From Differential Geometry to Non-commutative Geometry and Topology

406 Pages·2019·4.303 MB·English

by  Neculai S. Teleman

Neculai S. Teleman
From Diff erential
Geometry to
Non-commutative
Geometry and
Topology
From Differential Geometry to Non-commutative
Geometry and Topology
Neculai S. Teleman
From Differential Geometry
to Non-commutative
Geometry and Topology
NeculaiS.Teleman
DipartimentodiScienzeMatematiche
Universita`PolitecnicadelleMarche
Ancona,Italy
ISBN978-3-030-28432-9 ISBN978-3-030-28433-6 (eBook)
https://doi.org/10.1007/978-3-030-28433-6
MathematicsSubjectClassification(2010):53-XX
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Thisbookisa tributetothememoryof
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with deep esteem andgratitude.
Neculai S.Teleman
Foreword
Neculai S. Teleman takes the reader on a fascinating expedition exploring the
lands between the smooth and the continuous domains. For a very long time a
continuous function was assumed to be differentiable perhaps with the exception
ofafiniteornumerablebutdiscretesetofpoints.TheWeierstrassexample(1872)
of a continuous but nowhere differentiable function came as a great shock to
mathematicians of the nineteenth century. A similar attitude was prevalent in the
middle of the previous century when a topological manifold was considered to
carry the unique smooth structure inducing the initial topology of the manifold.
Then in 1956 John Milnor showed that on the 7-sphere there are several different
smooth exotic structures. But in 1963 Michael F. Atiyah and Isadore M. Singer
announcedtheirindextheoremwhichassertedthattheindexoftheLaplaceoperator
(associated with the smooth Riemannian metric) is equal to the topologicalindex
of the smooth manifold. Therefore, the index obtained from the data provided by
the smooth structure is the same as the one provided by the data derived from
the underlying topological structure. These two important and highly non-trivial
results turned attention of many a fine mathematician of the 1960s to the regions
where the smooth meets the continuous. A few years later (1965) S. P. Novikow
publishedhisfamousresultthattherationalPontrjaginclassesofasmoothmanifold
are topological invariants. By the end of the decade, it was known that rational
Pontrjaginclasses canbedefinedforanytopologicalmanifold(R.KirbyandL.C.
Siebelmann,1969).
In1971,I.M.SingerinFutureextensionsofindextheoryandellipticoperators
wrote
I was asked to speak on “possible future developments, give conjectures, and speculate
about futureadvances.” ....probablyatthisverymomentagraduatestudent isbusilyat
workonatheoremthatmightchangepresenttrendsdrastically.However,inalimitedway,
I will try and fulfill the charge – at least within the context of index theory for elliptic
operators.Mymainthemeisthis:byandlargethistheoryassignsanintegertoanelliptic
operatoronacompactsmoothmanifold;Iexpectextensionsofthepresenttheorytonew
situations; i.e., non-smooth manifolds, non-manifolds of special type, and to a context
whereitisnaturalthatintegerbereplacedbyarealnumber.
vii
viii Foreword
A fewyearslater NeculaiS. TelemanarrivedatMIT andin 1977defendedhis
Ph.D. thesis in which he extended the index theorem to combinatorialmanifolds.
Ever since he dedicated all his efforts to developing new tools and results which
permit to extend the validity of the index theorem such as combinatorial Hodge
theoryonPLmanifolds,(TAMS,1979,Inv.Math.,1980),“ellipticmachinery”and
signatureoperatorsonLipschitzmanifolds(Publ.IHES,1983).Thefollowingyear
hepublishedaversionoftheindextheoremforgeneraltopologicalmanifoldsusing
new tools, i.e. Kasparov’s realisation of the analytic K-homology group K (M)
0
andFredholmoperatorsonnon-compactmanifolds.Hisquesttogeneralisevarious
“smooth”constructionsledhimtousetheexistenceofthequasiconformalstructure
onanytopologicalmanifoldwhichfinallyturnedhisattentiontothetechniquesand
ideasofnon-commutativegeometry(1991).
The reader of the book is asked to embark together with the author on this
fascinating journey which leads to the boundaries of our topological knowledge.
NeculaiS.Telemanintroducesthereaderstepbysteptothebasicandthenadvanced
tools of this fascinating theory: classical differential geometry, e.g. Riemannian
metrics,Laplacian,Hodgetheoryandthesmoothversionoftheindextheorem.Then
to go beyondthat the basic elements of non-commutativegeometryare presented
asamethodofdelocalisationofourmathematicalperceptionandnaturalextension
of the classical geometricaltools. Havingcoveredthe knownregionsof geometry
and topology of manifolds, the knowledgeableand experienced guide shows new
fascinatingvistasofnon-commutativetopology.
It is a veryimportantvolumewhich gathersand presentsin an orderlymanner
thestoryoftheindextheorem.Thetoolsandtheresultsarescatteredinnumerous
publications.Buthavingtheminoneplaceisnottheonlyadvantageofthisbook.We
getmuchmoresincethenarrativeisbothobjectiveandpassionateatthesametime
asthenarratorisoneoftheprincipalprotagonistsinthisfascinatingmathematical
epic.
Kraków,Poland RobertWolak
April2019
Preface
This book combines some of the author’s research contributions in index theory
withelementsofnon-commutativegeometry.Thebookendswithnon-commutative
topology.
Thisworkshowsthattheindexformulaisatopologicalstatement;ontheother
hand, this book gives a re-formulation of the index formula. The re-formulation
shouldhaveimportantconsequences.
From the non-commutative geometry side, this book explains how differ-
ent results in index theory, obtained by classical geometry methods (Riemann–
Roch, Hirzebruch [32], Atiyah and Singer [53], Teleman [83, 84]) and by non-
commutative geometry (Connes [82], Connes and Moscovici [96], Connes et
al. [101]), are connected. This has to do with the possibility, offered by different
theories,toexhibit/describethetopologicalindexofellipticoperators.Thepassage
fromclassicaldifferentialgeometrytonon-commutativegeometry,exemplifiedby
theindexformuladoneinthisbook,describesalsothewayhowtheauthorbecame
interestedin,andapproached,non-commutativegeometry.Forthisreason,thisbook
doesnotintendtogiveacompletereviewofallitsresultsortogiveacompletelist
of its contributors. This book offers the reader a natural path starting with basic
problems of differential geometry and leading to non-commutativegeometry and
topology.However,althoughthisbookdoesnotintendtopresenttheverysignificant
developments of non-commutative geometry in all directions, it accompanies the
readertowardspresentresearchproblemsinindextheory.
Thisbookhasmultiplegoals:
1. TheauthordidhisPh.D.withProf.I.M.Singer(1977)workingonextendingthe
indextheoremtocombinatorialmanifolds—aproblemthatwasformulatedbyI.
M. Singer in his programme article “Prospects of Mathematics” Symposium,
March 16–18, Princeton [58]. One goal of this book is to fill the existing gap
in the mathematicalliterature in reportingthe developmentswhich occurredin
index theory after I. M. Singer presented his 1970-research programme. This
reportiscertainlynotcomplete.
ix
x Preface
2. Itaimstoprovideafriendlyintroductiontonon-commutativegeometry.Itstudies
index theory from a classical differential geometry perspective up to the point
whereclassicaldifferentialgeometrymethodsbecomeinsufficient.
3. Anothergoalistopresentnon-commutativegeometryasanaturalcontinuation
ofclassicaldifferentialgeometry.Thisisbasedonthefollowingconcepts:
• Non-commutativegeometryisabstractindextheory.
• Non-commutativegeometry is a de-localised theory. In many cases the de-
∞
localisation occurs by replacing C -homomorphisms by operators on L .
2
ThischangereplacesexactrelationsbyL -localaverages.
2
• Non-commutative geometry is a multi-distribution theory. The multi-
distribution theory occurs by means of two processes: (1) polarisation of
algebraically higher order expressions; (2) each factor of the polarisation
becomesamono-distributiontheory.
• Hochschildhomology—anditsvariants—replacesthedeRhamhomology.
4. Beginning with Riemann and up to the present times, index theory has passed
through most chapters of mathematics: algebraic geometry, analytic geometry,
differential geometry, combinatorial geometry, Lipschitz and quasi-conformal
geometry as well as L -structures geometry. Another goal of this work is to
p
showthatwhiletheobjectsofnon-commutativegeometryofa(classical)space
X liveinallpowersXp ofX,classicaldifferentialgeometryobjectsassociated
withXliveonthemaindiagonalsofthosepowers.Thepossibilityofdescribing
an object classically depends on whether the non-commutative object has a
classicallimit,i.e.whetheritsrestrictionalongthemaindiagonalsaredefined;
thisdependsonwhethercertainproductsofdistributionscanbeperformed.
Thebookhasthefollowingmainmathematicalobjectives:
(i) itintendstoprovideanaturallinkbetweenclassicaldifferentialgeometryand
non-commutativegeometry.
(ii) itpresentsthebasic algebraicstructureandresultswhichstayatthefounda-
tionsofnon-commutativedifferentialgeometry.
(iii) itsummarisesthebasicanalyticalstructuresontopologicalmanifolds.
(iv) itshowsthattheindextheoremisatopologicalstatement.
(v) it summarises the results in index theory obtained with non-commutative
geometrymethods—localindextheorems.
(vi) itproposesprospectsinindextheory.
(vii) itproposesnon-commutativetopology.
This book presents the mathematical facts in their chronological order most of
the time. Both the de Rham isomorphism theorem and the Hodge decomposition
theorem show that real cohomology may be extracted analytically from differen-
tiablestructuresontopologicalmanifolds.Theseresultsconstitutetwoofthebasic
linksbetweendifferentiablestructuresandtopologicalstructures.Given,however,
that not all topological manifolds possess differentiable structures, and that the
sametopologicalstructuremightbecompatiblewithnon-equivalentdifferentiable
Preface xi
structures, there is a fundamental question to ask whether these relations could
be derived purely in topological terms, on all topological manifolds. S. Novikov
proved the fundamental result [41] stating that the rational Pontrjagin classes
of smooth manifolds are topological invariants. Kirby-Siebenmann results [56]
complete Novikov’s theorem by stating that all topological manifolds possess
rational Pontrjagin classes. Going along the same line of ideas, the following
fundamentalquestionarose:whethertheHodgedecompositiontheoremandindex
formulacouldbestatedanalltopologicalmanifolds(seee.g.,[58]).
This fundamental question was answered using Sullivan’s fundamental result
[75], which states that any topological manifold of dimension (cid:3)= 4 possesess
a Lipschitz or quasi-conformal structure, unique up to Lipschitz/quasi-conformal
isotopy. The first results on Hodge theory on combinatorial manifolds are due
to Teleman [72, 76–78] and to Cheeger [79] on pseudo-manifolds. These results
are successively extended using Sullivan’s result on Lipschitz manifolds[83, 84].
SuccessiveworkduetoDonaldsonandSullivan[92],Connesetal.[101]extendsthe
indexformulaonquasi-conformalmanifolds.Thisbooksummarisestheseresults.
The first problem the author of this book undertook in his Ph.D. thesis was
extending the Hodge decomposition theorem from smooth manifolds to pseudo-
manifolds,see Teleman[72] andSinger[58].Atthattime (1976)thissubjectwas
totallyunexplored.Infact,I.M.Singer’sprogramme[58],publishedin1971,poses
thefollowingtwoquestionstoo:
A “If M is a PL manifold,the L-polynomialsare still well defined(Thom[25]),
and one can define from them the rational Pontrjagin classes. The Hirzebruch
signature theorem still holds. Is there an associated elliptic operator (as in the
smoothcase)whoseindexisthesignature?Onwhatspacesdoesitoperate?Does
ithaveasymbolandwheredoesthesymbollie?”
B “...Are presentanalytictechniquesstrongenoughtoprovetheHodgetheorem
forsmoothmanifoldsbyapproximationsviacombinatorialLaplaciansassociated
with smooth triangulations? If so, this might shed some light on the PL-
problem.”
Looking retrospectively, it is important to remark that extending the Hodge
isomorphismtheoremandtheindexformulatospaceswhicharenotdifferentiable
manifoldswasadaringtask;theproofoftheHodgeisomorphismtheorem(seede
Rham[86]) andthe proofofthe Atiyah–Singerindexformula(see Palais seminar
[42]andAtiyahandSinger[53])heavilyusedthedifferentiablestructure.
Combinatorialmanifoldsaretheclosestrelativesofsmoothmanifolds.Forthis
reason, the first problem addressed in trying to extend the index formula to more
generalcontextswasto investigatewhetherHodgetheorycouldbeextendedfrom
smoothmanifoldstocombinatorialmanifolds.
Given that the Laplace operator is defined in terms of a chosen Riemannian
metric on the manifold, it was crucial to understand in which respect the special
characteristicsoftheRiemannianmetricshouldbesatisfiedifaHodgetyperesult
is desired. The de Rham [86] proofof the Hodge decompositiontheorem showed
thatifonewishestoextendtheHodgetheorytocombinatorialmanifoldsinsucha