THE COHOMOLOGY OF ku
ROBERT R. BRUNER
Recall that H ∗ HZ = A/ASq 1 . We wish to show that H ∗ ku = A/A(Sq 1 , Q1 ).
I don’t have my copy of Adams’ blue book handy, but this is what he does.
1. The k-invariant
We start with the (co)fiber sequence of spectra
v
Σ2 ku −→ ku −→ HZ
which follows from the ring structure of ku and π∗ ku = Z[v]. Iterating, we get the
Postnikov tower of ku, with first k-invariant the top row in the diagram
Σ−1 HZ
/ Σ2 ku
Σ2 (
/ Σ2 HZ
v
ku
/ HZ
We have to determine this k-invariant in
H 2 Σ−1 HZ = hQ1 i
To do this, recall that ku = (BU ×Z, U, BU, SU, BSU, SU h5i, BU h6i, . . .). Restricting to the 3rd spaces, Ω∞ Σ3 (X), we have the diagram
K(Z, 2)
/ SU h5i
Σ2 (
/ K(Z, 5)
/ K(Z, 3)
v
SU
The top row here is no help because H 5 K(Z, 2) = 0, but we can detect more by
looking instead at the boundary map for the fibration SU h5i −→ SU −→ K(Z, 3):
H 6 K(Z, 3) ←− H 5 SU h5i ←− H 5 K(Z, 5)
If we let x5 ∈ H 5 SU h5i and ι3 ∈ H 3 K(Z, 3) be the fundamental classes, then in
total degree 5, the Serre spectral sequence for this fibration has two generators,
Sq 2 ι3 and x5 . The only way that H 5 SU can be one dimensional is for d5 (x5 ) = ι23 ,
giving the k-invariant:
Q1 (ι3 ) = ι23 ←−[ x5 ←−[ ι5
1
2
ROBERT R. BRUNER
x5
x5 Sq 2 ι3 x5 ι23
x5 ι3
·
·
·
·
1
·
·
ι3
·
Sq 2 ι3
!
ι23
Figure 1. Low degree classes in the E2 term of the Serre spectral
sequence H ∗ K(Z, 3) ⊗ H ∗ SU h5i =⇒ H ∗ SU .
Alternatively, we could look at the fourth spaces, K(Z, 3) −→ BSU h6i −→ BSU ,
and compute
H 6 K(Z, 3) ←− H 6 BSU h6i ←− H 6 K(Z, 6)
In this fibration, we must have d3 (ι3 ) = c2 , and hence d5 (Sq 2 ι3 ) = Sq 2 c2 = c3 .
This leaves ι23 ∈ H 6 K(Z, 3) as the image of the generator of H 6 BSU h6i, giving the
k-invariant:
Q1 (ι3 ) = ι23 ←−[ x6 ←−[ ι6
THE COHOMOLOGY OF ku
#
ι23
Sq 2 ι3
3
#
ι23 c22
ι23 c2
ι23 c3
#
Sq 2 ι3 c2
Sq 2 ι3 c3
#
Sq 2 ι3 c22
ι3 c2
ι3 c3
ι3 c22
·
ι3
·
·
1
·
·
·
#c
2
·
c3
·
#
c22
Figure 2. Low degree classes in the E2 term of the Serre spectral
sequence H ∗ BSU ⊗ H ∗ K(Z, 3) =⇒ H ∗ BSU h6i. (Not all classes
are shown.)
ι23
ι23 c3
Sq 2 ι3
Sq 2 ι3 c3
·
·
·
·
1
·
·
·
·
·
!c
3
·
·
Figure 3. Low degree classes in the E4 term of the Serre spectral
sequence H ∗ BSU ⊗ H ∗ K(Z, 3) =⇒ H ∗ BSU h6i.
4
ROBERT R. BRUNER
2. The long exact sequence in cohomology
From the k-invariant Q1 ∈ H 3 HZ for ku we get a diagram
/ Σ2 ku
/ ku
/ HZ
/ Σ2 HZ
/ F Q1
/ HZ
Σ−1 HZ
Q1
Σ−1 HZ
/ Σ3 ku
Q1
/ Σ3 HZ
Q1
Factoring the homomorphism H ∗ HZ ←− Σ3 H ∗ HZ we get
A//E(1)
ee
p
Σ3 A//E(1)
gg
x
i
x
A//E(0) o
Σ3 p
Σ3 i
Σ6 A//E(1)
w
w
Σ3 A//E(0)
Q1
To see the exactness claims here it is simplest to induce up from E(1)-modules,
where the epi-mono factorization of Q1 : Σ3 E(1)//E(0) −→ E(1)//E(0) and the
short exact sequence are easy.
This allows us to factor the restriction H ∗ ku ←− H ∗ HZ as
H ∗ ku o
c
A//E(1) o o
p
A//E(0) = H ∗ HZ
This makes the left square below commute. The right outer square commutes
because it is induced by maps of spectra. Inserting the factorization of the restriction Σ3 H ∗ ku ←− Σ3 H ∗ HZ, the rest of the diagram commutes because Σ3 p is an
epimorphism.
H ∗O ku o
A//E(0) o
Σ38 HO ∗ ku
Σ3 c
c
i
A//E(1) o o
p
Σ3 A//E(1)
x
ff
x
A//E(0) o
Σ3 p
Q1
A//E(0)
We have finally constructed the map of exact sequences which allows us to inductively show that
c
A//E(1) −→ H ∗ ku
is an isomorphism.
H ∗ Σ2 ku o
H ∗O ku o
H ∗ HZ o
q
Σ3 c
c
A//E(1) o o
H ∗ ΣO 3 ku o
p
A//E(0) o
i
o Σ3 A//E(1)
OO
Σ3 p
A//E(0) o
Q1
Σ3 A//E(0)
H ∗ Σku o
THE COHOMOLOGY OF ku
5
Suppose that c is an isomorphism in degrees less than or equal to n. Then Σ3 c
is an isomorphism in degrees ≤ n + 3 and hence c is a mono in degrees ≤ n + 3,
as follows. For i ≤ n + 3, suppose x ∈ (A//E(1))i has c(x) = 0. Then x = p(x0 )
for some x0 ∈ (A//E(0))i so c(p(x0 )) = 0 and thus x0 = q(y). Since i ≤ n + 3,
y = Σ3 c(z) for some z ∈ (Σ3 A//E(1))i . Then x = pi(z) = 0.
Similarly, c is an epi in degrees ≤ n + 2: for i ≤ n + 2, suppose x ∈ H i ku. Then
its image x0 ∈ H i Σ2 ku = H i+1 Σ3 ku is Σ3 c(y) for some y ∈ (Σ3 A//E(1))i+1 . But
i(y) = q(x0 ) = 0 by exactness of the top row. Since i is mono, y = 0, so x0 = 0.
This shows x comes from H ∗ HZ, so is in the image of c.
The induction starts, trivially, so c is an isomorphism in all degrees.
References
[1] Frank Adams, Blue book, UofC Press.