Algebraic Models for Equivariant Homotopy Theory over Abelian

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MathematischeZeitschriftmanuscriptNo.
(willbeinsertedbytheeditor)
Algebraic Models for Equivariant Homotopy Theory
over Abelian Compact Lie Groups
MichaelA.Mandell1!,LauraScull2
1 DepartmentofMathematics,UniversityofChicago,Chicago,IL,
e-mail:[email protected]
2 DepartmentofMathematics,TheUniversityofMichigan,AnnArbor,MI,
e-mail:[email protected]
Received:February5,2001/Revisedversion:May21,2001– c Springer-Verlag2007
!
Abstract Wedescribesomealgebraicmodelsforequivariantrationalandp-adic
homotopytheoryoverAbeliancompactLiegroups.
Mathematics Subject Classification (2000): Primary 55P91; Secondary 55P62,
55P92
Introduction
Equivariantalgebraictopologyusesdiscreteinvariantstostudyspaceswithaspec-
ifiedgroupofsymmetries,typicallyassumingthatthesymmetrygroupisbothLie
(foractionsonmanifoldsandrelatedspaces)andcompact(tomakethetheoryac-
cessible).Sincetheinvariantsarediscrete,theydependonlyonhomotopyclasses
ofG-maps.Thedomainofequivarianthomotopytheorythereforeisthehomotopy
categoryofG-spacesandthegoalistodescribethiscategoryinalgebraicterms.
Inthispaper,wedescribealargepartofequivarianthomotopytheoryinalgebraic
termswhenthecompactLiegroupGisAbelian.
Westateourresultsinthelanguageoflocalization.AmapofG-spacesX Y
→
is an equivariant weak equivalence when the induced map on H-fixed spaces
XH YH is a (non-equivariant) weak equivalence for every closed subgroup
→
H G. An equivariant homotopy equivalence is therefore an equivariant weak
⊆
equivalence, and a fundamental result is that an equivariant weak equivalence is
anequivarianthomotopyequivalenceprovidedtheG-spacesaresuitablynice,that
is,homotopyequivalenttoG-complexes.MostG-spacesofgeometricinterest(in-
cluding all G-manifolds) satisfy this. If we localize the category of G-spaces by
formallyinvertingtheequivariantweakequivalences,theresultiscalledtheequiv-
arianthomotopycategory.
! ThefirstauthorwassupportedinpartbyNSFpostdoctoralfellowshipDMS9804421.
2 MichaelA.Mandell,LauraScull
We can also consider weaker notions of equivalence: A G-map X Y is
→
an equivariant rational or p-adic equivalence if each induced map XH YH
→
isa(non-equivariant)rationalorp-adicequivalence,thatis,ahomologyisomor-
phism with rational or Z/pZ coefficients. The category obtained from G-spaces
byformallyinvertingtheequivariantrationalorp-adicequivalencesiscalledthe
equivariantrationalorp-adichomotopycategory.
The equivariant rational and p-adic homotopy categories fracture equivariant
homotopytheoryintoarationalpieceandapieceforeachprimep.Bythe(non-
equivariant)Whiteheadtheorem,aG-mapX Y betweenG-spaceswithsimply
→
connected fixed-point spaces is an equivariant weak equivalence if and only if it
isanequivariantrationalequivalenceandequivariantp-adicequivalenceforallp.
Justasinnon-equivarianthomotopytheory,muchofequivarianthomotopytheory
can be recovered from rational homotopy theory, p-adic homotopy theory, and
patchinginformation.
We describe algebraic models for rational and p-adic equivariant homotopy
theory.Intherationalcase,ourdescriptionisintermsofdiagramsofcommutative
differential graded Q-algebras (CDGA’s) of a particular shape. In Section 2, we
describeacategory closelyrelatedtothelatticeofclosedsubgroupsofG.We
D
definea -CDGAtobeafunctorfrom intothecategoryofcommutativediffer-
D D
entialgradedQ-algebras.WedefineacontravariantfunctorA fromthecategory
0
of G-spaces to the category of -CDGAs; the value at each object of is the
D D
Thom–SullivanDeRhamalgebraofthesingularcomplexofaBorelconstruction
onafixedpointspaceofthegivenG-space;seeSection3fordetails.WeletP
0
beacofibrantapproximationofA ( ),where istheone-pointG-space;essen-
0 ∗ ∗
tially,thisisaminimalmodelforA ( ).ThenwecanregardA asafunctorfrom
0 ∗ 0
G-spaces to the category of -CDGAs under P . We will see that A converts
D 0 0
equivariant rational equivalences to (objectwise) quasi-isomorphisms, and so A
0
passestoafunctoronthehomotopycategories.Intherationalcase,ourmainresult
isthefollowing.
Theorem A. Let G be an Abelian compact Lie group. There is a functor A
0
fromtheequivariantrationalhomotopycategorytothehomotopycategoryof -
D
CDGAsunderP .OnthefullsubcategoryofG-simplyconnectedG-finiteQ-type
0
G-spaces,thisfunctorisfullandfaithful.
In the p-adic case, we use E F¯ -algebras in place of rational differential
p
∞
graded Q-algebras, and the cochain complex in place of the DeRham complex,
butotherwiseessentiallythesameoutlineholds.Weprovethefollowingtheorem.
Theorem B. Let G be an Abelian compact Lie group. There is a functor A
p
from the equivariant p-adic homotopy category to the homotopy category of -
D
E DGAs under P . On the full subcategory of G-simply connected G-finite p-
p
∞
typeG-spaces,thisfunctorisfullandfaithful.
Here G-simply connected means that each fixed-point space is simply con-
nectedinthenon-equivariantsense(andmaybeemptyornon-connected).Like-
wise, G-finite Q- or p- type means that each fixed-point space is (non-equivar-
iantly) finite Q- or p- type, that is, its homology with coefficients in Q or Z/pZ
AlgebraicModelsforEquivariantHomotopyTheory 3
isfinitelygeneratedineachdegree.Inthecaseofeachofthesetheorems,wecan
describeinhomologicaltermstheimageoftheG-simplyconnectedG-finitetype
G-spaces;seeSection3fordetails.
Unfortunately,wearelesssuccessfulatdescribingtheobjectsP andP ,and
0 p
so the models obtained from our main theorems are not very explicit in general.
When dimG = 0, that is, when G is a finite group, we can take P to be the
0
constantdiagramonQintherationalcaseandPptobetheconstantdiagramonF¯p
inthep-adiccase.Themodelsweobtainthiswayaremorecomplicatedthanthe
onesdescribedin[11](rationally)andin[7](p-adically),wherethemethodsapply
moregenerallytonon-Abelianfinitegroups.InthecasewhenGisthecirclegroup
T, we can also describe our rational models explicitly; we do this in Section 4.
Thisisthealgebraiccategorydescribedinthesecondauthor’s1999Universityof
Chicagothesis[9].
1 DiagramsofSpaces
Inordertoproducealgebraicmodels,wefirstreduceequivarianthomotopytheory
to the study of diagrams of spaces and fibrations, and then apply the algebraic
modelsof[10]and[6].Historicallythefirstdescriptionofequivarianthomotopy
theory in terms of diagrams is the theorem of Elmendorf [5] that explains how
to recover a G-space X up to equivariant weak equivalence from its system of
fixed-pointspacesXH andinclusionrelations.
To make this precise, consider the orbit category of G, whose objects
G
O
arethecanonicalorbitsG/H forallclosedsubgroupsH G,andwhosemaps
⊆
areequivariantmapsbetweenthem.Anymapin (G/H,G/K)isoftheform
G
O
gH gaKforsomea Gwithsuchthata 1Ha K,andmoreoverthataand
−
→ ∈ ⊆
brepresentthesamemapifandonlyifab 1 K.Sothemapsinthiscategory
−
∈
aregivenby
(G/H,G/K)=(G/K)H.
OG ∼
Weregard asatopologicalcategory,topologizingthemapsbytheaboveiso-
G
O
morphism.
AssociatedtoanyG-spaceX thereisacontravariantfixedpointfunctorΦX
from tospaces.ThisfunctorhasthevalueΦX(G/H)=XH,withmorphisms
G
O
induced by the G-action on the space X. This functor is continuous in the sense
thatthemap
(G/K,G/H) ΦX(G/H) ΦX(G/K)
G
O × →
is a continuous map for every pair of objects G/K, G/H of . Motivated by
G
O
this example we define the category of -spaces to be the category of
G G
O U O
continuous contravariant functors from to spaces. We define weak equiva-
G
O
lence, rational equivalence, and p-adic equivalence in objectwise; we say
G
O U
that a map X Y in is a weak (resp. rational, p-adic) equivalence if
G
→ O U
X(G/H) Y(G/H) is a non-equivariant weak (resp. rational, p-adic) equiva-
→
lenceforeveryobjectG/H of .Wedefinethehomotopycategory,therational
G
O
homotopycategory,andthep-adichomotopycategoryof -spacesbyformally
G
O
4 MichaelA.Mandell,LauraScull
inverting the weak equivalences, rational equivalences, and p-adic equivalences
respectively.
We can interpret Φ as a functor from the category of G-spaces to . The
G
O U
earlier definitions immediately imply that a map of G-spaces X Y is an
→
equivariant weak, rational, or p-adic equivalence if and only if the induced map
ΦX ΦY isaweak,rational,orp-adicequivalencein respectively.Itfol-
G
→ O U
lowsthatΦfactorsthroughthehomotopycategoriesweobtainbyinvertingthese
equivalences. The following result shows thatΦX contains exactly the same ho-
motopyinformationastheoriginalspaceX.
Theorem1.1.(Elmendorf)ThefunctorΦinducesanequivalenceof:
(i)Theequivarianthomotopycategoryandthehomotopycategoryof -spaces,
G
O
(ii)Theequivariantrationalhomotopycategoryandtherationalhomotopycate-
goryof -spaces,and
G
O
(iii)Theequivariantp-adichomotopycategoryandthep-adichomotopycategory
of -spacesforeachprimep.
G
O
Proof. Elmendorf [5] constructs a functor C from -spaces to G-spaces, and
G
O
naturalweakequivalencesCΦ IdandΦC Id.
→ →
For reasons explained in the next section, we find it convenient to work with
a variant of the orbit category that we call the subdivided orbit category ∆ .
G
O
Although Elmendorf’s theorem holds very generally, here we need to restrict to
thecasewhenGisAbelian.Inthiscase,theorbitG/Khasanactionbythegroup
G/K. The idea for the category ∆ is to have many objects corresponding to
G
O
each orbit G/K, one for each group G/H that can act on it. We use the symbol
G/K[H]todenotetheorbitG/KthoughtofasaG/Hspace.Followingthisidea,
for maps, we have ∆ (G/K[H],G/K[H]) = G/H. As K varies, we have a
G
O
canonicalquotientmapG/K [H] G/K[H]wheneverK K.Inaddition,as
# #
→ ⊆
H varies, we have a canonical group-change map G/K[H] G/K[H ] when-
#
→
ever H H. To understand the variance in H (which may appear backwards),
#
⊆
observethatforacontravariantfunctorX,themapX(G/K[H ]) X(G/K[H])
#
→
goesfromaspacewithalargergroupactingtoaspacewithasmallergroupacting.
Thisleadstotheprecisedefinitionof the category∆ .Weemphasizethatthe
G
O
varianceinH isoppositetothatofK.
Definition1.2.Let ∆ G be the category that has one object G/K[H] for each
O
pairofclosedsubgroupsH K.Mapsaredefinedby
⊆
G/H ifH H K K
1 2 1 1 2
∆ (G/K [H ],G/K [H ])= ⊆ ⊆ ⊆
G 1 1 2 2
O ! otherwise
∅
Composition of maps is induced by multiplication in G. We define the category
∆ of ∆ -spaces to be the category of continuous contravariant functors
G G
O U O
from∆ tospaces.
G
O
AlgebraicModelsforEquivariantHomotopyTheory 5
Againwedefineweakequivalences,rationalequivalences,andp-adicequiva-
lencesobjectwise,andformthehomotopycategory,rationalhomotopycategory,
and p-adic homotopy category of ∆ -spaces by formally inverting the corre-
G
O
spondingequivalences.
Regarding the object G/K[H] as the orbit G/K with a G/H-action defines
a (covariant) functor from ∆ to : The functor sends the object G/K[H]
G G
O O
toG/K,andsendsthemorphismspace∆ (G/K [H ],G/K [H ]) = G/H
G 1 1 2 2 1
O
to the morphism space (G/K ,G/K ) = G/K via the quotient map when
G 1 2 2
O
H H K K .Composingwiththefunctor∆ ,wetherefore
2 1 1 2 G G
⊆ ⊆ ⊆ O →O
obtainafunctorI from to∆ .ClearlyI preservesweakequivalences,
G G
O U O U
rational equivalences, and p-adic equivalences, and so it passes to the categories
obtainedbyformallyinvertingtheseequivalences.InSection7,weshow:
Theorem1.3.ThefunctorI inducesfullembeddingsof:
(i)The homotopy category of -spaces in the homotopy category of ∆ -
G G
O O
spaces,
(ii)Therationalhomotopycategoryof -spacesintherationalhomotopycate-
G
O
goryof∆ -spaces,and
G
O
(iii)Thep-adichomotopycategoryof -spacesinthep-adichomotopycategory
G
O
of∆ -spacesforeachprimep.
G
O
In order to understand the embeddings, we also want to identify the image
of I. If we start with an object X in , then by definition IX(G/K[H]) =
G
O U
X(G/K)foreveryH K.DefinethecategoryIm∆ tobethefullsubcategory
⊆
of the homotopy category of ∆ consisting of those objects for which any
G
O U
mapX(G/K[H]) X(G/K[K])isaweakequivalenceforallsubgroupsH
→ ⊆
K; likewise define Im∆ and Im∆ to be the respective full subcategories of the
0 p
rationalandp-adichomotopycategoriesof∆ consistingofthoseobjectsfor
G
O U
whichanymapX(G/K[H]) X(G/K[K])isarationalandp-adicequivalence.
→
InSection7,weprove:
Theorem1.4.AnobjectXinthehomotopycategory,rationalhomotopycategory,
or p-adic homotopy category of ∆ -spaces is isomorphic to an object in the
G
O
imageofI ifandonlyifX isinIm∆,Im∆,orIm∆respectively.
0 p
Thus, we have equivalences of categories between the equivariant homotopy
categoryandIm∆,theequivariantrationalhomotopycategoryandIm∆,andthe
0
equivariantp-adichomotopycategoryandIm∆.
p
2 DiagramsofBundles
Sincethecategories and∆ aretopologized, -spacesand∆ -spaces
G G G G
O O O O
cannotbeuseddirectlytoproducealgebraicmodelsunlessGisfinite.Dwyerand
Kan[3]showedthatequivarianthomotopytheorycanbereducedtothetheoryof
diagramsonacertaindiscretecategoryandthetheoryoffibrations.Inthissection,
wedescribeaslightlydifferentsuchreductionthatfollowssimilarideas.
6 MichaelA.Mandell,LauraScull
We say that a map of G-spaces X Y is an underlying weak equivalence
→
whenitisanon-equivariantweakequivalence.TheprojectionmapEG X
× →
X is an underlying weak equivalence, and so every G-space is underlying weak
equivalenttoafreeG-space.AbasictenetofbundletheoryisthataG-spaceX is
determineduptounderlyingweakequivalencebytheclassifyingmapq: EG
G
×
X BG.Precisely,X isunderlyingweakequivalenttotheG-spaceobtainedas
→
thepullbackalongqofthemapEG BG,asthisisagaintheG-spaceEG X.
→ ×
Inan -spaceX,theG/H-spaceX(G/H)isdeterminedonlyuptounderlying
G
O
weakequivalence,andthispieceofthestructurecanthereforeberecoveredfrom
aclassifyingmap.Thegoalistofittheseclassifyingmapstogetherinsuchaway
thatthewhole -spacecanberecovered.
G
O
Thefirstdifficultyisthatthedifferentobjectsinthe diagramhaveactions
G
O
of different groups. For an Abelian group G, the universal bundle E(G/H)
→
B(G/H) is covariant in G/H, whereas the spaces X(G/H) in an -space X
G
O
are contravariant. We have introduced the subdivided orbit category∆ in the
G
O
previous section precisely to deal with this problem. For a map G/K [H ]
1 1
→
G/K [H ],wehavecontravariancebothintheuniversalbundles
2 2
(E(G/H ) B(G/H )) (E(G/H ) B(G/H ))
1 1 2 2
→ ←− →
and the spaces X(G/K [H ]) X(G/K [H ]). In fact, in contrast to -
1 1 2 2 G
← O
spaces,wecandefinea∆ -spaceE by
G
O
(2.1) E(G/K[H])=E(G/H).
ThefinalmapE isaweakequivalence.
→∗
Beforeproceeding,weshouldnoteoneothercomplication.Wecanrecoverthe
G-spaceX uptounderlyingweakequivalencefromtheclassifyingmapEG
G
×
X BG, even if it is known only up to weak equivalence in the category of
→
spaces over BG. However, if we know the classifying map only up to rational
orp-adicequivalence,wecannotnecessarilyrecoverX uptounderlyingrational
or p-adic equivalence. An instructive example is the case of X = S2 with the
antipodalactionofG=Z/2Z.Here,themapS2 isnotarationalequivalence
→∗
but induces a rational equivalence on Borel constructions. The classifying map
thereforecannotdistinguishthefreeZ/2Z-spaceS2 fromthetrivialZ/2Z-space
.Thisproblemisdirectlyrelatedtothefactthatthegroupintheexampleisnot
∗
connected. When G is connected, BG is simply connected, and it follows from
the ideas of Eilenberg and Moore [4] that pullbacks along the fibration EG
→
BGpreservehomologyisomorphisms.Forthisreason,webaseourmainbundle
constructionsonanactionoftheidentitycomponentG andseparateouttheaction
e
of the finite group π G = G/G . Instead of working with the classifying space
0 e
BG, we work instead with the π G-space EG/G , which is (non-equivariantly)
0 e
equivalenttoB(G ).
e
Now we combine these two ideas to define a category of bundle diagrams
B
whosevarioushomotopycategoriesareequivalenttothecorrespondinghomotopy
categoriesof∆ -spaces.Motivatedbytheobservationsabove,thefunctorfrom
G
O
∆ -spacestothisnewcategorytakesX toadiagramofbundlesoftheform
G
O
E(G/H) X(G/K[H]) E(G/H)/(G/H) ,
×(G/H)e → e
AlgebraicModelsforEquivariantHomotopyTheory 7
inotherwords,oftheform
E(G/K[H]) X(G/K[H]) E(G/K[H]) .
×(G/H)e → ×(G/H)e ∗
Wewantourdiagramstobeindexedonadiscretecategory ,andexamininghow
D
thesebundlesfittogetherleadsustothefollowingdefinition.
Definition2.2.Let bethehomotopycategoryof∆ G:itsobjectsarethesame
D O
astheobjectsof∆ ,andmapsare
G
O
(G/K [H ],G/K [H ])=π (∆ (G/K [H ],G/K [H ]))=π (G/H )
1 1 2 2 0 G 1 1 2 2 0 1
D O
ifH H K K and otherwise.Wedefinethecategory of -spaces
2 1 1 2
⊆ ⊆ ⊆ ∅ DU D
tobethecategoryofcontravariantfunctorsfrom tospaces.
D
We emphasize that although is a homotopy category, a functor in has
D DU
codomainthecategoryofspacesasopposedtothehomotopycategoryofspaces.
Theprescription
QX(G/K[H])=E(G/K[H]) X(G/K[H])
×(G/H)e
definesafunctorQfrom∆ -spacesto -spaces.WewriteB forQ ,where
G
O D ∗ ∗
istheconstant∆ -spaceontheone-pointspace.WeregardQasafunctorfrom
G
O
to /B.
G
O U DU
Definition2.3.Thecategory isthecategory /Bof -spaceslyingoverB.
B DU D
Inotherwords,anobjectof consistsofanobjectY of togetherwitha
B DU
mapY B.Amapin isamapin thatcommuteswiththemapstoB.
→ B DU
We define weak equivalences, rational equivalences, and p-adic equivalences
in objectwise for the underlying -space. As always, we form the homotopy
B D
category,rationalhomotopycategory,andp-adichomotopycategoryof byfor-
B
mallyinvertingthosemapsthatareweakequivalences,rationalequivalences,and
p-adicequivalencesrespectively.Itfollowsfromclassicalbundletheory(andthe
Serrespectralsequence)thatthefunctorQpreservesweakequivalences,rational
equivalences,andp-adicequivalences,andsoweobtaininducedfunctorsbetween
thevarioushomotopycategoriesof∆ -spacesandthecorrespondinghomotopy
G
O
categoriesof .InSection6,weshow:
B
Theorem2.4.The functor Q induces equivalences between the homotopy cate-
gory,rationalhomotopycategoryandp-adichomotopycategoryof∆ -spaces
G
O
andthecorrespondinghomotopycategoriesof .
B
Once again, we also wish to identify those objects equivalent to G-spaces,
or equivalently, to -spaces. For an -space X, the ∆ -space IX has the
G G G
O O O
propertythat
IX(G/K[H])=X(G/K)=IX(G/K[K])
8 MichaelA.Mandell,LauraScull
andthestructuremapsG/K[K] G/K[H]inducetheidentityisomorphism.It
→
followsthatfortheobjectQIX of ,
B
QIX(G/K[H])=E(G/H) IX(G/K[H])
×(G/H)e
=E(G/H) X(G/K)
×(G/H)e
=B(G/H) E(G/K) X(G/K)
∼ ×B(G/K) ×(G/K)e
=B(G/K[H]) QIX(G/K[K]).
∼ ×B(G/K[K])
With this as motivation, we define Im , Im , and Im to be the full subcate-
B B0 Bp
gories of the homotopy category, rational homotopy category, and p-adic homo-
topycategoryof ,consistingofthoseobjectsY forwhichthemap
B
(2.5) Y(G/K[H]) B(G/K[H]) Y(G/K[K]).
B(G/K[K])
→ ×
inducedby[e] π G/K = (G/K[K],G/K[H])isaweakequivalence,ratio-
0
∈ D
nalequivalence,andp-adicequivalencerespectively.InSection6,weprove:
Theorem2.6.AnobjectY inthehomotopycategory,rationalhomotopycategory,
orp-adichomotopycategoryof isisomorphictoanobjectintheimageofQI if
B
andonlyifX isinIm ,Im ,orIm respectively.
B B0 Bp
3 TheAlgebraizationTheorems
Armed with descriptions of the equivariant rational and p-adic homotopy cate-
goriesintermsofdiagramsonadiscretecategory,weexplainthealgebraization
theorems. The basic idea is that any (contravariant) functor from spaces to some
category of algebras defines a functor from the over category to the category
B
of(covariant) -diagramsofalgebraslyingunderthediagramobtainedfromB.
D
WhenthefunctorpreservesrationalorZ/pZhomologyisomorphisms,itinduces
afunctorontherationalorp-adichomotopycategories.Weapplythisobservation
totheThom–SullivanDeRhamandsingularcochainfunctors.
Definition3.1.LetΩ∗ denotethefunctorfromspacestocommutativedifferential
gradedQ-algebrasobtainedbyapplyingthepolynomialDeRhamfunctorof[10]
tothesingularsimplicialsetofaspace.LetC denotethesingularcochainfunctor
∗
fromspacestoE F¯ -algebras.
p
∞
Weconsiderthecategoryof(covariant)functorsfrom tocommutativedif-
ferential graded Q-algebras and likewise to E F¯p-algebrDas. We call objects in
∞
these categories -CDGAsand -E DGAs respectively. Weak equivalences in
D D ∞
thesecategoriesaremapsthatarequasi-isomorphismsateachobject.Applyingthe
functorsΩ andC objectwisetoa -space,weobtainfunctorsfrom -spacesto
∗ ∗
D D
-algebras.
D
Definition3.2.LetΩ∗ bethefunctorfrom -spacesto -CDGAsdefinedbyap-
D D
plyingΩ∗objectwise.LetC∗bethefunctorfrom -spacesto -E DGAsdefined
D D ∞
byapplyingC objectwise.
∗
AlgebraicModelsforEquivariantHomotopyTheory 9
IfweapplythefunctorΩ∗toanobjectY of ,weobtainnotjusta -CDGA
B D
Ω∗Y,butalsoamapof -CDGAsΩ∗B Ω∗Y.SincethestructuremapY B
D → →
isnecessaryinthedefinitionofthecategory toobtaintheembeddingoftheratio-
B
nalequivarianthomotopycategory,weshouldexpectthattoobtainanembedding
oftherationalhomotopycategoryintoahomotopycategoryinalgebra,weshould
needsomeformofthestructuremapΩ∗B Ω∗Y.Wecouldlookatthecategory
→
of -CDGAsunderΩ∗B,butthismightnothavethecorrecthomotopycategory.
D
An analogy to keep in mind is that the category of spaces under a given space
X typically does not contain all the homotopy types expected from the homo-
topytypeofX ifX isnothomotopyequivalenttoaCWcomplex.Similarly,we
needtolookatthecategoryof -CDGAsunderasuitablynicequasi-isomorphic
D
replacementforΩ∗B.
In[7]itisshownthatadiagramcategoryofcommutativedifferentialgraded
Q-algebrasorofE F¯p-algebrasisaclosedmodelcategory.Wethereforehavea
∞
notionofcofibrantobject,andthefactoringaxiomsallowustochooseacofibrant
approximation P0 of Ω∗B and a cofibrant approximation Pp of C∗B. This is
oneprecisemeaningforthephrase“suitablynicequasi-isomorphicreplacement”.
However,toprovethetheoremsweareafter,wedonotneedtoputsuchastrin-
gentrequirementonP andP ;weonlyneedtochooseanobjectwisecofibrant
0 p
approximation.Sinceweassumethatthereaderinterestedinthep-adiccaseisfa-
miliarwith[6],wedonotreviewthedefinitionofacofibrantE F¯ -algebra.Itis
p
∞
entirelyanalogoustothedefinitionofacofibrantcommutativedifferentialgraded
Q-algebrafrom[1],whichwedoreview.
Definition3.3.Let R be a commutative differential graded Q-algebra (CDGA).
WesaythatRiscellularifthereexistgradedsubmodules(X =0),X ,X ,...of
0 1 2
theunderlyinggradedmoduleofRsuchthat
(i)ThedifferentialofanyelementinX isinthesub-gradedalgebragenerated
n
byX ,...,X .
0 n 1
−
(ii)The underlying graded algebra of R is the free graded commutative algebra
onX X .
1 2
⊕ ⊕···
WesaythatRiscofibrantifitisaretractofacellularCDGA.
Astandardtrickinmodelcategorytheoryforfindingaweakequivalencefrom
a cellular object to an arbitrary object is called Quillen’s small object argument,
anditisexplainedforCDGA’sin[1,pp.20–22].Inthiscontext,itgivesafunctor
Γ fromthecategoryofCDGA’stoitselfandanaturalquasi-isomorphismΓ Id
→
suchthatforanyCDGAR,ΓR isacellularCDGA.Wehaveananalogouscon-
structionforE DGA’s.Thefollowingpropositionisanimmediateconsequence.
∞
Proposition3.4.There exists a -CDGA P and a quasi-isomorphism P
D 0 0 →
Ω∗B, such that P0(G/K[H]) is a cofibrant CDGA for every G/K[H] in D.
There exists a D-E∞DGA Pp and a quasi-isomorphism Pp → C∗B, such that
P (G/K[H])isacofibrantE DGAforeveryG/K[H]in .
p ∞ D
WechooseandfixsuchobjectsP0andPp.ThefunctorsΩ∗andC∗nowtake
objectsin toobjectsunderP andP .Wedenotetheseunder-categoriesas
B 0 p A0
and respectively.
p
A
10 MichaelA.Mandell,LauraScull
Definition3.5.Let A0 denote the functor from to 0 induced by Ω∗; let Ap
B A
denotethefunctorfromBtoApinducedbyC∗.WedefinethefunctorA0fromG-
spacesto tobethecompositeA Q I Φ,andthefunctorA fromG-spaces
A0 0◦ ◦ ◦ p
to tobethecompositeA Q I Φ.
p p
A ◦ ◦ ◦
ThefunctorA convertsrationalequivalencestoquasi-isomorphismsandthe
0
functorA convertsp-adicequivalencestoquasi-isomorphisms.Wethereforeob-
p
tain functors A and A on the homotopy categories obtained by inverting these
0 p
equivalences.
We can now explain the main theorems of the introduction. We say that an
object Y of is simply connected if each component of Y(G/K[H]) is simply
B
connectedforallG/K[H].Likewise,wesaythatY isfiniteQ-orp-typeifeach
Y(G/K[H])isfiniteQ-orp-type.Clearly,whenaG-spaceX isG-simplycon-
nected and G-finite type, the object QIΦX of is simply connected and finite
B
type. Theorems 1.1, 1.3, and 2.4 show that the functorQ I Φ embeds the ra-
◦ ◦
tionalequivarianthomotopycategoryintherationalhomotopycategoryof and
B
embedsthep-adicequivarianthomotopycategoryinthep-adichomotopycategory
of .Therefore,themaintheoremsareimmediatecorollariesofthefollowingthe-
B
oremthatweproveinSection5.
Theorem3.6.
(i)ThefunctorA0 embedsthefullsubcategoryofsimplyconnectedfiniteQ-type
objects in the rational homotopy category of as a full subcategory of the
B
homotopycategoryof .
0
A
(ii)The functor A embeds the full subcategory of the simply connected finite p-
p
typeobjectsinthep-adichomotopycategoryof asafullsubcategoryofthe
B
homotopycategoryof .
p
A
AsaconsequenceofTheorems1.1,1.3,2.4,and3.6,weobtainanembedding
oftheequivariantrationalhomotopycategoryofG-simplyconnectedG-finitetype
G-spacesasafullsubcategoryofthehomotopycategoryof .Tocompletethe
0
A
pictureweneedanintrinsiccharacterizationofthe -CDGAsinthissubcategory.
D
For this, recall that a graded commutative algebra is said to be finite type if it is
finitelygeneratedasamoduleineachdegreeandsimplyconnectedifitisconcen-
tratedinnon-negativedegreesandzeroindegree1.Wesaythatitisspacelikeif
indegreezeroitisacartesianproductofcopiesofQ.InSection6,weprovethe
followingcharacterizationtheorem.
Theorem3.7.Let R be an object of 0. Then R is isomorphic in the homotopy
A
categoryofA0 toA0X forsomeG-simplyconnectedG-finiteQ-typespaceX if
andonlyif:
(i)H R(G/K[H])isfinitetype,simplyconnected,andspacelikeforallG/K[H]
∗
in ,
D
(ii)ThenaturalmapH0R(G/K[H]) H2P (G/K[H]) H2R(G/K[H])is
⊗ 0 →
injectiveforallG/K[H]in ,and
D