THE MOTIVIC SEGAL CONJECTURE, LECTURE 2
JOHN ROGNES
Abstract. Jeg vil vise hvordan Segalformodningen kan omformuleres, ved hjelp av
Tatekonstruksjonen, norm-restriksjonssekvensene, Warwick dualitet og Adams’ transferekvivalens, til en form som lettere lar seg bevise v.h.a. den algebraiske Singerkonstruksjonen. Dette er det andre i en serie foredrag som sikter mot å gi et bevis for en
versjon av Segalformodningen i motivisk homotopiteori.
1. Stable equivariant homotopy and cohomotopy
Let G be a finite group, and let X and Y be finite based G-CW complexes. Let
{X, Y }G = colim [X ∧ S V , Y ∧ S V ]G
V
be the abelian group of stable homotopy classes of G-maps X → Y . Here V ranges over
the finite dimensional G-representations contained in a fixed complete G-universe U .
The stable G-equivariant homotopy category of G-spectra (indexed on U ) is constructed
so that
∞
G
{X, Y }G = [Σ∞
G X, ΣG Y ] ,
where Σ∞
G X denotes the suspension G-spectrum on X, and the right hand side is the
∞
set of morphisms from Σ∞
G X to ΣG Y in the stable G-equivariant homotopy category. In
0
particular, the sphere G-spectrum is SG = Σ∞
GS .
When discussing the Segal conjecture, we considered the stable G-equivariant homotopy
groups
G
πnG (Y ) = {S n , Y }G and π−n
(Y ) = {S 0 , Y ∧ S n }G ,
and the stable G-equivariant cohomotopy groups
n
πG
(X) = {X, S n }G
−n
and πG
(X) = {X ∧ S n , S 0 }G ,
for n ≥ 0. In particular,
0
π0G (S 0 ) = πG
(S 0 ) = {S 0 , S 0 }G = [SG , SG ]G
is the Burnside ring.
2. The Wirthmüller equivalence
Let ι : H → G be the inclusion of a subgroup. The restriction functor ι∗ from based
G-spaces to based H-spaces has a left adjoint
ι∗ X = G+ ∧H X
and a right adjoint ι! X = Map(G+ , X)H . After stabilizing, the restriction functor ι∗
from G-spectra to H-spectra also has a left and a right adjoint, and the level of stable
homotopy categories these are naturally isomorphic.
Date: September 21st 2014.
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Theorem 2.1 (Wirthmüller [?Ada84, Thm. 5.1, 5.2]). Suppose that X is a finite based
H-CW complex and Y is a based finite G-CW complex.
(a) There is a natural isomorphism
{G+ ∧H X, Y }G ∼
= {X, ι∗ Y }H .
(b) There is a natural isomorphism
{ι∗ Y, X}H ∼
= {Y, G+ ∧H X}G .
The natural isomorphism between these left and right adjoints can be promoted to a
stable equivalence of G-spectra, following Lewis–May–Steinberger [?LMS86, §II.6]. The
restriction functor ι∗ from G-spectra to H-spectra has a left adjoint G nH (−) and a right
adjoint FH [G, −).
Theorem 2.2 ([?LMS86, Thm. II.6.2]). For H-spectra D, there is a natural equivalence
'
of G-spectra ω : FH [G, D) −→ G nH D.
We shall say something about the proofs in lecture 3.
3. The Adams transfer equivalence
Let ρ : G → G/N be the projection to a quotient group. The restriction functor ρ∗
from based G/N -spaces to based G-spaces has the left adjoint ρ∗ X = X/N given by the
orbit space, and the right adjoint ρ! X = X N given by the fixed point subspace.
Theorem 3.1 (Adams [?Ada84, Thm. 5.3, 5.4]). Suppose that X is a finite based G-CW
complex in which N acts freely away from the base point, and let Y be a finite based
G/N -CW complex.
(a) There is a natural isomorphism
{X/N, Y }G/N ∼
= {X, ρ∗ Y }G .
(b) There is a natural isomorphism
{ρ∗ Y, X}G ∼
= {Y, X/N }G/N .
[[The pullback ρ∗ Y is finite based G-CW complex with trivial N -action, and the orbit
space X/N is a finite based G/N -CW complex. Comment on the relation between the
unstable and stable results.]]
The Adams isomorphism can also be promoted to a stable equivalence of G/N -spectra,
following [?LMS86, §II.7]. Let U be a complete G-universe, and let U N be the N -fixed
subuniverse. It can also be viewed as a complete G/N -universe.
To each G-spectrum D indexed on U N , we can associate the orbit prespectrum
{W 7→ D(W )/N }
for W ⊂ U . The orbit spectrum D/N is the G/N -spectrum on U N obtained from this
prespectrum by spectrification. We can also associate to D the fixed point spectrum DN ,
with
(DN )(W ) = D(W )N
for W ⊂ U N . This is already a G/N -spectrum, indexed on U N .
Let i : U N → U be the inclusion of universes. The restriction functor i∗ from Gspectra on U to G-spectra on U N has a left adjoint, i∗ . There is a prespectrum
N
{V 7→ D(W ) ∧ S V −W }
for V ⊂ U , where W = V N is the maximal subrepresentation contained in U N , and i∗ D
is the G-spectrum on U obtained from this prespectrum by spectrification.
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It is obtained from D by building in deloopings with respect to the representations
in U , in addition to those already in U N . For example, if N = G and D = Σ∞ X is
the (naive) suspension spectrum of a based G-space X, indexed on the G-trivial universe
U G , then i∗ D = Σ∞
G X is the genuine suspension spectrum of X, indexed on the complete
G-universe U . In particular, i∗ (S) = SG .
The fixed point functor (−)N is right adjoint to the functor i∗ ρ∗ from G/N -spectra
on U N to G-spectra on U .
Theorem 3.2 ([?LMS86, Thm. 7.1]). Let D be an N -free G-spectrum indexed on U N .
There is a natural map
τ : i∗ ρ∗ (D/N ) −→ i∗ D
(in the stable homotopy category) of G-spectra on U , and the right adjoint
τ̃ : D/N −→ (i∗ D)N
of τ is a stable equivalence of G/N -spectra indexed on U N .
We shall say something about the proofs in lecture 3.
4. The norm–restriction sequences
Let X be any G-spectrum, for instance the G-equivariant sphere spectrum SG . The
homotopy cofiber sequence EG+ → S 0 → ẼG → ΣEG+ and the map X = F (S 0 , X) →
F (EG+ , X) induce a map of horizontal homotopy cofiber sequences
[EG+ ∧ X]G
/ XG
/ [ẼG ∧ X]G
/ [ΣEG+ ∧ X]G
[EG+ ∧ F (EG+ , X)]G
/ F (EG+ , X)G
/ [ẼG ∧ F (EG+ , X)]G
/ [ΣEG+ ∧ F (EG+ , X)]G
Let
XhG = EG+ ∧G X
, X hG = F (EG+ , X)G
and X tG = [ẼG ∧ F (EG+ , X)]G
denote the homotopy orbit spectrum, the homotopy fixed point spectrum and the Tate
construction, respectively.
The adjunction counit i∗ i∗ X → X is a G-map and a stable equivalence, hence induces
a G-equivalence
'G
i∗ (EG+ ∧ i∗ X) ∼
= EG+ ∧ i∗ i∗ X −→ EG+ ∧ X .
Since G acts freely on EG, D = EG+ ∧ i∗ X is a G-free G(-CW) spectrum, and there is
a natural Adams transfer equivalence
'
τ : (EG+ ∧ i∗ X)/G −→ (i∗ (EG+ ∧ i∗ X))G ' [EG+ ∧ X]G
in the stable homotopy category.
The G-map and equivalence X → F (EG+ , X) induces an equivalence XhG ' F (EG+ , X)hG ,
hence the left hand vertical map [EG+ ∧ X]G → [EG+ ∧ F (EG+ , X)]G is an equivalence.
[[Explain the geometric fixed point equivalence [ẼG ∧ X]G ' ΦG (X) and the (cyclotomic) equivalence ΦG (SG ) ' S.]]
We obtain the following norm–restriction homotopy cofiber sequences:
XhG
XhG
N
Nh
/ XG
/ ΦG (X)
R
γ
/ X hG
Rh
3
∂
/ ΣXhG
∂h
/ ΣXhG
γ̂
/ X tG
The Segal conjecture for X, that γ : X G → X hG is an equivalence (after J(G)-adic/padic completion), is therefore equivalent to the assertion that γ̂ : ΦG (X) → X tG is an
equivalence (after J(G)-adic/p-adic completion).
In the particular case of X = SG and G = Cp , when ΦCp (SG ) ' S, the diagram appears
as follows:
R /
∂
N /
/ ΣBG+
BG+
(SG )G
S
BG+
Nh
γ
/ S hG
Rh
γ̂
/ S tG
∂h
/ ΣBG+
where BG+ = Σ∞ BG+ = ShG (and S hG ' F (BG+ , S) = D(BG+ )). To prove that
γ : S G → S hG is a p-adic equivalence, we can instead prove that γ̂ : S → S tG is a p-adic
equivalence.
5. The Tate tower
We can view the Tate construction X tG as the homotopy limit of a tower of spectra
X tG −→ . . . −→ X tG (k) −→ . . . −→ X tG (1) −→ X tG (0) ' ΣXhG
factoring the boundary map ∂ h : X tG → ΣXhG in the lower norm-restriction sequence.
Granting this, our aim is to prove that
γ̂ : ΦG (X) −→ holim X tG (k)
k
is a p-adic equivalence.
The Tate tower can be defined for each finite group G, but there are certain simplifications in a class of examples that includes the groups Cp of prime order, so we only discuss
this case here. More specifically, we assume that that there exists a G-representation V
with the property that G acts freely on the unit sphere S(V ). These are the spherical
space form groups. [[They have periodic cohomology, and have been classified, starting
with Milnor.]] When G = C2 the minimal example is V = R(1) with the sign action;
when G = Cp with p odd we may take V = C(1) with the standard action. L
∞
V , and
In this case we may let EG = S(∞V ) be the unit sphere in ∞V =
0
identify the mapping cone ẼG of c : EG+ → S with the one-point compactification
S ∞V = colimk S kV . We get an increasing filtration
S 0 ⊂ S V ⊂ · · · ⊂ S kV ⊂ · · · ⊂ S ∞V = ẼG
of ẼG, in the category of based G-spaces. If we apply the suspension spectrum functor
to G-spectra, then we can extend this filtration to the left as follows:
∗ −→ . . . −→ S −kV −→ . . . −→ S −V −→ S 0 −→ S V −→ . . . −→ S kV −→ . . . −→ ẼG .
kV
Here S kV is notation for Σ∞
when k ≥ 0, and S −kV is the kV -th desuspension of
GS
SG . This is the spectrification of the G-prespectrum with W -th space equal to S W −kV if
kV ⊆ W , and ∗ otherwise.
Definition 5.1. Let
X tG (k) = [ẼG/S −kV ∧ F (EG+ , X)]G
for k ≥ 0. The sequence of G-spectra
ẼG −→ . . . −→ ẼG/S −kV −→ . . . −→ ẼG/S −V −→ ẼG/S 0 = ΣEG+
induces the Tate tower of spectra
X tG −→ . . . −→ X tG (k) −→ . . . −→ X tG (1) −→ X tG (0) ' ΣXhG .
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Lemma 5.2. (a) X tG (0) ' ΣXhG , (b) X tG (k) ' Σ(S −kV ∧ X)hG , and (c) X tG '
holimk X tG (k).
Proof. (a)
X tG (0) = [ẼG/S 0 ∧ F (EG+ , X)]G = [ΣEG+ ∧ F (EG+ , X)]G ' [ΣEG+ ∧ X]G ,
since G acts freely on ΣEG+ and X → F (EG+ , X) is a G-map and an equivalence. By the
Adams transfer equivalence, the latter is homotopy equivalent to Σ(EG+ ∧G X) = ΣXhG .
(b) Note that ẼG/S −kV ' (ẼG/S 0 ) ∧ S −kV = ΣEG+ ∧ S −kV . This follows from
(S ∞V /S 0 ) ∧ S −kV ∼
= (S ∞V ∧ S −kV )/(S 0 ∧ S −kV ) ' S ∞V /S −kV .
Here we use that S ∞V ∼
= S ∞V +kV ∼
= S ∞V ∧ S kV , and that S kV ∧ S −kV ' S. Hence
X tG (k) = [ẼG/S −kV ∧ F (EG+ , X)]G ' [ΣEG+ ∧ S −kV ∧ F (EG+ , X)]G .
Like in part (a) we can rewrite this as [ΣEG+ ∧ S −kV ∧ X]G ' Σ(S −kV ∧ X)hG .
(c) By considering the G-homotopy cofiber sequences S −kV → ẼG → ẼG/S −kV , it
suffices to prove that
holim [S −kV ∧ F (EG+ , X)]G ' ∗ .
k
By G-equivariant Spanier–Whitehead duality for the finite G-CW spectrum S kV there is
a natural G-equivalence
S −kV ∧ F (EG+ , X) 'G F (S kV ∧ EG+ , X) ,
so we can rewrite the homotopy limit above as
holim F (S kV ∧ EG+ , X)G ' F (S ∞V ∧ EG+ , X)G .
k
Here S ∞V ∧ EG+ ' ẼG ∧ EG+ is the mapping cone of the G-equivalence c ∧ 1 : EG+ ∧
EG+ → S 0 ∧ EG+ , hence is G-equivariantly contractible. Thus the function spectrum
above is G-equivariantly contractible, and the G-fixed point spectrum is also contractible.
In particular, when X = SG we have
S tG (k) ' ΣEG+ ∧G S −kV = Σ(S −kV )hG .
For example
interpolating between S tC2
S tC2 (k) ' Σ(S −kR(1) )hC2
and ΣBC2+ = ΣRP+∞ , and
S tCp (k) ' Σ(S −kC(1) )hCp
∞
interpolating between S tCp and ΣBCp+ = ΣL∞
= S ∞ /Cp denotes
+ for p odd, where L
the infinite dimensional lens space.
6. Thom spaces and Thom spectra
Let V be any G-representation, of dimension d. The Thom space of the vector bundle
ξ : EG ×G V → BG is the quotient space
D(ξ) ∼ EG ×G D(V ) ∼
T h(ξ) =
=
= EG+ ∧G S V = (S V )hG .
S(ξ)
EG ×G S(V )
When V = 0 this is BG+ , and when V = Rd is the trivial representation, it is the d-fold
suspension BG+ ∧ S d = Σd (BG+ ). In general one often writes
BGV = EG+ ∧G S V
5
for the Thom space above, and thinks of it as a (twisted) V -fold suspension of BG+ .
If each element g ∈ G acts on V by an orientation-preserving map, or if we work
with cohomology with F2 -coefficients, there is an orientation class Uξ ∈ H̃ d (T h(ξ)) =
H̃ d (BGV ), and a Thom isomorphism
∼
=
Φξ = (−) ∪ Uξ : H ∗ (BG) −→ H̃ ∗+d (BGV ) .
We typically use this to identify the cohomology of any Thom space with the cohomology
of the base, up to a degree shift.
The inclusion 0 ⊂ V induces a based G-map S 0 → S V and a based map s : BG+ =
BG0 → BGV , which we can identify with the inclusion of the 0-section BG → D(ξ) in
ξ, followed with the collapse map to T h(ξ) = BGV . Let the pullback of the orientation
class along this section,
eV = s∗ (Uξ ) ∈ H̃ d (BG+ ) = H d (BG) ,
be the Euler class of ξ. [[The corresponding construction for the tangent bundle T M →
M of a closed manifold yields a cohomology class whose value on the fundamental class
of the manifold equals the Euler characteristic χ(M ), hence the name.]] The composite
homomorphism
s∗
Φξ
H ∗ (BG) −→ H̃ ∗+d (BGV ) −→ H ∗+d (BG)
is given by the cup product with eV , sending x to x ∪ eV .
More generally, the inclusion kV ⊂ (k + 1)V also induces a map of Thom spaces
s : BGkV → BG(k+1)V , and the composite homomorphism
Φ(k+1)ξ
Φ−1
kξ
s∗
H ∗ (BG) −→ H̃ ∗+(k+1)d (BG(k+1)V ) −→ H̃ ∗+(k+1)d (BGkV ) −→ H ∗+d (BG)
is also given by multiplication with the Euler class eV .
Turning from kV to the virtual representation −kV , we only have a virtual vector
bundle −kξ, and no Thom space T h(−kξ). However, we can define a Thom spectrum
T h(−kξ) = EG+ ∧G S −kV = (S −kV )hG
as the homotopy orbits of the G-spectrum S −kV , and will also write BG−kV for this
spectrum.
Again there is a Thom isomorphism
∼
=
Φ−kξ : H ∗ (BG) −→ H ∗−kd (BG−kV )
The map of G-spectra S −(k+1)V → S −kV induces a spectrum map s : BG−(k+1)V →
BG−kV , and the composite homomorphism
Φ−1
−(k+1)ξ
s∗
Φ−kξ
H ∗ (BG) −→ H̃ ∗−kd (BG−kV ) −→ H̃ ∗−kd (BG−(k+1)V ) −→ H ∗+d (BG)
is still given by multiplication with the Euler class eV .
With this notation, the Tate tower
S tG −→ . . . −→ S tG (k + 1) −→ S tG (k) −→ . . . −→ ΣBG+
is stably equivalent to the tower of Thom spectra
s
S tG −→ . . . −→ ΣBG−(k+1)V −→ ΣBG−kV −→ . . . −→ ΣBG0 .
The cohomology of the finite part of this tower defines a sequence
s∗
. . . ←− ΣH ∗ (BG−(k+1)V ) ←− ΣH ∗ (BG−kV ) ←− . . . ←− ΣH ∗ (BG0 ) .
6
Under the Thom isomorphisms Φ−kV , this is isomorphic to the sequence
(−)∪eV
. . . ←− ΣH ∗+(k+1)d (BG) ←− ΣH ∗+kd (BG) ←− . . . ←− ΣH ∗ (BG)
where each homomorphism is given by multiplication by the Euler class.
Proposition 6.1. The colimit of this sequence,
colim H ∗ (S tG (k)) ∼
= ΣH ∗ (BG)[e−1 ] ,
V
k
is given by localization of ΣH ∗ (BG) away from the Euler class.
Note the cohomology does not usually convert homotopy colimits to colimits, so this
colimit is not the cohomology of S tG . We instead call it the continuous cohomology of
S tG (with respect to its presentation as the homotopy limit of the Tate tower), and write
Hc∗ (S tG ) = colim H ∗ (S tG (k)) .
k
In fact, the Segal conjecture will tell us that H ∗ (S tG ; Fp ) ∼
= H ∗ (S; Fp ) is just Fp concentrated in degree 0, which is often very different from ΣH ∗ (BG)[e−1
V ].
7. Limits of stunted real projective and lens spaces
Example 7.1. When G = C2 and V = R(1), the Thom space BGkV = (RP ∞ )kξ =
EG+ ∧G S kR(1) is homeomorphic to the stunted real projective space
RPk∞ = RP ∞ /RP k−1
with one cell in each dimension ∗ ≥ k, in addition to the base point. [[See Atiyah’s
paper on Thom complexes.]] The map BGkV → BG(k+1) V corresponds to the map
∞
RPk∞ → RPk+1
that collapses the (bottom) k-cell to a point.
It is therefore suggestive to introduce the notation
∞
RP−k
= (RP ∞ )−kξ = EG+ ∧G S −kR(1)
for the Thom spectrum of −kξ. The Tate tower for C2 then appears as
s
∞
∞
S tC2 −→ . . . −→ ΣRP−k−1
−→ ΣRP−k
−→ . . . −→ ΣRP+∞ .
One also introduces the notation
∞
∞
RP−∞
= holim RP−k
k
∞
so that S tC2 ' ΣRP−∞
. The Segal conjecture for C2 is then the assertion that there is a
∞
∞
2-adic equivalence γ̂ : S −→ ΣRP−∞
, or that RP−∞
'2 S −1 . In this form the conjecture
was also studied by Mahowald. [[Reference, priority?]]
In cohomology, H ∗ (BC2 ; F2 ) = H ∗ (RP ∞ ; F2 ) = F2 [x] is the polynomial algebra on a
generator x in degree |x| = 1. The (mod 2) Euler class of the tautological line bundle ξ is
±1
eV = x, so the localization H ∗ (BC2 ; F2 )[e−1
V ] = F2 [x ] is the ring of Laurent polynomials
in x. When the coefficient field F2 is implicitly understood, we often write P (x) = F2 [x]
and P (x±1 ) = F2 [x±1 ] for this polynomial ring and its localization.
Lemma 7.2. Hc∗ (S tC2 ; F2 ) = ΣP (x±1 ).
Example 7.3. When G = Cp and V = C(1), the Thom space BGkV = (L∞ )kξ = EG+ ∧G
S kC(1) is homeomorphic to the stunted lens space
∞
2k−1
L∞
2k = L /L
7
with one cell in each dimension ∗ ≥ 2k, in addition to the base point. Here L2k−1 =
S 2k−1 /Cp is the (2k−1)-dimensional lens space. The map BGkV → BG(k+1) V corresponds
∞
to the map L∞
2k → L2k+2 that collapses the 2k- and 2k + 1-cells.
We introduce the notation
∞ −kξ
= EG+ ∧G S −kC(1)
L∞
−2k = (L )
for the Thom spectrum of −kξ. The Tate tower for Cp then appears as
s
∞
∞
S tCp −→ . . . −→ ΣL∞
−2k−2 −→ ΣL−2k −→ . . . −→ ΣL+ .
Let
∞
L∞
−∞ = holim L−2k
k
tCp
ΣL∞
−∞ .
ΣL∞
−∞ .
∗
The Segal conjecture for Cp asserts that there is a p-adic equiva'
so that S
lence γ̂ : S −→
In cohomology, H (BCp ; Fp ) = H ∗ (L; F2 ) = Fp [x, y]/(x2 ) is the exterior algebra on a
generator x in degree |x| = 1, tensored with the polynomial algebra on a generator y
in degree |y| = 2. The (mod p) Euler class of the tautological complex line bundle ξ is
±1
2
eV = y, so the localization H ∗ (BCp ; Fp )[e−1
V ] = Fp [x, y ]/(x ) is the tensor product of
the exterior algebra on x and the Laurent polynomial algebra on y.
Lemma 7.4. Hc∗ (S tCp ; Fp ) = ΣE(x) ⊗ P (y ±1 ).
8. Limits of Adams spectral sequences
The plan is now to analyze γ̂ using towers of Adams spectral sequences. For each
bounded below spectrum Y with H∗ (Y ; Fp ) of finite type, there is a strongly convergent
Adams spectral sequence
∗
∧
E2∗,∗ (Y ) = Ext∗,∗
A (H (Y ; Fp ), Fp ) =⇒ π∗ (Yp ) ,
where A denotes the Steenrod algebra of stable mod p cohomology operations. For
Y = ΦG (SG ) ' S this takes the form
∗,∗
(Fp , Fp ) =⇒ π∗ (Sp∧ ) ,
E2∗,∗ (S) = ExtA
since H ∗ (S; Fp ) = Fp .
Suppose that G acts freely on S(V ), as before. For each k ≥ 0 the Thom spectrum
S tG (k) ' ΣBG−kV is bounded below and of finite type mod p, so there is a strongly
convergent spectral sequence
∗
−kV
E2∗,∗ (k) = Ext∗,∗
; Fp ), Fp ) =⇒ π∗ Σ(BG−kV )∧p .
A (ΣH (BG
These form a tower of spectral sequences, and it can be proved that their algebraic limit
∗
−kV
E2∗,∗ (∞) = lim E2∗,∗ (k) = lim Ext∗,∗
; Fp ), Fp )
A (ΣH (BG
k
k
∗,∗
∼
= ExtA (colim ΣH ∗ (BG−kV ; Fp ), Fp )
=
k
∗,∗
∗
ExtA (Hc (S tG ; Fp ), Fp )
is again a spectral sequence converging strongly to
π∗ (S tG )∧ ∼
= π∗ holim Σ(BG−kV )∧ .
p
k
p
We now have a map of spectral sequences
∗,∗
∗,∗
∗
tG
γ̂ : E2∗,∗ (S) = Ext∗,∗
A (Fp , Fp ) −→ ExtA (Hc (S ; Fp ), Fp ) = E2 (∞)
8
converging strongly to
γ̂ : π∗ (Sp∧ ) −→ π∗ (S tG )∧p .
The map of E2 -terms is induced by an A -module homomorphism
: Hc∗ (S tG ; Fp ) = ΣH ∗ (BG; Fp )[e−1
V ] −→ Fp .
When G = C2 this is the homomorphism
: ΣHc∗ (S tC2 ; F2 ) = ΣP (x±1 ) −→ F2
that maps Σx−1 to 1. When G = Cp it is the homomorphism
: ΣHc∗ (S tCp ; Fp ) = ΣE(x) ⊗ P (y ±1 ) −→ Fp
that maps Σxy −1 to 1.
The remaining step of the proof of the Segal conjecture for C2 and Cp is then provided by the following algebraic theorems, because a map of spectral sequences that
is an isomorphism at the E2 -term is an isomorphism at all later terms, and a map of
strongly convergent spectral sequences that is an isomorphism at the E∞ -term induces
an isomorphism of the abutments.
Theorem 8.1 (W.H. Lin). : ΣP (x±1 ) → F2 induces an isomorphism
∗,∗
±1
∗ : Ext∗,∗
A (F2 , F2 ) −→ ExtA (ΣP (x ), F2 ) .
Theorem 8.2 (J. Gunawardena). : ΣE(x) ⊗ P (y ±1 ) → Fp induces an isomorphism
∗,∗
±1
∗ : Ext∗,∗
A (Fp , Fp ) −→ ExtA (ΣE(x) ⊗ P (y ), Fp ) .
We shall discuss these theorems in a later lecture, using the algebraic Singer construction.
[[Add references to Atiyah, Lin, Gunawardena, Greenlees–May.]]
References
[Ada84] J. F. Adams, Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture, Algebraic
topology, Aarhus 1982 (Aarhus, 1982), Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984,
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Department of Mathematics, University of Oslo, Norway
E-mail address: rognes@math.uio.no
URL: http://folk.uio.no/rognes
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