Algebraic Topology: The Abel Symposium 2007

417 Pages·2009·8.16 MB·English

by  John C. Baez

ABEL SYMPOSIA
Edited by the Norwegian Mathematical Society
·
Nils A. Baas Eric M. Friedlander
·
Bjørn Jahren Paul Arne Østvær
Editors
Algebraic Topology
The Abel Symposium 2007
Proceedings of the Fourth Abel Symposium,
−
Oslo, Norway, August 5 10, 2007
Editors
NilsA.Baas EricM.Friedlander
DepartmentofMathematics DepartmentofMathematics
NTNU UniversityofSouthernCalifornia
7491Trondheim 3620VermontAvenue
Norway LosAngeles,CA90089-2532
e-mail:[email protected] USA
e-mail:[email protected]
BjørnJahren PaulArneØstvær
DepartmentofMathematics DepartmentofMathematics
UniversityofOslo UniversityofOslo
P.O.Box1053Blindern P.O.Box1053Blindern
0316Oslo 0316Oslo
Norway Norway
e-mail:[email protected] e-mail:[email protected]
ISBN978-3-642-01199-3 e-ISBN:978-3-642-01200-6
DOI:10.1007/978-3-642-01200-6
SpringerDodrechtHeidelbergLondonNewYork
LibraryofCongressControlNumber:2009926010
(cid:2)c Springer-VerlagBerlinHeidelberg2009
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Preface to the Series
The Niels Henrik Abel Memorial Fund was established by the Norwegian gov-
ernment on January 1, 2002. The main objective is to honor the great Norwegian
mathematicianNielsHenrikAbelbyawardinganinternationalprizeforoutstand-
ing scientific work in the field of mathematics. The prize shall contribute towards
raising the status of mathematics in society and stimulate the interest for science
amongschoolchildrenandstudents.Inkeepingwiththisobjectivetheboardofthe
Abel fundhas decided to finance one or two Abel Symposiaeach year. The topic
may be selected broadly in the area of pure and applied mathematics. The Sym-
posiashouldbeatthehighestinternationallevel,andservetobuildbridgesbetween
thenationalandinternationalresearchcommunities.TheNorwegianMathematical
Societyisresponsiblefortheevents.Ithasalsobeendecidedthatthecontributions
fromthese Symposiashouldbepresentedina seriesofproceedings,andSpringer
Verlaghasenthusiasticallyagreedtopublishtheseries.TheboardoftheNielsHen-
rikAbelMemorialFundisconfidentthattheserieswillbeavaluablecontribution
tothemathematicalliterature.
RagnarWinther
ChairmanoftheboardoftheNielsHenrikAbelMemorialFund
v
Preface
The2007AbelSymposiumtookplaceattheUniversityofOslofromAugust5to
August10,2007.Theaimofthesymposiumwastobringtogethermathematicians
whoseresearcheffortshaveledtorecentadvancesinalgebraicgeometry,algebraic
-theory,algebraictopology,andmathematicalphysics.Thecommonthemeofthis
(cid:0)symposiumwasthedevelopmentofnewperspectivesandnewconstructionswitha
categoricalflavor.Asthe lecturesatthesymposiumandthepapersofthisvolume
demonstrate,theseperspectivesandconstructionshaveenabledabroadeningofvis-
tas, a synergybetweenonce-differentiatedsubjects, andsolutionsto mathematical
problemsbotholdandnew.
This symposium was organized by two complementary groups: an external
organizing committee consisting of Eric Friedlander (Northwestern,University of
Southern California), Stefan Schwede (Bonn) and Graeme Segal (Oxford) and a
local organizing committee consisting of Nils A. Baas (Trondheim), Bjørn Ian
Dundas(Bergen),BjørnJahren(Oslo)andJohnRognes(Oslo).
The webpage of the symposium can be found at http://abelsymposium.no/
symp2007/info.html
Theinterestedreaderwillfindtitlesandabstractsofthetalkslistedhere
Monday6th Tuesday7th Wednesday8th Thursday9th
09.30 10.30 F.Morel S.Stolz J.Lurie N.Strickland
11.00 12.00 M.Hopkins A.Merkurjev J.Baez U.Jannsen
13.30 14.20 R.Cohen M.Behrens H.Esnault C.Rezk
14.40 15.30 L.Hesselholt M.Rost B.Toe¨n M.Levine
16.30 17.20 M.Ando U.Tillmann D.Freed
17.40 18.30 D.Sullivan V.Voevodsky
aswellasafewonlinelecturenotes.Moreover,onewillfindhereusefulinformation
aboutourgracioushostcity,Oslo.
The present volume consists of 12 papers written by participants (and their
collaborators).Wegiveaverybriefoverviewofeachofthesepapers.
vii
viii Preface
“Theclassifyingspaceofatopological2-group”byJ.C.BaezandD.Stevenson:
Recentworkinhighergaugetheoryhasrevealedtheimportanceofcategorisingthe
theoryofbundlesandconsideringvariousnotionsof2-bundles.Thepresentpaper
givesasurveyonrecentworkonsuchgeneralizedbundlesandtheirclassification.
“String topology in dimensions two and three” by M. Chas and D. Sullivan
describessomeapplicationsofstringtopologyinlow dimensionsforsurfacesand
3-manifolds.TheauthorsrelatetheirresultstoatheoremofW.JacoandJ.Stallings
and a group theoretical statement equivalent to the three-dimensional Poincare
conjecture.
Thepaper“Floerhomotopytheory,realizingchaincomplexesbymodulespectra,
andmanifoldswithcorners”,byR.Cohen,extendsideasoftheauthor,J.D.S.Jones
and G. Segal on realizing the Floer complex as the cellular complex of a natural
stable homotopy type. Crucial in their work was a framing condition on certain
moduli spaces, and Cohen shows that by replacing this by a certain orientability
withrespecttoageneralizedcohomologytheory ,thereisanaturaldefinitionof
(cid:0)
Floer –homology. (cid:2)
In“(cid:2)R(cid:0)elativeCherncharactersfornilpotentideals”byG.Cortin˜asandC.Weibel,
theequalityoftworelativeCherncharactersfrom -theorytocyclichomologyis
shown in the case of nilpotent ideals. This import(cid:0)ant equality has been assumed
withoutproofinvariouspapersinthepast20years,includingrecentinvestigations
ofnegative -theory.
H. Esnau(cid:0)lt’s paper “Algebraic differential characters of flat connections with
nilpotent residues” shows that characteristic classes of flat bundles on quasi-
projective varieties lift canonically to classes over a projective completion if the
localmonodromyatinfinityisunipotent.Thisshouldfacilitatethecomputationsin
somesituations,foritissometimeseasiertocomputesuchcharacteristicclassesfor
flatbundlesonquasi-projectivevarieties.
“NormvarietiesandtheChainLemma(afterMarkusRost)”byC.Haesemeyer
and C. Weibel gives detailed proofs of two results of Markus Rost known as the
“Chain Lemma”and the “Normprinciple”.The authorsplace these two resultsin
thecontextoftheoverallproofoftheBloch–Katoconjecture.Theproofsarebased
onlecturesbyRost.
In“OntheWhiteheadspectrumof thecircle”L. Hesselholtextendsthe known
computations of homotopy groups WhTop of the Whitehead spectrum of
(cid:2)
thecircle.Thisproblemisoffundam(cid:3)e(cid:0)n(cid:4)talimp(cid:4)o(cid:5)rta(cid:6)(cid:6)nceforunderstandingthehome-
omorphismgroupsofmanifolds,inparticularthoseadmittingRiemannianmetrics
of negative curvature. The author achieves complete and explicit computations
for .
J(cid:7).F. J(cid:8)ardine’s paper “Cocycle categories” presents a new approach to defining
(cid:0)
andmanipulatingcocyclesinrightpropermodelcategories.Thesecocyclemethods
provide simple new proofs of homotopy classification results for torsors, abelian
sheaf cohomology groups, group extensions and gerbes. It is also shown that the
algebraicK-theorypresheafofspacesisasimplicialstackassociatedtoa suitably
definedparabolicgroupoid.
Preface ix
“Asurveyofellipticcohomology”byJ.Lurieisanexpositoryaccountoftherela-
tionship between elliptic cohomologyand derived algebraic geometry.This paper
lies attheintersectionofhomotopytheoryandalgebraicgeometry.Precursorsare
kept to a minimum, making the paper accessible to readers with a basic back-
groundinalgebraicgeometry,particularlywiththetheoryofellipticcurves.Amore
comprehensiveaccountwithcompletedefinitionsandproofs,willappearelsewhere.
“On Voevodsky’s algebraic -theory spectrum” by I. Panin, K. Pimenov and
O. Ro¨ndigs resolves some -th(cid:0)eoreticquestionsin the modernsetting of motivic
homotopy theory. They sho(cid:0)w that the motivic spectrum , which represents
algebraic -theoryinthemotivicstablehomotopycategory(cid:9),h(cid:10)a(cid:11)sauniqueringstruc-
tureover(cid:0)theintegers.Forgeneralbaseschemesthisstructurepullsbacktogivea
distinguishedmonoidalstructurewhichtheauthorshaveemployedintheirproofof
amotivicConner–Floydtheorem.
The paper “Chern character, loop spaces and derived algebraic geometry”
authored by B. Toe¨n and G. Vezzosi presents work in progress dealing with a
categorised version of sheaf theory. Central to their work is the new notion of
“derivedcategoricalsheaves”,whichcategorisesthenotionofcomplexesofsheaves
ofmodulesonschemes.Ideasoriginatinginderivedalgebraicgeometryandhigher
categorytheoryareusedtointroduce“derivedloopspaces”andtoconstructaChern
characterforcategoricalsheaveswithvaluesincyclichomology.Thisworkcanbe
seenasanattempttodefinealgebraicanalogsofelliptic objectsandcharacteristic
classes.
“Voevodsky’slecturesonmotiviccohomology2000/2001”consistsoffourparts,
each dealingwith foundationalmaterialrevolvingaroundthe proofof the Bloch–
Kato conjecture. A motivic equivariant homotopy theory is introduced with the
specificaimofextendingnon-additivefunctors,suchassymmetricproducts,from
schemestothemotivichomotopycategory.ThetextiswrittenbyP.Deligne.
We gratefully acknowledge the generous support of the Board for the Niels
Henrik Abel Memorial Fund and the Norwegian Mathematical Society. We also
thank Ruth Allewelt and Springer-Verlagfor constantencouragementand support
duringtheeditingoftheseproceedings.
Trondheim,LosAngelesandOslo NilsA.Baas
March2009 EricM.Friedlander
BjørnJahren
PaulArneØstvær
Contents
TheClassifyingSpaceofaTopological2-Group 1
JohnC.BaezandDannyStevenson (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
StringTopologyinDimensionsTwoandThree 33
MoiraChasandDennisSullivan (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
Floer HomotopyTheory, Realizing Chain Complexesby
ModuleSpectra,andManifoldswithCorners 39
RalphL.Cohen (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
RelativeChernCharactersforNilpotentIdeals 61
G.Cortin˜asandC.Weibel (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
AlgebraicDifferentialCharactersofFlat
ConnectionswithNilpotentResidues 83
He´le`neEsnault (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
NormVarietiesandtheChainLemma(AfterMarkusRost) 95
ChristianHaesemeyerandChuckWeibel (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
OntheWhiteheadSpectrumoftheCircle 131
LarsHesselholt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
CocycleCategories 185
J.F.Jardine (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
ASurveyofEllipticCohomology 219
J.Lurie (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
OnVoevodsky’sAlgebraic -TheorySpectrum 279
IvanPanin,KonstantinPimen(cid:0)ov,andOliverRo¨ndig(cid:12)(cid:12)s(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
xi
xii Contents
Chern Character, Loop Spaces and Derived Algebraic
Geometry 331
BertrandTo(cid:12)e¨(cid:12)n(cid:12)(cid:12)a(cid:12)n(cid:12)d(cid:12)(cid:12)G(cid:12)(cid:12)a(cid:12)b(cid:12)(cid:12)r(cid:12)ie(cid:12)(cid:12)le(cid:12)(cid:12)V(cid:12)(cid:12)e(cid:12)z(cid:12)z(cid:12)o(cid:12)s(cid:12)i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
Voevodsky’sLecturesonMotivicCohomology2000/2001 355
PierreDeligne (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)