Global homotopy theory

847 Pages·2018·3.374 MB·English

by  Schwede, Stefan

GlobalHomotopyTheory
Equivariant homotopy theory started from geometrically motivated questions about
symmetriesofmanifolds.Severalimportantequivariantphenomenaoccurnotjustfor
a particular group, but in a uniform way for all groups. Prominent examples include
stable homotopy, K-theory or bordism. Global equivariant homotopy theory studies
such uniform phenomena, i.e., universal symmetries encoded by simultaneous and
compatibleactionsofallcompactLiegroups.
Thisbookintroducesgraduatestudentsandresearcherstoglobalequivarianthomo-
topy theory. The framework is based on the new notion of global equivalences for
orthogonalspectra,amuchfinernotionofequivalencethanistraditionallyconsidered.
Thetreatmentislargelyself-containedandcontainsmanyexamples,makingitsuitable
as a textbook for an advanced graduate class. At the same time, the book is a
comprehensiveresearchmonographwithdetailedcalculationsthatrevealtheintrinsic
beautyofglobalequivariantphenomena.
Stefan Schwede is Professor in the Mathematical Institute at the University of
Bonn. His main area of expertise is algebraic topology, specifically stable homotopy
theory.
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Global Homotopy Theory
STEFAN SCHWEDE
RheinischeFriedrich-Wilhelms-Universita¨tBonn
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DOI:10.1017/9781108349161
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Contents
Preface pagevii
1 Unstableglobalhomotopytheory 1
1.1 Orthogonalspacesandglobalequivalences 2
1.2 Globalmodelstructurefororthogonalspaces 31
1.3 Monoidalstructures 54
1.4 Globalfamilies 64
1.5 Equivarianthomotopysets 77
2 Ultra-commutativemonoids 92
2.1 Globalmodelstructure 94
2.2 Globalpowermonoids 110
2.3 Examples 133
2.4 GlobalformsofBO 156
2.5 Globalgroupcompletionandunits 186
3 Equivariantstablehomotopytheory 227
3.1 Equivariantorthogonalspectra 228
3.2 TheWirthmu¨llerisomorphismandtransfers 261
3.3 Geometricfixedpoints 288
3.4 Thedoublecosetformula 305
3.5 Products 332
4 Globalstablehomotopytheory 348
4.1 Orthogonalspectraasglobalhomotopytypes 349
4.2 Globalfunctors 367
4.3 Globalmodelstructuresfororthogonalspectra 385
4.4 Triangulatedglobalstablehomotopycategories 409
4.5 Changeoffamilies 430
5 Ultra-commutativeringspectra 461
5.1 Poweroperations 462
v
vi Contents
5.2 Comonadicdescriptionofglobalpowerfunctors 481
5.3 Examples 506
5.4 Globalmodelstructure 524
6 GlobalThomandK-theoryspectra 543
6.1 GlobalThomspectra 544
6.2 Equivariantbordism 584
6.3 ConnectiveglobalK-theory 626
6.4 PeriodicglobalK-theory 661
AppendixA Compactlygeneratedspaces 687
AppendixB Equivariantspaces 735
AppendixC Enrichedfunctorcategories 793
References 809
Indexofsymbols 819
Index 822
Preface
Equivariant stable homotopy theory has a long tradition, starting from
geometrically motivated questions about symmetries of manifolds. The
homotopy-theoretic foundations of the subject were laid by tom Dieck, Se-
gal and May and their students and collaborators in the 1970s, and over the
intervening decades equivariant stable homotopy theory has been very use-
fulforsolvingcomputationalandconceptualproblemsinalgebraictopology,
geometrictopologyandalgebraicK-theory.Variousimportantequivariantthe-
ories naturally exist not just for a particular group, but in a uniform way for
all groups in a specific class. Prominent examples of this are equivariant sta-
blehomotopy,equivariantK-theoryorequivariantbordism.Globalequivariant
homotopytheorystudiessuchuniformphenomena,i.e.,theadjective‘global’
referstosimultaneousandcompatibleactionsofallcompactLiegroups.
This book introduces a new context for global homotopy theory. Various
ways of providing a home for global stable homotopy types have previously
beenexploredin[100,Ch.II],[68,Sec.5],[18]and[19].Weuseadifferentap-
proach:weworkwiththewell-knowncategoryoforthogonalspectra,butuse
anotionofequivalence,theglobalequivalence,whichismuchfinerthanwhat
istraditionallyconsidered.Thebasicunderlyingobservationisthatanorthog-
onal spectrum gives rise to an orthogonalG-spectrum for every compact Lie
groupG, and the fact that all these individual equivariant objects come from
oneorthogonalspectrumimplicitlyencodesstrongcompatibilityconditionsas
thegroupGvaries.AnorthogonalspectrumthushasG-equivarianthomotopy
groupsforeverycompactLiegroup,andaglobalequivalenceisamorphism
oforthogonalspectrathatinducesisomorphismsforallequivarianthomotopy
groupsforallcompactLiegroups(basedon‘completeG-universes’,compare
Definition4.1.3).
The structure of the equivariant homotopy groups of an orthogonal spec-
trum gives an idea of the information encoded in a global homotopy type in
our sense: the equivariant homotopy groups πG(X) are contravariantly func-
k
vii
viii Preface
torialforcontinuousgrouphomomorphisms(‘restrictionmaps’),andtheyare
covariantlyfunctorialforinclusionsofclosedsubgroups(‘transfermaps’).The
restrictionandtransfermapsenjoyvarioustransitivitypropertiesandinteract
via a double coset formula. This kind of algebraic structure has been studied
before under different names, e.g., ‘global Mackey functor’, ‘inflation func-
tor’, .... From a purely algebraic perspective, there are various parameters
herethatonecanvary,namelytheclassofgroupstowhichavalueisassigned
andtheclassesofhomomorphismstowhichrestrictionmapsortransfermaps
areassigned,andlotsofvariationshavebeenexplored.However,thedecision
toworkwithorthogonalspectraandequivarianthomotopygroupsoncomplete
universesdictatesacanonicalchoice:weproveinTheorem4.2.6thatthealge-
braofnaturaloperationsbetweentheequivarianthomotopygroupsoforthog-
onal spectra is freely generated by restriction maps along continuous group
homomorphisms and transfer maps along closed subgroup inclusion, subject
toexplicitlyunderstoodrelations.
WedefinetheglobalstablehomotopycategoryGH bylocalizingthecate-
gory of orthogonal spectra at the class of global equivalences. Every global
equivalenceisinparticularanon-equivariantstableequivalence,sothereisa
‘forgetful’ functor U: GH −→ SH on localizations, where SH denotes the
traditional non-equivariant stable homotopy category. By Theorem 4.5.1 this
forgetfulfunctorhasaleftadjoint LandarightadjointR,bothfullyfaithful,
whichparticipateinarecollementoftriangulatedcategories:
(cid:4)(cid:4) i∗ (cid:4)(cid:4) L
GH+ i∗ (cid:2)(cid:2)GH U (cid:2)(cid:2)SH .
(cid:3)(cid:3) (cid:3)(cid:3)
i! R
HereGH+denotesthefullsubcategoryoftheglobalstablehomotopycategory
spannedbytheorthogonalspectrathatarestablycontractibleinthetraditional,
non-equivariantsense.
Theglobalspherespectrumandsuspensionspectraareintheimageofthe
left adjoint (Example 4.5.11). Global Borel cohomology theories are the im-
age of the right adjoint (Example 4.5.19). The ‘natural’ global versions of
Eilenberg–MacLanespectra(Construction5.3.8),Thomspectra(Section6.1),
ortopologicalK-theoryspectra(Sections6.3and6.4)arenotintheimageof
eitherofthetwoadjoints.PeriodicglobalK-theory,however,isrightinduced
fromfinitecyclicgroups,i.e.,intheimageoftheanalogousrightadjointfrom
anintermediateglobalhomotopycategoryGH basedonfinitecyclicgroups
cyc
(Example6.4.27).
Looking at orthogonal spectra through the eyes of global equivalences is
like using a prism: the latter breaks up white light into a spectrum of colors,
Preface ix
andglobalequivalencessplitatraditional,non-equivarianthomotopytypeinto
manydifferentglobalhomotopytypes.Thefirstexampleofthisphenomenon
that we will encounter refines the classifying space of a compact Lie group
G.Ontheonehand,thereistheconstantorthogonalspacewithvalueanon-
equivariantmodelfor BG;andthereistheglobalclassifyingspace B G (see
gl
Definition1.1.27).Theglobalclassifyingspaceisanalogoustothegeometric
classifyingspaceofalinearalgebraicgroupinmotivichomotopytheory[123,
4.2],anditisthecounterparttothestackofG-principalbundlesintheworld
ofstacks.
Anothergoodexample isthesplittingupofthenon-equivariant homotopy
type of the classifying space of the infinite orthogonal group. Again there is
the constant orthogonal space with value BO, the Grassmannian model BO
(Example 2.4.1), a different Grassmannian model bO (Example 2.4.18), the
bar construction model B◦O (Example 2.4.14), and finally a certain ‘cofree’
orthogonalspaceR(BO).TheorthogonalspacebOisalsoahomotopycolimit,
asngoestoinfinity,oftheglobalclassifyingspacesB O(n).Wediscussthese
gl
differentglobalformsof BOinsomedetailinSection2.4,andtheassociated
ThomspectrainSection6.1.
In the stable global world, every non-equivariant homotopy type has two
extremeglobalrefinements,the‘leftinduced’(theglobalanalogofaconstant
orthogonalspace,seeExample4.5.10)andthe‘rightinduced’globalhomotopy
type (representing Borel cohomology theories, see Example 4.5.19). Many
important stable homotopy types have other natural global forms. The non-
equivariant Eilenberg–MacLane spectrum of the integers has a ‘free abelian
groupfunctor’model(Construction5.3.8),andanotherincarnationistheEilen-
berg–MacLanespectrumoftheconstantglobalfunctorwithvalueZ(Remark
4.4.12).ThesetwoglobalrefinementsoftheintegralEilenberg–MacLanespec-
trum agree on finite groups, but differ for compact Lie groups of positive di-
mensions.
Asalreadyindicated,thereisagreatvarietyoforthogonalThomspectra,in
real(orunoriented)flavorsasmOandMO,ascomplex(orunitary)versions
mU and MU, and there are periodic versions mOP, MOP, mUP and MUP
ofthese;wediscussthesespectrainSection6.1.Thetheoriesrepresentedby
mO and mU have the closest ties to geometry; for example, the equivariant
homotopy groups of mO receive Thom–Pontryagin maps from equivariant
bordismrings,andtheseareisomorphismsforproductsoffinitegroupsandtori
(compareTheorem6.2.33).ThetheoriesrepresentedbyMOaretomDieck’s
homotopicalequivariantbordism,isomorphicto‘stableequivariantbordism’.
Connective topological K-theory also has two fairly natural global refine-
ments, in addition to the left and right induced ones. The ‘orthogonal sub-
space’ model ku (Construction 6.3.9) represents connective equivariant