HIGHER SINGER CONSTRUCTIONS AND THE SEGAL CONJECTURE FOR
SMASH POWERS
HÅKON SCHAD BERGSAKER
Introduction
Around 1970, Segal (unpublished) formulated his famous conjecture about the stable cohomotopy of classifying spaces, which says the following. For a finite group G, let A(G) denote
the Burnside ring of G, which by definition is the group completion of the semi-ring of isomorphism classes of finite G-sets, under disjoint union and cartesian product. Segal defined a map
A(G) → πs0 (BG+ ) into the zeroth stable cohomotopy group of BG+ , and the conjecture states
that this map is an isomorphism after completing A(G) at its augmentation ideal.
∗
(S 0 ) →
In 1984, Carlsson published a proof [2] of a more general result, namely that the map πG
0
∗
πs (BG+ ) induced by the non-trivial based map BG+ → S is an isomorphism, in all degrees,
∗
again after a suitable completion of the source. Here πG
denotes stable G-equivariant cohomotopy,
0
∼
and Segal’s original conjecture follows since A(G) = πG
(S 0 ). Carlsson proof used new techniques
in equivariant stable homotopy, among other things, to reduce to the case when G is elementary
abelian. Calculations by Adams, Gunawardena and Miller [1] could then be applied to conclude
the argument.
Another formulation of Carlsson’s result is the statement that the map SG → ShG from fixed
points to homotopy fixed points of the G-equivariant sphere spectrum, is an equivalence after some
completion. When G is a p-group, p-completion suffices. This formulation prompts the question
of whether the map X G → X hG is (close to) an equivalence for more general G-spectra X. This
can not be expected to hold in full generality, but one family of examples is given by the following
result by Lunøe-Nielsen and Rognes [8].
Theorem 0.1 (Lunøe-Nielsen-Rognes). Let p be a prime, and let B be a bounded below spectrum
with H∗ (B; Fp ) of finite type. Then
(B ∧p )Cp → (B ∧p )hCp
is an equivalence after p-completion.
The proof of this theorem uses a topological Singer construction which realizes the classical
Singer construction on continuous cohomology. Recall that the Singer construction for A -modules
is an endofunctor R+ on the category of A -modules, equipped with an augmentation R+ M → M
which induces an isomorphism of Ext groups over A . This makes it very useful in combination
with the Adams spectral sequence, and another main ingredient in the proof of the above theorem
is an Adams spectral sequence for continuous cohomology.
The purpose of this paper is to construct certain higher rank versions of these Singer constructions, both algebraic and topological, and use them to prove a generalization of Theorem 0.1 to
elementary abelien groups. An argument similar to the one used by Carlsson let us go from elementary abelian groups to general p-groups, resulting in the following main result, appearing as
Corollary 3.9.
Date: April 1, 2016.
Parts of this research was done while the author was supported by the ”Topology in Norway” project FRINATEK
ES479962.
1
2
HÅKON SCHAD BERGSAKER
Theorem 0.2. Let G be a finite p-group and let B be a bounded below spectrum with H∗ (B; Fp )
of finite type. Then the map
γ : B ∧G → F (EG+ , B ∧G )
is a G-equivalence after p-completion. In particular, the map
γ G : (B ∧G )G → (B ∧G )hG
is an equivalence after p-completion.
As another application of the techniques used in this paper is the following result for iterated
topological Hochschild homology. Recall that the topological Hochschild homology T HH(B) of
an associative (structured) ring spectrum B is a T -equivariant spectrum defined by mimicking the
construction of Hochschild homology for rings. Here T denotes the circle group. The iterated, or
higher, topological Hochschild homology T HH n (B) is a T n -equivariant spectrum, where T n is the
n-torus, which can be thought of as applying T HH n times.
Theorem 0.3. Let B be a bounded below associative ring spectrum with H∗ (B; Fp ) of finite type,
and assume that
T HH n (B)A → T HH n (B)hA
is an equivalence after p-completion for all elementary abelian p-subgroups A ⊂ T n . Then
T HH n (B)G → T HH n (B)hG
is an equivalence after p-completion for all finite p-subgroups G ⊂ T n .
The reduction argument from general p-groups to elementary abelian p-groups works whenever
the various geomtric fixed points of the spectra in question is again of the “same type”. The precise
statement of this is given by the results in Section 3. The smash power and T HH spectra are
examples of this since ΦH (B ∧G ) = B ∧G/H for H ⊂ G a normal subgroup, and ΦH (T HH n (B)) =
T HH n (B) for H ⊂ T n finite.
Now for a more detailed table of contents for each Section. In Seciotn 1 we define our higher
algebraic Singer constructions, after setting up some A -module and A∗ -comodule algebra. More
precisely, for each integer k ≥ 1 we construct an endofunctor Rk : MA → MA on the category of
left A -modules, where A denotes the Steenrod algebra. This functor comes with an augmentation
: Rk → Id to the identity functor, such that the induced map on Ext-groups
(1)
∗,∗
ExtA
(M, Fp ) → Ext∗,∗
A (Rk (M ), Fp )
is an isomorphism for all A -modules M . This is the main result in Section 1.
In Subsection 1.2 we compare our Singer construction to other constructions bearing the same
name. For k = 1 the functor Rk coincides with the classical Singer construction, and for p = 2
and k ≥ 1 it coincides with the functor considered by Kulich in his Northwestern thesis [5]. Our
construction generalizes Kulich’s functor to odd primes, but we use a more conceptual construction
similar to the one used by Adams-Gunawardena-Miller in the case k = 1.
In Subsection 1.3 we identify the coefficient module Rk (Fp ). Here some modular representation
theory come into play, and in particular the interplay between the Steinberg idempotent and the
A -module structure on the algebra Pk = E(x1 , . . . , xk ) ⊗ P (y1 , . . . , yk ), where the xi are in degree
1 and the yi are in degree 2. The general linear group GLk = GLk (Fp ) acts on Pk in a natural
way. Let lk ∈ Pk be the product of all non-zero linear forms in the yi , and let ek ∈ Fp [GLk ] be the
Steinberg idempotent. The main result in this subsection is that Rk (Fp ) ∼
= Σk Pk [lk−1 ]ek .
In Section 2 we construct topological versions of the algebraic Singer constructions from Section
1. These topological Singer constructions are naturally given as inverse limits of certain towers
of spectra, and their continuous cohomology is shown to be isomorphic to the algebraic Singer
construction applied to the cohomology of the original spectrum. In other words, Hc∗ (Rk (B)) ∼
=
Rk (H ∗ (B)), for a spectrum B. Before show this, we have to review Carlsson’s theory of S-functors,
and his filtration of the singular set functor.
HIGHER SINGER CONSTRUCTIONS AND THE SEGAL CONJECTURE FOR SMASH POWERS
3
Finally, in Section 3 we prove our results about smash powers and topological Hochschild homology.
1. Algebraic Singer constructions
1.1. Definitions and first properties. Write A for the mod p Steenrod algebra and A∗ for its
dual. Recall that as an algebra,
A∗ = E(τ0 , τ1 , . . . ) ⊗ P (ξ1 , ξ2 , . . . )
i
with τi in degree 2p − 1 and ξi in degree 2pi − 2, assuming p is odd. For p = 2,
A∗ = P (ξ1 , ξ2 , . . . )
i
with ξi in degree 2 − 1.
Definition 1.1. Let n ≥ −1. For p = 2 let
n+1
I(n) = (ξ12
n
2
, ξ22 , . . . , ξn+1
, ξs |s ≥ n + 2) ⊂ A∗
be the ideal in A∗ generated by the displayed elements. Similarly for odd p,
n
I(n) = (ξ1p , ξ2p
n−1
, . . . , ξnp , ξs , τt |s ≥ n + 1, t ≥ n + 1) ⊂ A∗ .
Let A(n)∗ = A∗ /I(n) be the quotient algebra.
The ideal I(n) is easily seen to be a Hopf ideal, so A(n)∗ is a quotient Hopf algebra of A∗ . Its
dual Hopf algebra, which is denoted A(n), is a sub-Hopf algebra of A . It can be shown that A(n)
i
i
i
is generated as an algebra by Sq 2 with 0 ≤ i ≤ n at the prime two, and by P p and βP p with
0 ≤ i ≤ n − 1 at odd primes. Note that A(n)∗ , and hence A(n), are finite dimensional.
Definition 1.2. Let k ≥ 0 and n ≥ k − 1. For p = 2 let
n+1−k
2
J(n, k) = (ξk+1
2
, . . . , ξn+1
, ξs |s ≥ n + 2) ⊂ A∗
be the ideal generated by the same elements as I(n), except for the first k generators. Similarly
for odd primes,
n−k
p
J(n, k) = (ξk+1
, . . . , ξnp , ξs , τt |s ≥ n + 1, t ≥ n + 1) ⊂ A∗ .
Let C(n, k)∗ = A∗ /J(n, k).
Note that C(n, 0)∗ = A(n)∗ .
Lemma 1.3. C(n, k)∗ is an A(n)∗ − A(n − k)∗ -bicomodule algebra.
Proof. The result will follow once we show that the coproduct ψ : A∗ → A∗ ⊗ A∗ satisfies the two
conditions
(2)
ψ(J(n, k)) ⊆ I(n) ⊗ A∗ + A∗ ⊗ J(n, k)
(3)
ψ(J(n, k)) ⊆ A∗ ⊗ I(n − 1) + J(n, k) ⊗ A∗ .
For odd primes, the formulas for ψ give
e
ψ(ξsp ) =
X
ξip
j+e
⊗ ξjp
e
i+j=s
ψ(τs ) =
X
j
ξip ⊗ τj + τs ⊗ 1
i+j=s
and the inclusions (2) and (3) follow. Similiar formulas hold for the prime 2.
Definition 1.4.
B(n, k)∗ = C(n, k)∗ [ξk−1 ] .
4
HÅKON SCHAD BERGSAKER
Let
ξ∗ : Σ p
(4)
n+1−k
(2pk −2)
A∗ → A ∗
n−k+1
n−k+2
be multiplication by ξk2
for p = 2, and multiplication by ξkp
the same symbol the map restricted to (a suspension of) C(n, k)∗ .
n+1−k
for odd primes. Denote by
k
(2p −2)
Lemma 1.5. The map ξ∗ : Σp
C(n, k)∗ → C(n, k)∗ is an A(n)∗ − A(n − k)∗ -bicomodule
homomorphism, and there is a short exact sequence
(5)
0
Σ(2p
k
−2)(pn−k+1 )
C(n, k)∗
ξ∗
C(n, k)∗
C(n, k − 1)∗
0
of A(n)∗ − A(n − 1)∗ -bicomodules.
Proof. The exactness of the sequence follows from the definition of C(n, k)∗ . That ξ∗ is a bicon+1−k
module map follows from the formulas for the coproduct, which show that ξkp
(assuming p is
odd) is both a left A(n)∗ -comodule primitive and a right A(n − k)∗ -comodule primitive. Similarly
for p = 2.
Lemma 1.6. The canonical map
C(n, k)∗ → B(n, k)∗
is a map of A(n)∗ − A(n − k)∗ -bicomodules.
Proof. Since B(n, k)∗ can be written as the colimit
B(n, k)∗ = colimj (Σ−j(2p
k
−2)(pn−k+1 )
C(n, k)∗ )
over iterations of the map ξ∗ , it has the structure of a A(n)∗ − A(n − k)∗ -bicomodule induced from
C(n, k)∗ , by Lemma 1.5. This also shows that the inclusion C(n, k)∗ → B(n, k)∗ is a bicomodule
map.
To ease the notation we introduce the following subalgebras. Let PA (a1 , . . . , am ) ⊆ A(n)∗ denote
the subalgebra generated by some elements a1 , . . . , am ∈ A(n)∗ . Similarly define the subalgebras
PB (b1 , . . . , bm ) ⊆ B(n, k)∗ for bi ∈ B(n, k)∗ and PC (c1 , . . . , cm ) ⊆ B(n, k)∗ for ci ∈ B(n, k)∗ . Also
write ā for the conjugate χ(a) of an element a ∈ A∗ .
Lemma 1.7. At the prime 2,
n+1−k
n−k
, ξ¯22 , . . . , ξ¯n+2−k , . . . , ξ¯n+1 )
A(n)∗ A(n−k)∗ F2 = PA (ξ¯12
as subalgebras of A(n)∗ . For odd primes p,
A(n)∗ A(n−k)∗ Fp = PA (ξ¯1p
n−k
, ξ¯2p
n−k−1
, . . . , ξ¯n+1−k , . . . , ξ¯n , τ̄n+1−k , . . . , τ̄n ) .
Proof. We do the case for odd primes; the proof at the prime two is similar. The elements
ξ¯1p
n−k
, ξ¯2p
n−k−1
, . . . , ξ¯n , τ̄n+1−k , . . . , τ̄n
all lie in A(n)∗ A(n−1)∗ Fp , hence we have an inclusion
(6)
PA (ξ¯1p
n−k
, ξ¯2p
n−k−1
, . . . , ξ¯n , τ̄n+1−k , . . . , τ̄n ) ⊆ A(n)∗ A(n−k)∗ Fp
n+1
2
of the subalgebra generated by these elements. The dimension of A(n)∗ is 2n+1 p(
dual A(n) is free over A(n − k), so the quotient A(n) ⊗A(n−k) Fp has dimension
) , and the
n+1
k
2n+1 p( 2 )
= 2k pk(n+1−k) p(2) .
n−k+1
(
)
n−k+1
2
p 2
Dualizing back to A(n)∗ A(n−k)∗ Fp and comparing dimensions in (6), we see that the inclusion
has to be an equality.
HIGHER SINGER CONSTRUCTIONS AND THE SEGAL CONJECTURE FOR SMASH POWERS
5
Lemma 1.8. As subalgebras of C(n, k)∗ at the prime two,
n+1
F2 A(n)∗ C(n, k)∗ = PC (ξ12
n+2−k
, . . . , ξk2
)
and
n+1−k
2
C(n, k)∗ A(n−k)∗ F2 = PC (ξ¯12
, . . . , ξ¯n+1−k
, ξ¯n+2−k , . . . , ξ¯n+1 ) .
For odd primes,
n
Fp A(n)∗ C(n, k)∗ = PC (ξ1p , . . . , ξkp
n+1−k
)
and
C(n, k)∗ A(n−k)∗ Fp = PC (ξ¯1p
n−k
p
, . . . , ξ¯n−k
, ξ¯n+1−k , . . . , ξ¯n , τ̄n+1−k , . . . , τ̄n ) .
Proof. First we introduce some notation to make the proof more readable. Write L(n, k)∗ for
Fp A(n)∗ C(n, k)∗ and R(n, k)∗ for C(n, k)∗ A(n−k)∗ Fp . Also let
n
P (n, k)∗ = PC (ξ1p , . . . , ξkp
n+1−k
)
and
Q(n, k)∗ = PC (ξ¯1p
n−k
p
, . . . , ξ¯n−k
, ξ¯n+1−k , . . . , ξ¯n , τ̄n+1−k , . . . , τ̄n ) .
For the first claim, left cotensoring the exact sequence in Lemma 1.5 with Fp over A(n)∗ yields
the exact sequence
0
Σ(2p
k
−2)(pn−k+1 )
L(n, k)∗
ξ∗
L(n, k − 1)∗
L(n, k)∗
0.
n−i+1
Using the explicit formulas for the coproduct, we see that ξip
is in L(n, k)∗ for 1 ≤ i ≤ n, so
P (n, k)∗ ⊆ L(n, k)∗ . By induction on k we assume L(n, k − 1)∗ = P (n, k − 1)∗ ; the base case k = 0
is covered by Lemma 1.7. The above exact sequence and an induction on the degree now shows
that L(n, k)∗ = P (n, k)∗ .
For the second claim, we instead right cotensor the sequence in Lemma 1.5 with Fp over A(n−k)∗
and get the exact sequence
0
Σ(2p
k
−2)(pn−k+1 )
R(n, k)∗
ξ∗
R(n, k − 1)∗
R(n, k)∗
0.
We will proceed by induction on k, and to make the induction work we need to know both
C(n, k)∗ A(n−k)∗ Fp and C(n, k)∗ A(n−k−1)∗ Fp at each step. Again the base case k = 0 is covered
by Lemma 1.7, and as before we can lift elements from R(n, k − 1)∗ to R(n, k)∗ . Induction on the
degree shows that the subalgebras generated by these lifted elements are in fact all of R(n, k)∗ . Proposition 1.9. C(n, k)∗ is injective both as a left A(n)∗ -comodule and as a right A(n − k)∗ comodule.
Proof. A result of Milnor-Moore [10, Theorem 4.7] gives splittings
C(n, k)∗ ∼
= A(n)∗ ⊗ (Fp A(n) C(n, k)∗ )
∗
C(n, k)∗ ∼
= (C(n, k)∗ A(n−k)∗ Fp ) ⊗ A(n − k)∗ ,
as left A(n)∗ -comodules and right A(n − k)∗ -comodules, respectively. This implies the injectivity
of C(n, k)∗ .
Consider the additive retraction map
n+1
P (ξ1 , . . . , ξk ) → P (ξ12
n+2−k
, . . . , ξk2
)
for the prime 2, and the corresponding retraction for odd primes. Composing the canonical map
C(n, k)∗ → C(k − 1, k)∗ = P (ξ1 , . . . , ξk ) with this retraction yields a map
n+1
γ∗0 : C(n, k)∗ → P (ξ12
n+2−k
, . . . , ξk2
).
6
HÅKON SCHAD BERGSAKER
Write ψC for the coaction C(n, k)∗ → A(n)∗ ⊗ C(n, k)∗ and ψB for the coaction B(n, k)∗ →
A(n)∗ ⊗ B(n, k)∗ . If we consider the target of the coaction maps as extended comodules, then the
coaction maps are A(n)∗ -comodule maps. Let γ∗ = (1 ⊗ γ∗0 ) ◦ ψC and β∗ = (1 ⊗ β∗0 ) ◦ ψB .
Proposition 1.10. At the prime 2, the maps
n+1
γl∗ : C(n, k)∗ → A(n)∗ ⊗ P (ξ12
βl∗ : B(n, k)∗ → A(n)∗ ⊗
n+2−k
, . . . , ξk2
)
n+2−k
n+1
n+3−k
P (ξ12 , . . . , ξk2
, ξk±2
)
are isomorphisms of left A(n)∗ -comodules, where the targets are considered as extended A(n)∗ comodules. At odd primes, we have the left A(n)∗ -comodule isomorphisms
n
γl∗ : C(n, k)∗ → A(n)∗ ⊗ P (ξ1p , . . . , ξkp
βl∗ : B(n, k)∗ → A(n)∗ ⊗
n+1−k
)
n
n+2−k
n+1−k
P (ξ1p , . . . , ξkp
, ξk±p
).
n
Proof. We do this for odd primes. Write P∗ for PC (ξ1p , . . . , ξkp
n+1−k
). We first show that
1γ∗ : Fp A(n)∗ C(n, k)∗ → Fp A(n)∗ (A(n)∗ ⊗ P∗ )
is an isomorphism. The induced map 1ψC is isomorphic to the inclusion Fp A(n)∗ C(n, k)∗ ⊆
C(n, k)∗ , and the induced map 1(1⊗γ∗0 ) is isomorphic to just γ∗0 . Since 1γ∗ = 1(1⊗γ∗0 )◦1ψC ,
it follows that 1γ∗ is isomorphic to the identity of Lemma 1.8, hence is an isomorphism.
Since A(n)∗ is connected, this implies that γ∗ is injective by [10, Proposition 2.5]. Now since
C(n, k)∗ is an injective A(n)∗ -comodule by Lemma 1.9, the sequence
0
Fp A(n)∗ C(n, k)∗
1γ∗
Fp A(n)∗ P∗
Fp A(n)∗ coker(γ∗ )
0
is exact. We have Fp A(n)∗ coker(γ∗ ) = 0, so coker(γ∗ ) = 0 by [10, Proposition 2.4] and γ∗ is an
isomorphism. By inverting ξkp
n+1−k
we immediately get that β∗ is an isomorphism as well.
Proposition 1.11. At the prime 2, the maps
γr∗ : C(n, k)∗ → ((...)) ⊗ A(n − k)∗
βr∗ : B(n, k)∗ → B(k − 1, k)∗ ⊗ A(n − k)∗
are isomorphisms of right A(n − k)∗ -comodules, where the targets are considered as extended A(n −
k)∗ -comodules. At odd primes, we have the right A(n − k)∗ -comodule isomorphisms
γr∗ : C(n, k)∗ → ((...)) ⊗ A(n − k)∗
βr∗ : B(n, k)∗ → B(k, k)∗ ⊗ A(n − k)∗ .
Proof. Again we consider the case for odd primes.
Now consider the composite map
B(n, k)∗ → B(n, k)∗ ⊗ A(n − k)∗ → B(k, k)∗ ⊗ A(n − k)∗ .
It sits in the following commutative diagram.
Σ−j(2p
k
−2)(pn−k+1 )
B(n, k)∗
C(n, k)∗
Σ−j(2p
k
−2)(pn−k+1 )
C(k, k)∗ ⊗ A(n − k)∗
B(k, k)∗ ⊗ A(n − k)∗
HIGHER SINGER CONSTRUCTIONS AND THE SEGAL CONJECTURE FOR SMASH POWERS
7
We now want to dualize these results. Since B(n, k)∗ is not of finite type, we put a topology on
it and instead consider the continuous dual. Let Fd B(n, k)∗ ⊂ B(n, k)∗ denote the vector space
spanned by the monomials ξ1s1 · · · ξksk with sk ≤ d, for all integers d. This defines an increasing
filtration F∗ B(n, k)∗ on B(n, k)∗ which is exhaustive and complete Hausdorff, and we give B(n, k)∗
the linear topology defined by taking the subspaces Fd B(n, k)∗ as the fundamental system of
neighborhoods.
Definition 1.12. Let
C(n, k) = (C(n, k)∗ )∗
be the dual A(n) − A(n − k)-bimodule coalgebra of C(n, k)∗ , and let
B(n, k) = (B(n, k)∗ )∗c
be the continuous dual of B(n, k)∗ with respect to the topology defined by the filtration F∗ B(n, k)∗ .
If we define a filtration on A(n)∗ ⊗ B(n, k)∗ by
Fd (A(n)∗ ⊗ B(n, k)∗ ) = A(n)∗ ⊗ Fd B(n, k)∗ ,
then the coaction map B(n, k)∗ → A(n)∗ ⊗ B(n, k)∗ becomes a filtered, hence continuous, map.
Similarly for the right coaction. We also need the filtration on B(n, k)∗ ⊗ B(n, k)∗ defined by
M
Fd (B(n, k)∗ ⊗ B(n, k)∗ ) =
Fs B(n, k)∗ ⊗ Ft B(n, k)∗ .
s+t=d
Lemma 1.13. We have the following identifications of continuous duals.
(1) (A(n)∗ ⊗ B(n, k)∗ )∗c = A(n) ⊗ B(n, k)
(2) (B(n, k)∗ ⊗ A(n − k)∗ )∗c = B(n, k) ⊗ A(n − k)
(3) (B(n, k)∗ ⊗ B(n, k)∗ )∗c = B(n, k) ⊗ B(n, k)
Hence B(n, k) is an A(n) − A(n − k)-bimodule coalgebra.
Recall that the Milnor basis for A is dual to the monomial basis for A∗ . Let R = (r1 , . . . , rs )
be a sequence of non-negative integers. Write Sq R ∈ A for the element dual to ξ1r1 . . . ξsrs ∈ A∗ for
the prime 2. For odd primes, write P R ∈ A for the element defined in the same way. Also write
Sq (r1 ,...,rk ) ∈ B(n, k) for the element dual to ξ1r1 . . . ξkrk ∈ B(n, k)∗ , where now rk ∈ Z. Similarly
for P R ∈ B(n, k). Taking continuous duals of the maps in Proposition 1.10 and Proposition 1.11
we have the following results.
Proposition 1.14. At the prime 2, the maps
γl : A(n) ⊗ F2 {Sq(j1 2n+1 , . . . , jk 2n+2−k ) | j1 , . . . , jk ≥ 0} → C(n, k)
βl : A(n) ⊗ F2 {Sq(j1 2n+1 , . . . , jk 2n+2−k ) | j1 , . . . , jk−1 ≥ 0, jk ∈ Z} → B(n, k)
are isomorphisms of left A(n)-modules, where the left hand sides are extended A(n)-modules. At
odd primes, we have the left A(n)-module isomorphisms
γl : A(n) ⊗ Fp {P (j1 pn , . . . , jk pn+1−k ) | j1 , . . . , jk ≥ 0} → C(n, k)
βl : A(n) ⊗ Fp {P (j1 pn , . . . , jk pn+1−k ) | j1 , . . . , jk−1 ≥ 0, jk ∈ Z} → B(n, k)
Proposition 1.15. At the prime 2, the maps
γr : C(k − 1, k) ⊗ A(n − k) → C(n, k)
βr : B(k − 1, k) ⊗ A(n − k) → B(n, k)
are isomorphisms of right A(n − k)-modules. At odd primes, we have the right A(n − k)-module
isomorphisms
γr : C(k, k) ⊗ A(n − k) → C(n, k)
βr : B(k, k) ⊗ A(n − k) → B(n, k)
8
HÅKON SCHAD BERGSAKER
For any A(n − k)-module M , the tensor product B(n, k) ⊗A(n−k) M is a left A(n)-module, and
the map B(n, k) → B(n + 1, k) induces a map
B(n, k) ⊗A(n−k) M → B(n + 1, k) ⊗A(n+1−k) M
(7)
of left A(n)-modules. If we start with an A -module M we can form the direct limit over the maps
in (7), and we get an induced A -module structure on the limit. Note that by Proposition 1.15
these maps are in fact isomorphisms.
The composite map B(n, k) → C(n, k) → A induces a map
B(n, k) ⊗A(n−k) M → A ⊗A(n−k) M
which is compatible with the maps in the direct systems. Taking the colimit over n of both sides
yields a map : Rk (M ) → M .
Definition 1.16. Let M be an A -module. The A -module
Rk (M ) = colimn (B(n, k) ⊗A(n−k) M )
is the (cohomological) Singer construction on M . The induced A -linear map
: Rk (M ) → M
is the augmentation.
Proposition 1.17. Let M be a free A -module. Then Rk (M ) is A -flat, and the map
1 ⊗ : Fp ⊗A Rk (M ) −→ Fp ⊗A M
is an isomorphism. Furthermore, Rk is an exact functor.
Proof. That Rk is exact follows from the fact that B(n, k) is a free right A(n − k)-module and that
colim is exact. Now let M be a free A -module. Since M is free over A , it is free over A(n − k).
So B(n, k) ⊗A(n−k) M is a direct sum of copies of B(n, k), hence is a free A(n)-module by Lemma
1.14. So Rk (M ) is free as a left A(n)-module for all n, and
A(n)
TorA
s,∗ (L, Rk (M )) = colimn Tors,∗ (L, Rk (M )) = 0
for all A -modules L and s ≥ 1. Hence Rk (M ) is flat as an A -module.
To establish that 1 ⊗ is an isomorphism, we need only consider the case M = A , since Rk is
exact. We now have
Fp ⊗A Rk (A ) = colimn (Fp ⊗A(n) Rk (A ))
= colimn (Fp ⊗A(n) B(n, k) ⊗A(n−k) A )
= colimn colimm≥n−k (Fp ⊗A(n) B(n, k) ⊗A(n−k) A(m)
= colimn (Fp ⊗A(n) B(n, k) ⊗A(n−k) A(n − k)
= colimn (Fp ⊗A(n) B(n, k)) .
By Proposition 1.14, B(n, k) is free over A(n) on generators P R with R = (j1 pn , . . . , jk pn+1−k ).
The maps in the direct limit
Fp ⊗A(n) B(n, k) → Fp ⊗A(n+1) B(n + 1, k)
are isomorphic to the maps
Fp {P (j1 p
R
R
n
,...,jk pn+1−k )
n+1
} → Fp {P (j1 p
,...,jk pn+2−k )
}
which send P to P when R = (0, . . . , 0) or when p divides js for 1 ≤ s ≤ k. The other basis
elements map to zero. Hence in the direct limit everything except for P 0 eventually dies, and
Fp ⊗A Rk (A ) ∼
= Fp . The generator 1 ⊗ P 0 maps under 1 ⊗ to P 0 (1) = 1 in Fp ⊗A A ∼
= Fp , so
1 ⊗ is an isomorphism.
HIGHER SINGER CONSTRUCTIONS AND THE SEGAL CONJECTURE FOR SMASH POWERS
9
Theorem 1.18. Let M be an A -module. The induced map
A
∗ : TorA
∗,∗ (Fp , Rk (M )) −→ Tor∗,∗ (Fp , M )
is an isomorphism.
Proof. Let
. . . −→ Fs −→ · · · −→ F1 −→ F0 −→ M
be a free resolution of M . Applying Rk gives a flat resolution
. . . −→ Rk (Fs ) −→ · · · −→ Rk (F1 ) −→ Rk (F0 ) −→ Rk (M )
of Rk (M ), by Proposition 1.17. We have the following commutative diagram
···
Rk (Ps )
···
Rk (F1 )
···
Fs
···
Rk (F0 )
F1
F0 ,
and after applying the indecomposables functor Fp ⊗A (−), the vertical maps become isomorphisms by 1.17. Hence the induced map
A
∗ : TorA
∗,∗ (Fp , Rk (M )) −→ Tor∗,∗ (Fp , M )
on left derived functors is an isomorphism.
Corollary 1.19. Let M be an A -module. The induced map
∗,∗
∗ : ExtA
(M, Fp ) −→ Ext∗,∗
A (Rk (M ), Fp )
is an isomorphism.
Proof. This follows from the previous theorem and the natural isomorphism
A
∗
∼
Ext∗,∗
A (L, Fp ) = (Tor∗,∗ (Fp , L))
established in [7, Lemma 4.3].
1.2. Relation to other Singer constructions. Here we will compare our Singer construction
to other constructions in the literature that bear the same name. In the rank 1 case, our definition
of R1 coincides with the functor T used by Adams-Gunawardena-Miller in their work on the Segal
conjecture for elementary abelian p-groups [1]. We will now give a description of our Rk in terms
of the Cartan-Serre basis, and in doing so, show that our R1 also coincides with Singer’s (and
Li-Singer’s for odd primes) original functor R+ constructued in [16, 6].
Assume p is odd. Let I = (0 , i1 , 1 , i2 , . . . , ik ) be a sequence of integers such that s = 0, 1 and
is ≥ pis+1 + i . Note that the is are now allowed to be negative. In this section we call such a
sequence an admissible sequence. Let Fp {P I | I admissible} be the vector space with basis given
by all symbols P I with I admissible, and define
Tk (M ) = Fp {P I | I admissible} ⊗ M
for an A -module M .
We will define an A -module structure on Tk (M ). Let P I ⊗ m ∈ Tk (M ). For the Bockstein β,
define β(P I ⊗ m) = (βP I ) ⊗ m. Now let P a be a Steenrod power, and write I + pe for the sequence
(8)
(0 , i1 + pe−1 , 1 , i2 + pe−2 , . . . , ik + pe−k , k ) .
Choose e sufficiently large. By the Adem relations we have
X
e
P a P I+p =
cJ P J
J
10
HÅKON SCHAD BERGSAKER
where the sum runs over admissible sequences J = (0 , j1 , . . . ) with l(J) ≤ k + 1, and cJ ∈ Fp . We
now define P a (P I ⊗ m) to be
X
e
e−k+1
cJ β 0 P j1 −p . . . P jk −p
β k ⊗ P jk+1 (m) .
J
This gives the required A -module structure on Tk (M ). Just as for Rk , there is an A -linear
augmentation map : Tk (M ) → M , now given by
(
P I (m) if is ≥ 0 for all s
(P I ⊗ m) =
0
otherwise .
For p = 2, there is a similar construction of an A -module structure on
Tk (M ) = F2 {Sq I | I admissible} ⊗ M ,
where now I = (i1 , . . . , ik ) is an admissible sequence with possibly negative entries. At the prime
2, Tk is the (higher) Singer construction defined by Kulich in his thesis [5].
Proposition 1.20. There is a natural isomorphism of A -modules
φk : Rk (M ) → Tk (M )
such that ◦ φk = . In particular
R1 (M ) ∼
= R+ (M )
with compatible augmentation maps.
To prove this we need some preliminary results on the relation between the Cartan-Serre basis
and the Milnor basis. Let R denote the set of all finite sequences I = (0 , i1 , 1 , i2 , . . . ) of integers
where is ≥ 0 and s = 0, 1. Write P I for the element β 0 P i1 β 1 P i2 · · · ∈ A . Recall that I is called
admissible if is ≥ pis+1 + s for all s, and let I ⊂ R denote the subset of all admissible sequences.
We define the length l(I) of I by l(I) ≤ m if rs = 0 for s > m and s = 0 for s ≥ m. We then set
l(I) = m if l(I) ≤ m and l(I) m − 1.
For I ∈ R, write ξ I for the monomial τ00 ξ1i1 τ11 ξ2i2 . . . in A∗ . The collectioen of these monomials
constitute a basis for A∗ , and the dual basis is the Milnor basis for A . Write P (I) for the basis
element in A dual to ξ I .
Define γ : I → R by
φ(0 , i1 , 1 , i2 , . . . ) = (0 , i1 − pi2 − 1 , 1 , i2 − pi3 − 2 , . . . ) ;
this is a bijection. We also define Rm ⊂ R to be the subset of sequences I with l(I) ≤ m.
Similarly, Im ⊂ I is the subset of admissible sequences I with l(I) ≤ m. The bijection γ restricts
to a bijection γm : Im → Rm .
We lexicographically order sequences I ∈ R from the right, and give I, Rm and Im the induced
orderings. Let h, i : A ⊗ A∗ → Fp denote the canonical pairing. Then we have
(
±1 if I < J
I γ(J)
hP , ξ
i=
0
if I = J
for admissible sequences I, J. This is proved (for odd primes) in [9, Lemma 8].
Now define two increasing filtrations F∗ and F∗0 of A by
Fm = Fp {P I | I ∈ Im }
0
Fm
= Fp {P (R) | R ∈ Rm } .
Lemma 1.21. The two filtrations F∗ and F∗0 of A coincide.
HIGHER SINGER CONSTRUCTIONS AND THE SEGAL CONJECTURE FOR SMASH POWERS
11
0
Proof. Let P (R) ∈ Fm
.
The bijection γ restricts to a bijection φm : Im → Rm , where Im ⊂ I and Rm ⊂ R denote
subsets of sequences of length at most m. Hence there is a one-to-one correspondence between
0
0
basis elements in Fm and Fm
, so the inclusion Fm
⊆ Fm has to be an equality.
Corollary 1.22. As vector spaces,
C(k − 1, k) ∼
= F2 {Sq I | I ∈ Ik , ik ≥ 0}
∼ F2 {Sq I | I ∈ Ik }
B(k − 1, k) =
at the prime 2. For odd primes,
C(k, k) ∼
= Fp {P I | I ∈ Ik , ik ≥ 0}
B(k, k) ∼
= Fp {P I | I ∈ Ik } .
Proof. Since
C(k, k)∗ = P (ξ1 , . . . , ξk ) ⊗ E(τ0 , . . . , τk ) ,
we have C(k, k) =
Fk0
= Fk .
Let
ξ : A → Σp
n+1−k
(2pk −2)
A
be the dual of the map ξ∗ in (4).
Lemma 1.23. The map ξ is given by
(
I
ξ(Sq ) =
Sq I−2
0
n+k
if I − 2n+k ≥ 0
.
otherwise
when p = 2, and
(
I
ξ(P ) =
P I−p
0
n+k−1
if I − pn+k−1 ≥ 0
.
otherwise
for p odd.
Proof. Let P I be a monomial in A , and x ∈ A∗ any element. Then
hξ(P I ), xi = hP I , ξ∗ (x)i
= hψ(P I ), ξkn−k+1 ⊗ xi
X
=
hP J , ξkn−k+1 ihP K , xi
J+K=I
= hP I−p
n−k+1
, xi
and the result for odd p follows. As usual the same proof with notational adjustments works for
p = 2.
Proof of Proposition 1.20. The inclusion B(k, k) → B(n, k) induces an isomorphism of vector
spaces
B(k, k) ⊗ M → B(n, k) ⊗A(n−k) M
for all n. Passing to the direct limit over n, we obtain the map
φk : Tk (M ) ∼
= B(k, k) ⊗ M → Rk (M )
which is also a vector space isomorphism. We have to show that it commutes with the A -actions
defined on Tk (M ) and Rk (M ).
For k = 1, the definition of T1 (M ) as an A -module reduces to the definition of R+ (M ) given by
Li-Singer in [6, (3.4)–(3.7)]. Here our β 0 P i1 corresponds to their u0 v i1 . Also the augmentation
maps coincide by [6, (1.1)].
12
HÅKON SCHAD BERGSAKER
1.3. The coefficient module Rk (Fp ). First we recall some modular representation theory and in
particular the Steinberg idempotent. Let GLk = GLk (Fp ) be the general linear group over Fp and
Fp [GLk ] its associated group ring. For a subgroup H ⊆ GLk we define an element H ∈ Fp [GLk ]
as follows.
(P
sgn(h)h if H ⊆ Σk
,
H = Ph∈H
if H * Σk
h∈H h
where sgn : Σk → {±1} is the sign map. Here Σk ⊂ GLk is the subgroup of permutation matrices.
Also let Bk be the subgroup of upper triangler matrices, and Uk the subgroup of triangular matrices
with diagonal entries all equal to 1.
The Steinberg idempotent ek ∈ Fp [GLk ] is defined to be the element
ek =
Σk B k
.
[GLk : Uk ]
Steinberg shows in [17] that ek is idempotent. The associated module St = ek Fp [GLk ] is called
the Steinberg module.
Let
Pk = P (y1 , . . . , yk )
for p = 2, and
Pk = E(x1 , . . . , xk ) ⊗ P (y1 , . . . , yk )
for odd primes. Here xi is in degree 1 and yi is in degree 2. GLk acts on the right on the vector
space spanned by x1 , . . . , xk , and on the vector space spanned by y1 , . . . , yk . Extend this action to
a (right) algebra action on Pk . Let lk ∈ Pk be the product of all non-zero linear combinations of
the yi . The element lk is GLk -invariant, so we can invert lk to form the A -algebra Pk [lk−1 ] with
its induced GLk -action. The A -action on Pk [lk−1 ] is the unique extension of the A -action on Pk ,
given by the Cartan formula. See [?] for more details.
We will need the following results by Mitchell-Priddy. Let
Gk = Fp {P I | I ∈ I, l(I) > m} .
Using the Adem relations this is easily seen to be a left ideal in A . Now write
Xk = x1 . . . xk ∈ Pk
Yk = y1 . . . yk ∈ Pk ,
and define a map ψk : Gk−1 →
Pk [lk−1 ]
by
ψk (P I ) = P I (Xk Yk−1 ) .
It follows from [11, Lemma 3.6] that ψk lands in Pk ⊂ Pk [lk−1 ], and also that ψk vanishes on Gk .
A special case of [11, Lemma 5.9] tells us that (Xk Yk−1 )ek = Xk Yk−1 , so ψk (P I ) is invariant under
ek . Combining [11, Proposition 3.5] with [11, Theorem 5.8] gives the following result.
Lemma 1.24 (Mitchell-Priddy). The map
ψ̄k : Gk−1 /Gk → Pk ek
induced by ψk is an isomorphism of A -modules.
Lemma 1.25. Let I ∈ R with l(I) ≤ k − 1, and let = 0 + . . . n−2 . Then
(x1 y1−1 P I (x2 y2−1 . . . xk yk−1 ))ek = (−1) P I (x1 y1−1 . . . xk yk−1 ) .
j
k−1
Proof. Let J = (j0 , . . . , jk−1 ) with js = 0, 1, and set j = j0 + . . . jk−1 . Write QJ = Qj00 . . . Qk−1
,
where the Qs ∈ A denote the Milnor primitives. Let I0 ∈ R be a sequence with l(I0 ) ≤ k − 1
where all the s -terms are zero. Lemma 5.9 in [11] states that
(9)
(x1 y1−1 QJ P I0 (x2 y2−1 . . . xk yk−1 ))ek = (−1)j QJ P I0 (x1 y1−1 . . . xk yk−1 ) .
HIGHER SINGER CONSTRUCTIONS AND THE SEGAL CONJECTURE FOR SMASH POWERS
13
To apply this to P I we use [9, Theorem 4a] to rewrite as
X
(10)
PI =
QJi P Ii (Xk Yk−1 ) ,
i
where Ji has exactly non-zero terms and Ii has all s -terms equal to zero. This follows by an
s
s
easy induction and the formula P i Qs = Qs P i + Qs+1 P i−p , where P i−p is interpreted as zero
when i < ps . We can now apply (9) to each term in the sum (10), and the lemma follows.
We will need an explicit description of the action of the Steenrod powers on lk . To this end, we
introduce the Moore determinants
k−1
k−1
k−1
p
y2p
. . . ykp y1
pk−2
k−2
k−2 y
y2p
. . . ykp Mk = 1 .
..
.. .
..
.
. y1
y2
...
yk More generally, for 0 ≤ i ≤ k, let
Mk,i
pk
y
1
.
..
pi+1
= y1 i−1
y p
1
.
..
yp
1
y2p
..
.
k
i+1
...
y2p
i−1
y2p
..
.
...
...
y2p
...
k
i+1 ykp .
i−1 ykp .. . yp ykp
..
.
k
Note that Mk = Mk,k . In [4] Dickson used these determinants to describe certain GLk -invariants
in Pk , and in particular our lk can be so described.
Lemma 1.26. The element lk ∈ Pk can be written as
lk = Mkp−1 .
The action of the Steenrod powers on lk is given by
p−1
ik
i0
pk+1 −1−i
P
(lk ) =
Mk,k
. . . Mk,0
,
ik . . . i0
where ik + · · · + i0 = p − 1 and i = ik pk + ik−1 pk−1 + · · · + i0 . The remaining P j (lk ) are all zero.
Proof. By definition,
lk =
Y
(a1 y1 + · · · + ak yk ) ,
a6=0
where a = (a1 , . . . , ak ) runs through all non-zero vectors in F×k
p .
Let P = P 0 + P 1 + . . . be the the total Steenrod power. We first describe P (Mk ). Since P is
an algebra map and P (yi ) = yi + yip , we get
k−1
k−1
k
k
k−1
k
p
+ y1p
y2p
+ y2p
...
ykp
+ ykp y1
pk−2
k−1
k−2
k−1 k−1
k−2
y
+ y1p
y2p
+ y2p
. . . ykp
+ ykp P (Mk ) = 1
.
..
..
..
.
.
.
p
p
y1 + y p
y
+
y
.
.
.
y
+
y
k
2
1
2
k
Since the determinant is multilinear in its rows, P (Mk ) expands as a sum of 2k determinants. The
only determinants in this sum which do not have two identical rows are the ones dubbed Mk,i .
Hence
P (Mk ) = Mk,k + · · · + Mk,0 .
14
HÅKON SCHAD BERGSAKER
Now we have
(11)
P (lk ) = P (Mk )
p−1
X
=
p−1
ik
i0
Mk,k
. . . Mk,0
ik . . . i0
ik +···+i0 =p−1
k
i+1
The degree of Mk,i is 2(p + · · · + p
+ pi−1 + · · · + 1), and a quick calculation shows that
ik
i0
k+1
Mk,k . . . Mk,0 lie in degree 2(p
− 1 − i), with i as in the statement of the lemma. In particular
the terms in (11) are all in different degrees, and the lemma follows.
Define the map ζ : Gk−1 /Gk → Gk−1 /Gk in the following way. Let P I ∈ Gk−1 , where I is an
m
admissible sequence with l(I) = m. Then ζ sends the class of P I to the class of P I+p . (Recall
the notation defined in (8).)
Lemma 1.27. The diagram
ψ̄k
Gk−1 /Gk
P k ek
lk ·
ζ
ψ̄k
Gk−1 /Gk
P k ek
commutes.
Proof. We use induction on k. For k = 1, let I = (0 , i1 ). The Steenrod powers act on y1−1 by
i(p−1)−1
P i (y1−1 ) = (−1)i y1
(12)
.
On the one hand, we have
(i +1)(p−1)−1
ψ̄1 (ζ(P I )) = β 0 P i1 +1 (x1 y1−1 ) = (−1)i1 +1 β 0 x1 y1 1
,
while on the other
i (p−1)−1
l1 ψ̄1 (P I ) = l1 β 0 P i1 (x1 y1−1 ) = (−1)i1 l1 β 0 (x1 y11
).
−y1p−1 .
These expressions are equal since l1 =
For the inductive step, assume the lemma holds for k − 1, and let I = (0 , i1 , . . . , k−1 , ik ) and
0
I 0 = (1 , i2 , . . . , k−1 , ik ). Also let lk−1
be the product of all non-zero linear forms in the variables
y2 , . . . , yk . We have
k
k−1
P I+p (Xk Yk−1 ) = β 0 P i1 +p
(x1 y1−1 P I
0
+pk−1
(x2 y2−1 . . . xk yk−1 ))ek
0
k−1
0
= β 0 P i1 +p (x1 y1−1 lk−1
P I (x2 y2−1 . . . xk yk−1 ))ek
X
k−1
0
0
=
(β 0 P i1 +p −m (x1 y1−1 lk−1
)P m P I (x2 y2−1 . . . xk yk−1 ))ek
(13)
m
by Lemma 1.25, the inductive hypothesis, and the Cartan formula. On the other hand,
0
(14)
lk P I (Xk Yk−1 ) = lk β 0 P i1 (x1 y1−1 P I (x2 y2−1 . . . xk yk−1 ))ek
X
0
=
(lk β 0 P i1 −m (x1 y1−1 )P m P I (x2 y2−1 . . . xk yk−1 ))ek
m
again by Lemma 1.25 and the Cartan formula. (Note that multiplication by lk commutes with ek
since lk is GLk -invariant.) Comparing (13) and (14), and noting that β commutes with lk and the
P i commutes with x1 , it remains to prove that
P i+p
(15)
k−1
0
(y1−1 lk−1
) = lk P i (y1−1 )
for all i. Now
P
i+pk−1
0
(y1−1 lk−1
)
=
pk−1
X−1
j=0
P i+p
k−1
−j
0
(y1−1 )P j (lk−1
)
HIGHER SINGER CONSTRUCTIONS AND THE SEGAL CONJECTURE FOR SMASH POWERS
15
by the Cartan formula, and expanding the action on y1−1 using (12) transforms the equality (15)
into
pk−1
X−1
(pk−1 −j)(p−1)
(−1)j y1
(16)
0
P j (lk−1
) = lk .
j=0
0
Note that the maximal j in these sums is pk−1 − 1, since P j (lk−1
) vanishes for j ≥ pk−1 .
We show (16) by substituting the determinant expressions given in Lemma 1.26. The left hand
side can now be written as
X p − 1 ik−1
(pk−1 −j)(p−1)
i0
(−1)j y1
Mk−1,k−1
. . . Mk−1,0
,
i
.
.
.
i
k−1
0
j
while the right hand side is

p−1
k−1
X
j
lk = Mkp−1 =  (−1)j y1p Mk−1,j 
j=0
=
X
ik−1 +···+i0 =p−1
p−1
ik−1
i
pk−1 +···+i0
i0
Mk−1,k−1
. . . Mk−1,0
,
y k−1
ik−1 . . . i0 1
where we expand the determinant Mk along the first coloumn. Since the two sides are now seen
to be equal, we are done.
Write colimζ Gk−1 /Gk for the colimit
colim(Gk−1 /Gk
ζ
ζ
Gk−1 /Gk
...).
Similarly we have colimlk Pk ek . There are maps of A -modules fm : Gk−1 /Gk → Tk (Fp ) which
m
sends the class of P I to P I−p ⊗ 1. These are compatible under the map ζ, hence defines a map
f : colimζ Gk−1 /Gk → Tk (Fp ).
Lemma 1.28. The induced map
f : colimζ Gk−1 /Gk → Tk (Fp )
is an isomorphism of A -modules.
By Lemma 1.27 and Lemma 1.28 we have the composite isomorphism
Tk (Fp ) ∼
= colimζ Gk−1 /Gk ∼
= coliml Pk ek ∼
= Pk [l−1 ]ek ,
k
k
which combined with Proposition 1.20 yields the following result.
Theorem 1.29. There is an isomorphism of A -modules
Rk (Fp ) ∼
= Σk Pk [l−1 ]ek .
k
Corollary 1.19 combined with Theorem 1.29 results in the following, originally proved by PriddyWilkerson in [15].
Corollary 1.30. There is an isomorphism
∗,∗
Ext∗,∗ (Fp , Fp ) ∼
= Ext (Σk Pk [l−1 ]ek , Fp ) .
A
A
k
2. Topological Singer constructions
In this section we will geometrically realize the algebraic Singer constructions defined in the
previous section. Given a spectrum X, we seek a spectrum Rk (X) such that H ∗ (Rk (X)) ∼
=
Rk (H ∗ (X)). Unfortunately, this is not possible as it stands. Instead we will construct a tower of
spectra whose continuous cohomology is isomorphic to Rk (H ∗ (X)).
16
HÅKON SCHAD BERGSAKER
2.1. Models for equivariant smash powers. Here we discuss suitable models for the equivariant smash powers of spectra. Since most of our arguments will take place in the (equivariant or
non-equivariant) stable homotopy category, we don’t want to be too model specific.
2.2. S-functors. We review Carlsson’s theory of S-functors, both for completeness and the fact
that we need a refinement where we keep track of the GLk -action in the case where the group is
elementary abelian. We follow Caruso-May-Priddy’s exposition and refer to [?, Section 3-4], [?, ?]
for more details.
An S-functor is an endofunctor T on the category of G-complexes together with a natural
transformation
τ : T (X ∧ Y ) → T (X) ∧ Y ,
natural in both X and Y , such that τ is the identity for Y = S 0 and a homeomorphism when G
acts trivially on Y . In addition we require
T (X ∧ Y ∧ Z)
T (X ∧ Y ) ∧ Z
T (X) ∧ Y ∧ Z
to commute for all X, Y, Z. Maps of S-functors are natural transformations, with cofibrations and
equivalences defined componentwise.
The singular set functor
[
S(X) =
XH
16=H⊆G
with τ as the inclusion is the prototypical example of an S-functor. An important family of
examples is given by the following construction. Given subgroups K ⊆ H ⊆ G, with K normal in
H, let
C(K, H)(X) = G+ ∧H X K .
Define maps
τ : G+ ∧H (X K ∧ Y K ) → (G+ ∧H X K ) ∧ Y
by induction on the H-equivariant inclusion X K ∧ Y K → (G+ ∧H X K ) ∧ Y . Then C(K, H) with
τ is easily seen to be an S-functor.
The following theorem describes a model for the singular set functor which comes with a convenient filtration. Part 1 and 2 constitutes [?, Theorem 4.1]. Recall that the p-rank of a finite
p-group is the maximal length of a chain of non-trivial elementary abelian subgroups.
Theorem 2.1. Let G be a finite p-group of p-rank k.
(i) There is an S-functor A and an equivalence A → S. The functor A has a filtration
F0 A ⊂ F1 A ⊂ · · · ⊂ Fk−1 A = A
by successive cofibrations.
Fq A/Fq−1 A ∼
=
_
Σq C(A(ω), H(ω))
(ω)
(ii) If G =
Cp×k ,
e with filtration
there is an S-functor A
e ⊂ F1 A
e ⊂ · · · ⊂ Fk−2 A
e=A
e
F0 A
and filtration quotients
e q−1 A
e∼
Fq A/F
=
_
Σq C(A(ω), H(ω))
(ω)
Cp×k
e has induced
(iii) If G =
and X has an action of Affk extending the G-action, both A and A
actions of Affk such that ... are equivariant.
HIGHER SINGER CONSTRUCTIONS AND THE SEGAL CONJECTURE FOR SMASH POWERS
Sketch of proof.
17
2.3. Cohomology of the topological Singer construction.
Definition 2.2. Let X be a spectrum with H∗ (X) bounded below, and define
Ek (X) = holimn Σn DG (Σ−n X)
Fk (X) = holimn ek Σn DG (Σ−n X) ,
where ek ∈ Fp [GLk ] is the Steinberg idempotent defined in 1.3.
Note that E1 and F1 coincide; this construction was used by Jones and Wegmann in their
geometric version of Lin’s theorem [?].
Lemma 2.3. Let X be a spectrum with H∗ (X) bounded below. Then
H∗ (DG (X)) ∼
= Fp {}
k
k
Proof. By [?] we have H∗ (DG (X)) ∼
= H∗ (G; H∗ (X)⊗p ), where G ⊂ Σpk acts on H∗ (X)⊗p by
permuting factors. The right hand side can be computed by standard results in group homology.
For k = 1 one can use the complex
0.
The case k ≥ 2 follows from this by induction and the Künneth formula
H∗ (G1 × G2 ; M1 ⊗ M2 ) ∼
= H∗ (G1 ; M1 ) ⊗ H∗ (G2 ; M2 )
valid for any G1 -module M1 and G2 -module M2 .
Proposition 2.4.
Rk (H ∗ (X)) ∼
= Hc∗ (Fk (X))
Proof.
One possible definition of a topological Singer construction would be Fk (X). However, with the
application in the next section in mind, we find it is better to make the following definition. This
is also a more direct generalization of the rank 1 case defined by Lunøe-Nielsen-Rognes [?].
Definition 2.5. Let G = Cp×k be an elementary abelian p-group, and let X be a spectrum. The
Singer construction of rank k on X is given by
Rk (X) = ΦG (F (EG+ , X ∧G )) .
Our main goal in this section is to express Rk (X) as a tower of spectra and identify its continuous
cohomology.
Theorem 2.6.
Hc∗ (Rk (X)) ∼
= Rk (H ∗ (X)) .
3. The Segal conjecture for smash powers and higher T HH
First we recall the Adams spectral sequence for towers of spectra, which will be of paramount
importance to us.
Theorem 3.1. Let {Xn }n be a tower of bounded below spectra with H∗ (Xn ) of finite type, and let
X = holimn Xn . There is a strongly convergent spectral sequence
∗
∧
Exts,t
A (Hc (X), Fp ) =⇒ πt−s (Xp ) .
18
HÅKON SCHAD BERGSAKER
3.1. Reduction to elementary abelian groups.
Lemma 3.2. Let G be a finite p-group which is not elementary abelian, and let X be a bounded
below spectrum with H∗ (X) of finite type. Then
e EG+ ∧ X) ' ∗ .
F (EP,
e Y ) ' ∗, and so F (EP,
e Y )H ' ∗ for
Proof. First note that for any G-spectrum Y , EP+ ∧ F (EP,
any proper subgroup H ⊂ G. ((mention this in a general recap of families?)) In particular, to
e EG+ ∧ X)G ' ∗.
prove the lemma it suffices to show that F (EP,
nρ
e = colimn S . We have
Recall that EP
e EG+ ∧ X)G ' holimn F (S nρ , EG+ ∧ X)G ' holimn (S −nρ ∧ EG+ ∧ X)G
(17)
F (EP,
and
(S −nρ ∧ EG+ ∧ X)G ' EG+ ∧G (Σ−nρ X) = DG (Σ−nρ X)
by the Adams isomorphism. Looking at the homotopy orbit spectral sequence
H∗ (G; π∗ (Σ−nρ X)) =⇒ π∗ (DG (Σ−nρ X))
we see that DG (Σ−nρ X) is bounded below, and the corresponding spectral sequence in homology
shows that H∗ (DG (Σ−nρ X)) is of finite type. Thus the spectral sequence in Theorem 3.1 applies
to the inverse system in (17); we will show that the E2 term
∗,∗
∗
nV
Ext∗,∗
, EG+ ∧ X)G ), Fp ) ∼
= ExtA (Hc∗ (DG (Σ−nρ X)), Fp )
A (Hc (F (S
e EG+ ∧ X)G is contractible.
vanishes, hence proving that F (EP,
In fact, the continuous cohomology Hc∗ (DG (Σ−nρ X)) is zero, before taking
Tsalidis shows that there is a commutative diagram
H ∗ (BG−nV ) ⊗H ∗ (BG) H ∗ (DG (X))
∼
=
H ∗ (DG (Σ−nV X))
∼
=
H ∗ (DG (Σ−(n+1)V X))
χ(V )⊗1
H ∗ (BG−(n+1)V ) ⊗H ∗ (BG) H ∗ (DG (X))
of
A result by Quillen and Venkov says that the Euler class χ(V ) ∈ H ∗ (BG) is nilpotent when G
is not elementary abelian.
Lemma 3.3. Let K and L be G-complexes, and F be a family of subgroups of G. The inclusions
e induce bijections
KF → K and S 0 → EF
e ∧ L]G ∼
e ∧ L]G ∼
[K, EF
= [KF , EF
= [KF , L]G .
Lemma 3.4. Let X be a G-spectrum, and assume that for each proper subgroup H ⊂ G, the
spectrum F (K, X) is H-contractible for all H-spaces K which are non-equivariantly contractible.
e X) is G-contractible, then F (K, X) is G-contractible for all non-equivariantly
If in addition F (EP,
contractible G-spaces K.
Proposition 3.5. Let G be a finite p-group which is not elementary abelian, and let X be a
bounded below spectrum with H∗ (X) of finite type. Assume that the maps
γP(J) : ΦK (X) → F (EP(J)+ , ΦK (X))
are equivalences of J-spectra for all proper subquotients J = H/A of G, with A an elementary
abelian Then the map
γP : X → F (EP+ , X)
is an equivalence of G-spectra.
HIGHER SINGER CONSTRUCTIONS AND THE SEGAL CONJECTURE FOR SMASH POWERS
19
Proof. The assumptions on γP(J) is equivalent to the statement that the fibers
e
F (EP(J),
ΦA (X))
are J-contractible for each proper sub-quotient J. Using induction on the order of J, it follows by
Lemma 3.4 that in fact, F (K, ΦA (X)) is J-contractible for all J-spaces K that are non-equivariantly
e X) of γP is a contractible G-spectrum. This
contractible. We want to show that the fiber F (EP,
fiber is part of the cofiber sequence
e EG+ ∧ X)
F (EP,
(18)
e X)
F (EP,
e EG
e ∧ X) ,
F (EP,
where the first term is contractible by Lemma 3.2. It remains to show that the last term in (18)
is also contractible, and for this it suffices to show that
e EG
e ∧ X)G ' ∗ .
F (EP,
S
e = S nρ̄
We use the explicit model EP
e EG
e ∧ X)G ' holimn F (S nρ̄ , EG
e ∧ X)G
F (EP,
e ∧ X)G is contractible for all n, or equivalently, that
It suffices to show that F (S nρ̄ , EG
e ∧ X)) = 0
π∗G (F (S nρ̄ , EG
for all n. We have
e ∧ X)) = [S k , F (S nρ̄ , EG
e ∧ X)]G = [S k+nρ̄ , EG
e ∧ X]G ,
πkG (F (S nρ̄ , EG
and since S k+nρ̄ is a compact CW -complex,
e ∧ X]G = colimV [S k+nρ̄+V , EG
e ∧ XV ]G
[S k+nρ̄ , EG
= colimV [S(S k+nρ̄+V ), XV ]G ,
where S(−) denotes the singular set functor. The last identification follows from Lemma 3.3
by choosing F to be the collection consisting of the trivial subgroup. Theorem 2.1 provides an
equivalent functor A(−) equipped with a filtration {Fq A}q , and identifies the filtration quotients
Bq = Fq A/Fq−1 A. By 2.1 (i),
M
[Bq (S k+nρ̄+V ), XV ]G =
[Σq C(A(ω), H(ω))(S k+nρ̄+V ), XV ]G
ω
=
M
=
M
[Σq G+ ∧H(ω) (S k+nρ̄+V )A(ω) , XV ]G
ω
[Σq+k (S nρ̄+V )A(ω) , XV ]H(ω) .
ω
Now, the filtration {Fq A}q induces short exact sequences ... and since all the terms ... are zero,
we conclude that
[S(S k+nρ̄+V ), XV ]G = [A(S k+nρ̄+V ), XV ]G = 0 ,
e ∧ X]G = 0 and finally that F (EP,
e EG
e ∧ X) ' ∗.
which implies that [S k+nρ̄ , EG
Corollary 3.6. Let G be a finite p-group which is not elementary abelian, and let X be a bounded
below G-spectrum with H∗ (X) of finite type. Assume that the map
γ J : ΦA (X)J → ΦA (X)hJ
is an equivalence for all proper subquotioents J = H/A with A an elementary abelian subgroups
A ⊂ G. Then
γ G : X G → X hG
is an equivalence.
Proof.
20
HÅKON SCHAD BERGSAKER
Corollary 3.7. Let B be an associative ring spectrum with H∗ (B) of finite type, and assume that
T HH n (B)A → T HH n (B)hA
is an equivalence for all elementary abelian p-subgroups A ⊂ T n . Then
T HH n (B)G → T HH n (B)hG
is an equivalence for all finite p-subgroups G ⊂ T n .
Proof.
3.2. The case of elementary abelian groups for smash powers.
Theorem 3.8. Let G be an elementary abelian p-group and let B be a bounded below spectrum
with H∗ (B) of finite type. Then the map
γ : B ∧G → F (EG+ , B ∧G )
is an equivalence after p-completion.
Proof. Consider the Tate diagram
EP+ ∧ X ∧G
X ∧G
e ∧ X ∧G
EP
γ
EP+ ∧ F (EG+ , X ∧G )
F (EG+ , X ∧G )
of horizontal cofiber sequences.
e ∧ F (EG+ , X ∧G )
EP
Corollary 3.9. Let G be a finite p-group and let B be a bounded below spectrum with H∗ (B) of
finite type. Then the map
γ : B ∧G → F (EG+ , B ∧G )
is an equivalence. In particular, the map
γ G : (B ∧G )G → (B ∧G )hG
is an equivalence.
Proof. If G is elementary abelian, this result is Theorem 3.8. If not, we apply Corollary 3.6 with
X = B ∧G .
3.3. Application to Goodwillie towers?
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E-mail address: hakonsb@math.uio.no