Table Of ContentTHE STABLE GALOIS CORRESPONDENCE
FOR REAL CLOSED FIELDS
7
J.HELLERANDK.ORMSBY
1
0
For Ivo and Irving.
2
n
a Abstract. Inpreviouswork[6],theauthorsconstructedandstudiedaliftoftheGalois
J correspondence to stable homotopy categories. In particular, if L/k is a finite Galois
1 extension of fields withGaloisgroup G,there isafunctor c∗L/k :SHG →SHk from the
3 G-equivariantstablehomotopycategorytothestablemotivichomotopycategoryoverk
] sculocshedthfiaetldc∗La/nkd(GL/=H+k[)i]=,tShpeerce(sLtrHic)t+io.nTohfec∗maintoththeoerηem-coomfp[6le]tseayssubthcaattewgohreyniksfisulalarenadl
T faithful. Here we“uncomplete” this theoremL/skothat itapplies to c∗ itself. Ourmain
L/k
A
tools are Bachmann’s theorem on the (2,η)-periodic stable motivic homotopy category
h. andanisomorphismrangeforthemapπ⋆RSR→π⋆C2SC2 inducedbyC2-equivariantBetti
realization.
t
a
m
[
1. Introduction
1
v In[8], Levine showedthat the “constant” functor c∗ :SH→SHk fromthe classicalstable
9 homotopy category to the motivic stable homotopy category over an algebraically closed
9
field of characteristic zero is a full and faithful embedding. Inspired by his result, in [6] we
0
introduced and studied functors c∗ : SHG → SH , where L/k is a Galois extension with
9 L/k k
0 Galois groupG. We showedthat if k is realclosed and L=k[i], then after completing at a
. prime p andatη, ifp6=2,the functor c∗ is fullandfaithful. The needfor the completion
1 L/k
0 arosefromourlackofknowledgeaboutcertainhomotopygroupsofthe motivic sphereover
7 R. In the meantime advances have been made. Ananyevskiy-Levine-Panin[1] established a
1 motivicversionofSerre’sfinitenesstheorem,whichinparticularimpliesthatc∗ isfulland
: L/k
v faithful after η-completion. The purpose of this paper is to use Bachmann’s recent results
i [2], about a localizationof πR(S ), to remove the η-completions in the main theorem of [6].
X ⋆ R
r Theorem 1.1. Let k be a real closed field and L = k[i] be its algebraic closure. Then the
a
functor
c∗ :SHC2 →SH
L/k k
is a full and faithful embedding.
One of the primary tools in the proof of Theorem 1.1 is the C -equivariant Betti real-
2
ization functor ReC2 : SH → SHC2 which extends the functor taking a smooth R-scheme
B R
to its C-points with complex conjugation action. In [5], Dugger-Isaksen study the effect of
this functor on the bigraded homotopy groups of the real motivic sphere spectrum, prov-
ing that ReC2 : πR (S )∧ → πC2 (S )∧ is an isomorphism when m ≥ 2n−5. We
B m+nα R 2 m+nσ C2 2
2010 Mathematics Subject Classification. Primary: 14F42,55P91Secondary: 11E81,19E15.
Keywords and phrases. Equivariantandmotivicstablehomotopytheory, equivariantBettirealization.
Thesecondauthor waspartiallysupportedbyNationalScienceFoundationAwardDMS-1406327.
1
2 J.HELLERANDK.ORMSBY
use odd-primary Adams spectral sequences and Bachmann’s results to produce a range of
bigradings in which the integral version of this map is an isomorphism. In particular, we
prove the following.
Theorem 1.2 (Theorem 3.11). The map πR (S )→πC2 (S ) is
m+nα R m+nσ C2
(i) an injection if m=2n−6 and n≥0, and
(ii) an isomorphism if m≥2n−5 and n≥0.
Outline. In Section 2 we recall Bachmann’s theorem and deduce some consequences for
the C -equivariantBetti realizationfunctor and Morel’s ±-splitting of the 2-periodic stable
2
motivichomotopycategory. InSection 3weadaptthemethodsof[5]toodd-primaryAdams
spectralsequences. ViaanarithmeticfracturesquareandtheresultsofSection 2,wededuce
Theorem 3.11. Finally, in Section 4 we recall how to bootstrap Theorem 3.11 into a proof
of Theorem 1.1.
Notation. We use the following notation throughout the paper.
» k is a field and L/k is a finite Galois extension of fields with Galois group G.
» SH is Voevodsky’s stable motivic homotopy category [11]; hom sets in SH are
k k
denoted [ , ] .
k
» SHG is the G-equivariant stable homotopy category in the sense of [9]; hom sets in
SHG are denoted [ , ] .
G
» c∗ : SHG → SH is the functor induced by the classical Galois correspondence
L/k k
G/H 7→Spec(LH), constructed in [6, Section 4.3].1
» S is the sphere spectrum in SH and S is the sphere spectrum in SHG.
k k G
» For integers m,n, Sm+nα = (S1)∧m ∧(A1r0)∧n. If G = C , Sm+nσ = (S1)∧m ∧
2
(Sσ)∧n where Sσ is the one-point compactification of the real sign representation.
» For X ∈SH , πk X :=[Sm+nα,X] is the (m+nα)-th homotopy group of X.
k m+nα k
» ForX ∈SHC2,πC2 X :=[Sm+nσ,X] isthe(m+nσ)-thhomotopygroupofX.
m+nσ C2
» For X ∈SH , πkX := π X and similarly for πC2Y when Y ∈SHG.
k ∗ m∈Z m+0α ∗
» Given an embedding φ:k ֒→R, ReC2 =ReC2 :SH →SHC2 is the C -equivariant
L B,φ B k 2
Betti realization which extends the functor taking a smooth k-scheme X to X(C)
with the conjugation action [6, §4.4].
» Depending on context, η is either the motivic Hopf map arising from A2r0 →P1
or the C -equivariant Hopf map.
2
» ( )∧ is the 2-completion functor and ( )∧ is the η-completion functor. If a=2 or η
2 η
we have X∧ =holimX/an.
a
» We write X[a−1] or X[1/a] for the homotopy colimit of X −→a X −→a ···. If X ≃
X[a−1] we say that X is a-periodic.
2. Preliminaries
The main new input we use in this paper is a recent theorem of Bachmann [2, Theorem
31]. HistheoremcomparesalocalizationofSH withtheclassicalstablehomotopycategory.
R
The most convenient form of his result for us is the following recasting.
1
Inloc.cit.westatethatthecategoryGsSetofG-simplicialsetsisequivalenttothecategorysPre(OrG)
of simplicial presheaves on the orbit category. This isn’t quite true: GsSet ⊆ sPre(OrG) is only a full
subcategory. (Thanks to Tom Bachmann for drawing our attention to this inaccuracy.) Thishas noeffect
on any of the subsequent mathematics in loc. cit. because what is used is that the associated homotopy
categories areequivalent andthisistruebyElmendorf’sTheorem.
THE STABLE GALOIS CORRESPONDENCE FOR REAL CLOSED FIELDS 3
Theorem 2.1 (Bachmann). Betti realization induces an equivalence of triangulated cate-
gories
ReC2 :SH [1/2,η−1]−≃→SHC2[1/2,η−1].
B R
Proof. Consider the functor Re : SH → SH which extends the functor Sm → Top,
R R R
sending X to X(R). Consider as well the composite ΦC2 ◦ReC2, where ΦC2 : SHC2 → SH
B
is the geometricfixedpoints functor. BothRe andΦC2◦ReC2 preservehomotopycolimits
R B
and if X ∈ Sm and n ∈ Z, they both send Σn Σ∞X to the spectrum ΣnΣ∞X(R) . It
R P1 P1 + +
follows that Re =ΦC2 ◦ReC2 and so we have the commutative triangle of functors
R B
SH [1/2,η−1] ReCB2// SHC2[1/2,η−1]
R ◗◗◗◗R◗◗e◗R◗◗◗◗◗◗(( (cid:15)(cid:15) ΦC2
SH[1/2].
Write ρ : S0 → Sσ for the standard inclusion. Since η2ρ = −2η, we have the equiva-
lence of categories SHC2[1/2,η−1] ≃ SHC2[1/2,ρ−1]. It follows that ΦC2 is an equivalence
SHC2[1/2,η−1] ≃ SH[1/2]. A specialization of Bachmann’s theorem [2, Theorem 31] says
that Re in the above diagram is an equivalence. We conclude that ReC2 is an equivalence
R B
as well.
(cid:3)
Recall Morel’s ±-operations in motivic homotopy theory (see [3, Section 16.2]). Let ε
denote the stable map induced by the twist isomorphism Sα∧Sα ≃ Sα∧Sα. In the C -
2
equivariant setting, let ε denote the twist Sσ ∧Sσ ≃ Sσ ∧Sσ, and note that ReC2(ε) = ε.
B
In either the motivic or equivariant setting, invert 2 and note that e := (ε−1)/2 and
+
e− :=(ε+1)/2 are orthogonal idempotents. Let the operation ( )+ denote inversion of e+
and let ( )− denote inversion of e−, i.e., ( )± is the cofiber of the operation e±. For any
2-periodicmotivicorC -equivariantspectrumX thereisanaturalsplitting X ≃X+∨X−.
2
Lemma 2.2. If X is a 2-periodic motivic or C -equivariant spectrum, then X+ ≃X∧ and
2 η
X− ≃X[η−1].
Proof. We prove the motivic version of this statement, which easily adapts to the C -
2
equivariant setting. Let X be a spectrum such that X ≃ X[1/2]. We have ε = −1−ρη
whencee− = −1ρη. Thusinvertinge− invertsbothρandη. Since(2+ρη)η =0,thisis the
2
same as inverting 2 and η, whence X− ≃X[η−1]. Now apply η-completion to the splitting
X ≃X+∨X− to get X∧ ≃(X+)∧∨(X−)∧. Since X− ≃X[η−1], the second summand is
η η η
trivial. Since e η =0, X+ is η-complete, i.e., (X+)∧ ≃X+. We conclude that X∧ ≃X+,
+ η η
as desired. (cid:3)
Asaninterestingcorollary(whichwewillnotuseintheremainderofthispaper)wenote
the following.
Proposition 2.3. The natural map ReC2((S )∧)→(S )∧ is an equivalence.
B R η C2 η
4 J.HELLERANDK.ORMSBY
Proof. Let S = S and η-complete the 2-primary fracture square for S in order to produce
R
the bicartesian square
S∧ // S[1/2]∧
η η
(cid:15)(cid:15) (cid:15)(cid:15)
S∧ //(S∧[1/2])∧.
2,η 2 η
Applying ReC2 results in a homotopy pullback square which maps to the corresponding
B
fracture square for (S )∧. The maps between the vertices on the right edge of the square
C2 η
are equivalences by Lemma 2.2. Thus it suffices to show that ReC2(S∧ ) → (S )∧ is
B 2,η C2 2,η
an equivalence. This may be checked on the level of Mackey functor homotopy groups by
comparing the motivic and C -equivariant Adams spectral sequences as in [6, Proposition
2
2.4], concluding the proof. (cid:3)
3. Comparing stable stems
In this section we establish a range of bidegrees in which the map on stable stems
πR (S )→πC2 (S ) induced by equivariant Betti realization is an isomorphism.
m+nα R m+nσ C2
Recall that Dugger-Isaksenestablish a range in which 2-complete stems are isomorphic.
Theorem 3.1 ([5, Theorem 4.1]). The map πR ((S )∧) → πC2 ((S )∧) is an iso-
m+nα R 2 m+nα C2 2
morphism if m≥2n−5 and an injection if m=2n−6.
Remark 3.2. There are isomorphisms πm+nα((SR)∧2)∼=πm+nα((SR)∧2,η) for all m,n by [7,
Theorem 1]. Similarly there are isomorphisms πC2 ((S )∧) ∼= πC2 ((S )∧ ) for all
m+nσ C2 2 m+nσ C2 2,η
m,n by [6, Theorem 2.10]. Dugger-Isaksen’s result can thus equivalently be stated as a
comparison between (2,η)-complete stable stems.
Recall the discussion of motivic and C -equivariantcobar complexes from [5, §3], noting
2
that all of these constructions may be made with HF , p odd, in place of HF . The
p 2
significance of the p-primary cobar complex is that it forms the E -page of the p-primary
1
Adams spectral sequence (in the motivic or equivariant context). In order to concisely
express the properties of these spectral sequences, let S be S (k a field) or S , let M
k C2 p
denote the homology of a point with coefficients in F or F , let A denote the p-primary
p p p
motivicorC -equivariantdualSteenrodalgebra,letC∗ denotethep-primarymotivicorC -
2 p 2
equivariantcobarcomplex,letγ beαorσ,dependingoncontext,andletExt∗,∗+∗γ(M ,M )
A p p
p
denote the homology of C∗ (which is also Ext in the category of A -comodules).
p p
Theorem 3.3 ([7, Theorem 1] and [6, Theorem 2.10]). The (motivic or C -equivariant)
2
p-primary Adams spectral sequence has E -page C∗ and E -page Ext∗,∗+∗γ(M ,M ); it is
1 p 2 Ap p p
strongly convergent with
E2s,m+nγ =⇒ πm−s+nγS∧p,η.
When p=2, we have S∧ ≃S∧ as long as cd (k[i])<∞ (in the motivic case).
2,η 2 2
The Dugger-Isaksen result is obtained by comparing cobar complexes. We first extend
this method to odd p to obtain an isomorphism range on (p,η)-complete stems. We begin
by recallingsome facts about the motivic and equivariantSteenrodalgebrasat oddprimes.
Write A for the classical mod-p dual Steenrod algebra. Recall that
Fp
A =Sym (τ ,τ ,...,ξ ,ξ ,...)
Fp Fp 0 1 1 2
THE STABLE GALOIS CORRESPONDENCE FOR REAL CLOSED FIELDS 5
is a free graded commutative algebra, where |τ |=pi and |ξ |=pi−1.
i i
Recall that MR := πR HF = F [θ] where θ has degree 2−2α. This follows
p m,n∈Z m+nα p p
fromtheaffirmativeresolutionoftheBloch-Katoconjecture[13,Theorem6.1]togetherwith
L
[10,Theorem7.4]. IntheequivariantcasewehaveMC2 := πC2 HF =F [θ,θ−1],
p m,n∈Z m+nσ p p
see e.g. [4, Theorem 2.8]. The dual motivic Steenrod algebra over R is equal to
L
AR ∼=MRp ⊗Fp AFp =MRp[τ0,τ1,...,ξ1,ξ2,···]/(τ02,τ12,...),
where the elements of A are considered to be bigraded by assigning weights so that
Fp
|τ |=pi+(pi−1)α and |ξ |=(pi−1)+(pi−1)α, see [12, Remark 12.12].
i i
Similarly, the dual C -equivariant Steenrod algebra is equal to
2
AC2 ∼=MCp2 ⊗Fp AFp =MCp2[τ0,τ1,...,ξ1,ξ2,···]/(τ02,τ12,...),
where in this case elements of A are considered to be bigraded by assigning weights so
Fp
that |τ |=pi+(pi−1)σ and |ξ |=(pi−1)+(pi−1)σ.
i i
EquivariantBetti realizationReC2 induces maps MR →MC2 andA →A which have
B p p R C2
the obvious effects on the above named elements, i.e., θ 7→θ, τ 7→τ and ξ 7→ξ .
i i i i
Write M for either MR or MC2. In both cases, the dual Steenrod algebra is free over
p p p
M , and a basis is given by monomials τǫ0τǫ1···ξn1ξn2···, where ǫ ∈ {0,1} and n is a
p 0 1 1 2 i i
nonnegative integer. We write τǫξn for such a monomial.
p
Lemma 3.4. Suppose that |τǫξn|=k+ℓα. Then k ≤ ℓ+1.
p−1
Proof. The bidegree of ξ satisfies k < p ℓ and the bidegree of τ satisfies k ≤ p ℓ
i p−1 i p−1
provided i ≥ 1. It follows that if ǫ = 0 then the bidegree of τǫξn satisfies k ≤ p ℓ. If
0 p−1
ǫ = 1 then write τǫξn = τ τǫ′ξn, where ǫ′ = 0. This element thus satisfies the inequality
0 0 0
k ≤ p ℓ+1. (cid:3)
p−1
Lemma 3.5. The map Cf →Cf is
R C2
(i) an injection in all degrees, and
(ii) an isomorphism if k−f ≥ p ℓ− 4p−2 +1.
p−1 p−1
j k
Proof. We have that CCf2 ∼= MCp2 ⊗MpR CRf, that CRf is free over MRp, and that the map
Mp →Mp is injective. It follows that Cf →Cf is injective.
R C2 R C2
Let Z = [z |z |···|z ] be a cobar element of Adams filtration f, where each z is of the
1 2 f i
formτǫξn, ofbidegree k +ℓ α. By Lemma 3.4 we havethat k ≤ p ℓ +1. Summing over
i i i p−1 i
i, we find that if |Z| = k+ℓα, then k ≤ p ℓ+f. The cokernel of Cf → Cf consists of
p−1 R C2
elementsoftheformθ−nZ forn≥1. Theelementsθ−n lieabovethelineofslope2(p−1)/p
passing through θ−1 and so we find that these satisfy the inequality k ≤ p ℓ− 4p−2. It
p−1 p−1
follows that the bidegree of θ−nZ satisfies the inequality k ≤ p ℓ+f − 4p−2. Thus the
p−1 p−1
cokernel is zero in bidegrees satisfying k > p ℓ+f − 4p−2.
p−1 p−1
(cid:3)
RecalltheformofthemotivicandC -equivariantAdamsspectralsequencesfromTheorem 3.3.
2
EquivariantBetti realizationinduces a map betweenthese spectralsequences whichwe can
now analyze.
2
Weusethestandardconvention inwhichthek-axisishorizontal andtheℓα-axisisvertical.
6 J.HELLERANDK.ORMSBY
Proposition 3.6. The map Ext(f,k+ℓα)(MR,MR)→Ext(f,k+ℓα)(MC2,MC2) is an injection
AR p p AC2 p p
if k−f = p ℓ− 4p−2 and an isomorphism if k−f ≥ p ℓ− 4p−2 +1.
p−1 p−1 p−1 p−1
Proof. Thijs follows fromk[5, Lemma 3.4] and Lemma 3.5. j k (cid:3)
Lemma 3.7. Ext(f,k+ℓα)(MR,MR) is a finite-dimensional F -vector space for all f,k,ℓ.
AR p p p
Proof. WritingCFp fortheclassicalcobarcomplex,wehaveCR ∼=MRp⊗FpCFp. Theuniversal
coefficient theorem then yields the isomorphism
ExtA∗,R∗+∗α(MRp,MRp)∼=Ext∗A,F∗p(Fp,Fp)⊗Fp MRp
up to a grading shift, and this is a finite-dimensional F -vector space in all degrees. (cid:3)
p
Theorem 3.8. The map πR ((S )∧ )→πC2 ((S )∧ ) is
m+nα R (p,η) m+nσ C2 (p,η)
(i) an injection if m= p n− 4p−2 , and
p−1 p−1
(ii) an isomorphism if mj≥ p n− 4kp−2 +1.
p−1 p−1
Proof. ThisfollowsfromPropjosition 3.6usinkgthesameargumentasin[5,Theorem4.1]. (cid:3)
Remark 3.9. In [5], Dugger-Isaksen establish a second isomorphism range on 2-complete
stems. Namely,π (S )∧ →π (S )∧ isanisomorphismifm+n≤−1. Aversionof
m+nα R 2 m+nσ C2 2
this rangeappearsto holdon(p,η)-complete stems, withthe difference thatthe mapmight
be only surjective when m +n = −1. However, this second isomorphism range doesn’t
extend to integral stems and we do not pursue it further here.
We now turn our attention to p-complete spheres.
Proposition 3.10. The map πR ((S )∧)→πC2 ((S )∧) is
m+nα R p m+nσ C2 p
(1) an injection if m= p n− 4p−2 , and
p−1 p−1
(2) an isomorphism if mj≥ p n− 4kp−2 +1.
p−1 p−1
j k
Proof. Write S foreither of S orS . Ifp=2,the statementofthe propositionis Dugger-
R C2
Isaksen’sTheorem 3.1, sowecanassumep is odd. Inthis case2 isinvertibleinS/pr andin
S∧. By Lemma 2.2 we have that (S∧)+ ≃S∧ and (S∧)− ≃S∧[η−1] and so we have that
p p (p,η) p p
S∧ ≃S∧ ∨S∧[η−1].
p (p,η) p
Note that we have an isomorphism (S∧)− ∼= lim ((S/pr)−). It follows from Theorem 2.1
p r
that the map πR ((S )∧[η−1])→πC2 ((S )∧[η−1]) is an isomorphism. The result thus
∗+∗α R p ∗+∗σ C2 p
follows from Theorem 3.8 and the direct sum decomposition of S∧ above.
p
(cid:3)
Theorem 3.11. The map πR (S )→πC2 (S ) is
m+nα R m+nσ C2
(i) an injection if m=2n−6 and n≥0, and
(ii) an isomorphism if m≥2n−5 and n≥0.
Proof. Consider the comparison of long exact sequences of homotopy groups induced by
cofiber sequences
S→ S∧ ∨S → S∧ .
p Q p
Q
(cid:16)Yp (cid:17) (cid:16)Yp (cid:17)
THE STABLE GALOIS CORRESPONDENCE FOR REAL CLOSED FIELDS 7
obtained from the arithmetic fracture squares for S and S . It suffices to show that
R C2
the comparison map πR (−) → πC2 (−) at the middle and the righthand terms are
m+nα m+nσ
injections if m=2n−6 and n≥0 and an isomorphism if m≥2n−5 and n≥0.
Theinequalitiesfortheisomorphismrangeofp-completestablestemsfromProposition 3.10
foroddparedominatedbyDugger-Isaksen’sinequalitiesinTheorem 3.1,forthe2-complete
stable stems. It follows that πR ( (S )∧)→πC2 ( (S )∧) is an injection if m=
m+nα p R p m+nα p C2 p
2n−6andanisomorphismifm≥2n−5. SincewehavethatthemapπR ( (S )∧) →
Q Q m+nα p R p Q
πC2 ( (S )∧) is a filtered colimit of these maps, it too is an injection if m = 2n−6
m+nα p C2 p Q Q
and an isomorphism if m≥2n−5.
By TQheorem 2.1 and Lemma 2.2, πR (S )− →πC2 (S )− is an isomorphism. By [3,
∗+∗α R Q ∗+∗α C2 Q
Theorems 11 and 16.2.13], (S )+ ≃ HQ, where HQ is the rationalized motivic cohomology
R Q
spectrum. Thus, we have that πR (S )+ =0 whenever j >0. When j =0, we have that
i+jα R Q
πR(S )+ =0 if i6=0 and πR(S )+ =Q. We have
i R Q 0 R Q
Q j is even and i+j =0
πC2 (S )+ =
i+jα C2 Q (0 else.
Notethatifj ≥0,thisvanishingregionsatisfiesi≥2j−5. ThemapπR(S )+ →πC2(S )+
0 R Q 0 C2 Q
is an ismorphism. We conclude that πR (S )+ → πC2 (S )+ is an injection if m =
m+nα R Q m+nσ C2 Q
2n−6 and n≥0, and an isomorphism if m≥2n−5 and n≥0. (cid:3)
4. Proof of Theorem 1.1
Wefinishbyexplaininghowthecomparisonofstablestemsintheprevioussectionimplies
the embedding theorem.
Proposition 4.1. If
∼
(i) ReC2 :[Sn,S ] −=→[Sn,S ] , and
B R R C2 C2
∼
(ii) ReC2 :[Spec(C) ∧Sn,S ] −=→[C ∧Sn,S ]
B + R R 2+ C2 C2
are isomorphisms for all n∈Z, then Theorem 1.1 is true for any real closed field k.
Proof. Let k be a real closed field and L =k[i]. To prove Theorem 1.1, it suffices to prove
that
∼
(a) c∗ :[Sn,S ] −=→[Sn,S ] , and
L/k C2 C2 k k
∼
(b) c∗ :[C ∧Sn,S ] −=→[Spec(L) ∧Sn,S ]
L/k 2+ C2 C2 + k k
are isomorphisms for all n ∈ Z, by the same argument as in the beginning of the proof of
[6, Theorem 2.21].
To prove that the maps in (a), (b) are isomorphisms, we can assume that k = R and
L=C,bythesameargumentasin[6,Proposition2.20]. WenowconsidertheC -equivariant
2
Betti realization functor ReC2 : SH → SHC2. Since ReC2 ◦c∗ = id, it follows that (a)
B R B C/R
and (b) are isomorphisms. (cid:3)
Corollary 4.2 (Theorem 1.1). Let k be a real closed field and L = k[i] be its algebraic
closure. Then the functor
c∗ :SHC2 →SH
L/k k
is a full and faithful embedding.
8 J.HELLERANDK.ORMSBY
Proof. If i<0 then πR(S )=πC2(S )=0 and so the map in 4.1(i) is an isomorphism for
i R i C2
i<0. It is an isomorphism for i≥0 by setting n=0 in Theorem 3.11. The map in 4.1(ii)
is identical to the map [Sn,S ] →[Sn,S] induced by complex Betti realization. This is an
C C
isomorphism by Levine’s theorem [8]. (cid:3)
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