontopresentaresultofTuraevwhichinessencesaysthattherefinedtorsionofa
3-manifoldwithpositiveb isuniquelydeterminedbytheAlexanderpolynomialsof
1
linksinS3 andtheMayer–Vietorisprinciple.
Thisuniquenessresultdoesnotincluderationalhomologyspheres,andthusoffers
no indication on how to approach this class of manifolds. We spend the last part of
thischapteranalyzingthisclassof3-manifolds.
In §3.8 we describe a very powerful method for computing the torsion of such
3-manifolds, based on the complex Fourier transform results in Chapter 1, and an
extremelyversatileholomorphicregularizationtechnique. Theseleadtoexplicitfor-
mulæfortheFouriertransformofthetorsionofarationalhomologysphereintermsof
surgerydata. Theseformulæstillhavethetwoexpectedambiguities: asignambiguity
andaspinc ambiguity. In§3.9wedescribeaverysimplealgorithmforremovingthe
spincambiguity. Thisrequiresaquitelongtopologicaldetourintheworldofquadratic
functionsonfiniteAbeliangroups, andsurgerydescriptionsofspin andspinc struc-
tures, but the payoff is worth the trouble. The sign ambiguity is finally removed in
§3.10 in the case of plumbed rational homology spheres, relying on an idea in [75],
basedontheFouriertransform,andarelationshipbetweenthetorsionandthelinking
formdiscoveredbyTuraev.
Chapter 4 discusses more analytic descriptions of the Reidemeister torsion: in
terms of gauge theory, in terms of Morse theory, and in terms of Hodge theory. We
discuss Meng–Taubes theorem and the improvements due to Turaev. We also out-
line our recent proof [83] of the extension of the Meng–Taubes–Turaev theorem to
rational homology spheres. As an immediate consequence of this result, we give a
new description of the Brumfiel–Morgan [7] correspondence for rational homology
3-sphereswhichassociatestoeachspinc structurearefinementofthelinkingform.
On the Morse theoretic side we describe Hutchings–Lee–Pajitnov results which
give a Morse theoretic interpretation of the Reidemeister torsion. We barely scratch
the Hodge theoretic approach to torsion. We only provide some motivation for the
ζ-functiondescriptionoftheanalytictorsionandtheCheeger–Müllertheoremwhich
identifiesthisspectralquantitywiththeReidemeistertorsion.
∗∗∗
Acknowledgements.1 IspentalmosttwoyearsthinkingabouttheseissuesandIwas
helpedalongthewaybymanypeople. IamgreatlyindebtedtoFrankConnollywho
1ThisworkwaspartiallysupportedbyNSFgrantDMS-0071820.