# The connective $K$-theory of the Eilenberg-MacLane space
$K({\mathbb{Z}}_{p},2)$

Donald M. Davis  Department of Mathematics, Lehigh University
Bethlehem, PA 18015, USA<EMAIL_ADDRESS>and  W. Stephen Wilson  Department
of Mathematics, Johns Hopkins University
Baltimore, MD 01220, USA<EMAIL_ADDRESS>
(Date: February 10, 2023)

###### Abstract.

We compute $ku^{*}(K({\mathbb{Z}}_{p},2))$ and
$ku_{*}(K({\mathbb{Z}}_{p},2))$, the connective $KU$-cohomology and connective
$KU$-homology groups of the mod-$p$ Eilenberg-MacLane space
$K({\mathbb{Z}}_{p},2)$, using the Adams spectral sequence. We obtain a
striking interaction between $h_{0}$-extensions and exotic extensions. The
mod-$p$ connective $KU$-cohomology groups, computed elsewhere, are needed in
order to establish higher differentials and exotic extensions in the integral
groups.

###### Key words and phrases:

Adams spectral sequence, connective $K$-theory, Eilenberg-MacLane spaces

2000 Mathematics Subject Classification: 55T15 55N20, 55N15.

##  1\. Main results

In [12] and [5], the authors initiated a partial computation of the connective
$KU$-homology groups, $ku_{*}(K({\mathbb{Z}}_{2},2))$, of the mod-2 Eilenberg-
MacLane space $K({\mathbb{Z}}_{2},2)$ in separate studies of Stiefel-Whitney
classes of manifolds. We eventually turned to the associated cohomology
groups, $ku^{*}(K({\mathbb{Z}}_{2},2))$, and were able to give a complete
determination, via the Adams spectral sequence (ASS). This generalized nicely
to the odd primes, and then we found a duality result ([4]) relating these
homology and cohomology groups which enabled us to determine the homology
groups $ku_{*}(K({\mathbb{Z}}_{p},2))$.

Let $K_{2}=K({\mathbb{Z}}_{p},2)$, with the prime $p$ implicit. We begin with
a description of the $ku^{*}$-module $ku^{*}(K_{2})$. Note that
$ku^{*}={\mathbb{Z}}_{(p)}[v]$ with $|v|=-2(p-1)$. We find that depiction via
ASS charts is the most insightful way to envision the groups. There is a very
nice interplay between extensions (multiplication by $p$) seen in Ext
($h_{0}$-extensions) and exotic extensions. We depict the ASS with
cohomological (co)degrees increasing from right-to-left. We write $|x|=d$ if
$x\in ku^{d}(K_{2})$ or the associated $E_{2}$-term.

In $ku^{*}(K_{2})$, there is a trivial submodule whose Poincaré series when
$p=2$ is described at the end of Section 2. It plays no role and will be
ignored from now on. As a $ku^{*}$-module, $ku^{*}(K_{2})$ is generated by
certain products of elements of $E_{2}^{0}$, $y_{0}$, $y_{i}=y_{0}^{p^{i}}$,
with $|y_{i}|=2p^{i}$, $z_{j}$ for $j\geq 0$ with $|z_{j}|=2(p^{j+1}+1)$, and
$q$ with $|q|=9$ if $p=2$, and $|q|=4p-1$ if $p$ is odd.

The even-graded part $ku^{\text{ev}}(K_{2})$ is formed from shifted copies of
$ku^{*}$-modules $A_{k}$ and $B_{k}$, which can be defined inductively as
follows.

######  Definition 1.1.

Let $k_{0}=1$ if $p$ is odd, and $k_{0}=2$ if $p=2$. Let $B_{k_{0}-1}=0$. Let
$A_{0}=\langle z_{0}\rangle$ for all $p$. Inductively

$B_{k}\text{ is built from }z_{k-1}^{p-1}B_{k-1},\ TP_{p^{k}-k}[v]z_{k},\text{
and }y_{k-1}^{p-1}B_{k-1},\text{ if }k\geq k_{0}$

and

$A_{k}\text{ is built from }z_{k-1}^{p-1}B_{k-1},\ TP_{p^{k}}[v]z_{k},\text{
and }y_{k-1}^{p-1}A_{k-1},\text{ if }k\geq 1$

with extensions determined by

$pz_{k}=vz_{k-1}^{p}\text{ for $k\geq 2$, and
}py_{k-1}^{p-1}z_{k-1}=v^{p^{k-1}(p-1)}z_{k}.$ (1.2)

Here $TP_{i}[v]$ is the truncated polynomial algebra over ${\mathbb{Z}}_{p}$
with generator $v$ and relation $v^{i}$. When we write something like $zB$, we
mean that all elements of $B$ are multiplied by the element $z$. Saying “is
built from” means that these are successive quotients in a filtration as a
$ku^{*}$-module. The extension formulas are only asserted up to multiplication
by a unit is ${\mathbb{Z}}_{p}$, and can both occur on an element. For
example, in Figure 1.6, we have, in grading 116 when $p=2$,
$2y_{3}z_{3}z_{4}=vy_{3}z_{2}^{2}z_{4}+v^{8}z_{4}^{2}$.

Figure 1.6 should enable the reader to envision $A_{k}$ and $B_{k}$ for $p=2$
and $k\leq 5$, and, by extrapolating, for all $k$. Elements connected by
dashed lines are in $A_{5}$ but not in $B_{5}$. The long red lines, sometimes
slightly curved, are the exotic extensions. The portion in gradings $\leq
102$, not including the top $v$-tower or the extensions to it, is $y_{4}A_{4}$
(or $y_{4}B_{4}$ if the dashed part is omitted). The portion in gradings $\geq
106$, not including the $v$-tower on $z_{5}$ or the $h_{0}$-extensions from
it, is $z_{4}B_{4}$. The reader is encouraged to understand how the case $k=5$
of Definition 1.1 is embodied in Figure 1.6.

The portion in the lower right corner of Figure 1.6 in grading $\leq 84$ and
height $\leq 7$ is $y_{3}y_{4}A_{3}$, and $y_{2}y_{3}y_{4}A_{2}$ is in
gradings $\leq 74$. In Figure 1.7, we present a schematic of $A_{3}$ and
$B_{3}$ at the odd primes. Again the dashed portion is in $A_{3}$, but not
$B_{3}$, and the triangle in the lower right portion is
$y_{1}^{p-1}y_{2}^{p-1}A_{1}$.

A generating set as a ${\mathbb{Z}}_{p}[v]$-module for $B_{k}$ is

$\biggl{\\{}z_{j}\prod_{i=j}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}:\ k_{0}\leq
j\leq k\biggr{\\}},$ (1.3)

while $A_{k}$ has additional generators

$\begin{cases}z_{1}y_{1}\cdots y_{k-1}&p=2\\\ z_{0}y_{0}^{p-1}\cdots
y_{k-1}^{p-1}&\text{all }p.\end{cases}$

The notation here means a product over all choices of one of the two elements
in each factor. For example,
$\prod_{i=1}^{2}\\{z_{i}^{p-1},y_{i}^{p-1}\\}=z_{1}^{p-1}z_{2}^{p-1}+z_{1}^{p-1}y_{2}^{p-1}+y_{1}^{p-1}z_{2}^{p-1}+y_{1}^{p-1}y_{2}^{p-1}$.
An empty product is defined to equal 1.

The following theorem explains how the portion of $ku^{*}(K_{2})$ in even
gradings is a direct sum of shifted versions of $A_{k}$ and $B_{k}$.

######  Theorem 1.4.

Let $M_{p}[S]$ denote the set of monomials in the elements of a set $S$ raised
to powers $<p$. Let

$\mathcal{M}_{k}=(M_{p}[z_{k},y_{k}]-\\{z_{k}^{p-1},y_{k}^{p-1}\\})\cdot
M_{p}[z_{i},y_{i}:\ i>k].$ (1.5)

Let $\mathcal{M}_{k}^{A}$ be the set of monomials in $\mathcal{M}_{k}$ with no
$z$-factors, and $\mathcal{M}_{k}^{B}=\mathcal{M}_{k}-\mathcal{M}_{k}^{A}$.
Then

$ku^{\text{ev}}(K_{2})=\bigoplus_{k\geq
1}\biggl{(}\bigoplus_{M\in\mathcal{M}_{k}^{A}}M\cdot
A_{k}\oplus\bigoplus_{M\in\mathcal{M}_{k}^{B}}M\cdot B_{k}\biggr{)}$

plus a trivial $ku^{*}$-module.

Note that the monomial 1 is in $\mathcal{M}_{k}^{A}$, so $A_{k}$ appears by
itself, but $B_{k}$ does not. For example, if $p=2$, copies of $B_{k}$ appear
multiplied by each monomial of the form

$z_{k}^{\varepsilon_{k}}y_{k}^{\delta_{k}}z_{k+1}^{\varepsilon_{k+1}}y_{k+1}^{\delta_{k+1}}\cdots\text{
such that }\varepsilon_{k}=\delta_{k}\text{ and}\sum\varepsilon_{i}\geq 1.$

######  Figure 1.6.

$B_{5}$ and $A_{5}$ when $p=2$.

$136$$130$$z_{2}^{2}z_{3}z_{4}$$z_{5}$$z_{4}^{2}$$126$$y_{2}z_{2}z_{3}z_{4}$$116$$y_{3}z_{3}z_{4}$$108$$y_{2}y_{3}z_{3}z_{4}$$102$$y_{4}z_{2}^{2}z_{3}$$98$$y_{4}z_{4}$$92$$y_{2}y_{4}z_{2}z_{3}$$y_{3}y_{4}z_{3}$$82$$74$$y_{2}y_{3}y_{4}z_{2}$$68$$y_{0}y_{1}y_{2}y_{3}y_{4}z_{0}$

######  Figure 1.7.

Schematic of $A_{3}$ and $B_{3}$ for odd $p$.

$p-2$$p^{2}-p$$p^{2}-3$$p^{3}-1$$p^{3}-p^{2}+p-2$$y_{1}^{p-1}z_{1}z_{2}^{p-1}$$y_{1}^{p-1}y_{2}^{p-1}z_{1}$$y_{2}^{p-1}z_{1}^{p}$$y_{2}^{p-1}z_{2}$$y_{0}^{p-1}y_{1}^{p-1}y_{2}^{p-1}z_{0}$$z_{2}^{p}$$z_{3}$$z_{1}^{p}z_{2}^{p-1}$

Now we describe the portion of $ku^{*}(K_{2})$ in odd gradings. Let $P[S]$
denote the polynomial algebra on a set $S$, and $TP_{i}[S]=P[S]/(s^{i}:\ s\in
S)$, the truncated polynomial algebra. Let $\Lambda_{j}=TP_{p}[z_{i}:\ i\geq
j]$. Note that if $p=2$, $\Lambda_{j}$ is an exterior algebra. For $i\leq j$,
let

$z_{i,j}=z_{i}(z_{i}\cdots z_{j-1})^{p-1}.$ (1.8)

If $j=i$, then $z_{i,j}=z_{i}$.

######  Definition 1.9.

For $\ell>k\geq 1$, let $S_{k,\ell}=TP_{k+1}[v]\langle
z_{k_{0},\ell},\ldots,z_{\ell-k-1+k_{0},\ell}\rangle$ with
$pz_{i,\ell}=vz_{i-1,\ell}$ and $pz_{k_{0},\ell}=0$.

For example, $S_{5,8}$ with $p=2$ is depicted in Figure 1.10.

######  Figure 1.10.

$S_{5,8}$ if $p=2$

$1040$$1036$$z_{2,8}$$z_{4,8}$

The following result describes the portion of $ku^{*}(K_{2})$ in odd gradings.
The exponent of $p$ in an integer $i$ is denoted simply by $\nu(i)$; the prime
$p$ is implicit. The element $q$ here has grading $9$ or $4p-1$, as mentioned
earlier.

######  Theorem 1.11.

There is an isomorphism of $ku^{*}$-modules

$ku^{\text{odd}}(K_{2})\approx\bigoplus_{i\geq
1}\bigoplus_{\ell\geq\nu(i)+2}qy_{1}^{i-1}S_{\nu(i)+1,\ell}\otimes
TP_{p-1}[z_{\ell}]\otimes\Lambda_{\ell+1}.$

The non-visual, formulaic form of our result is as follows.

######  Theorem 1.12.

The $ku^{*}$-module $ku^{*}(K_{2})$ is isomorphic to a trivial $ku^{*}$-module
plus a module whose associated graded is

$\displaystyle P[y_{1}]y_{0}^{p-1}z_{0}\oplus\bigoplus_{t\geq
1}TP_{p^{t}}[v]\otimes P[y_{t}]z_{t}$ (1.13) $\displaystyle\oplus$
$\displaystyle\bigoplus_{t\geq k_{0}}TP_{p^{t}-t}[v]\otimes
P[y_{t}]z_{t}\Lambda_{t}$ (1.14) $\displaystyle\oplus$
$\displaystyle\bigoplus_{i\geq 1}\bigoplus_{\ell\geq
0}TP_{\nu(i)+2}[v]qy_{1}^{i-1}z_{k_{0}+\ell,\ell+\nu(i)+2}\overline{\Lambda}_{\ell+\nu(i)+2}.$
(1.15)

Multiplication by $p$ in (1.13) and (1.14) is determined by (1.2) and in
(1.15) as in Definition 1.9.

Our initial interest in this project was $ku_{*}(K_{2})$ ([12],[5]), but we
first achieved success in computing $ku^{*}(K_{2})$. In [4, Example 3.4], the
following result was proved.

######  Theorem 1.16.

There is an isomorphism of $ku_{*}$-modules
$ku_{*}(K_{2})\approx(ku^{*+2p}K_{2})^{\vee}$.

Here $M^{\vee}=\operatorname{Hom}(M,{\mathbb{Z}}/p^{\infty})$, the Pontryagin
dual, localized at $p$. A homotopy chart for $ku_{*}(K_{2})$ could be thought
of as a shifted version of the homotopy chart of $ku^{*}(K_{2})$ viewed
upside-down and backwards. For example, the element of
$ku^{108}(K_{2})^{\vee}$ dual to the element $v^{4}y_{3}z_{3}z_{4}$ in Figure
1.6 corresponds to the generator of a ${\mathbb{Z}}_{4}$ in $ku_{104}(K_{2})$
on which $v^{4}$ acts nontrivially. This element can be seen in Figure 1.18.

A remarkable property, for which one explanation is given in Section 7, is
that $B_{k}$ is self-dual as a $ku^{*}$-module. One way of stating this is to
let $\widetilde{B}_{k}$ denote $B_{k}$ with its indices negated. Then there is
an isomorphism of $ku_{*}$-modules

$\Sigma^{2(p^{k+1}+p^{k}+(k+1)p-k+1)}\widetilde{B}_{k}\approx B_{k}^{\vee}.$
(1.17)

For example, with $p=2$, the second smallest generator $Y$ of
$\Sigma^{208}\widetilde{B}_{5}$ is in grading $208-134=74$ and has $2Y\neq 0$
and $v^{4}Y\neq 0$. (See Figure 1.6.) The second generator $Z$ of
$B_{5}^{\vee}$ is dual to the class in position $(74,4)$ in Figure 1.6, and
also satisfies $2Z\neq 0$ and $v^{4}Z\neq 0$. The isomorphism (1.17) can be
proved by induction on $k$ using Definition 1.1.

A complete description of the $ku_{*}$-module $ku_{*}(K_{2})$ is immediate
from Theorems 1.4, 1.11, and 1.16. However, one might like a complete
description of its ASS. We can write formulas for the $E_{2}$-term and
differentials, but will not do so here. In Theorem 1.19 we give a complete
description of the $E_{\infty}$-term of the ASS of $ku_{*}(K_{2})$ with exotic
extensions included, in terms of the charts described in Section 1.

In [4], a comparison was made of a chart for $A_{3}$ and its $ku_{*}$
analogue. Here we present in Figure 1.18 the $ku_{*}$ analogue of Figure 1.6.
This presents the portion of the ASS of $ku_{*}(K_{2})$ dual to $A_{5}$ with
$p=2$ under the isomorphism of Theorem 1.16. The ASS chart dual to $B_{5}$ is
obtained from this by removing the classes connected by dashed lines, and
lowering the remaining tower so that the bottom is in filtration 0. The
resulting chart is isomorphic to the $B_{5}$ part of Figure 1.6.

######  Figure 1.18.

Portion of $ku_{*}(K_{2})$ corresponding to $B_{5}$ and $A_{5}$.

$64$$78$$88$$96$$102$$112$$120$$132$

We observe that in even gradings of the ASS for $ku_{*}(K_{2})$,
$h_{0}$-extensions exactly correspond to exotic extensions in the ASS of
$ku^{*+2p}(K_{2})$, and vice versa. As a typical example of the duality, the
summands of $ku^{82}(K_{2})$, $ku^{82}(K_{2})^{\vee}$, and $ku_{78}(K_{2})$ in
Figures 1.6 and 1.18 are all isomorphic to
${\mathbb{Z}}_{8}\oplus{\mathbb{Z}}_{2}$. But for the $ku_{*}$-module
structure, it is $ku^{82}(K_{2})^{\vee}$ and $ku_{78}(K_{2})$ that correspond,
since in both, the element that is divisible by 4, in position $(82,0)$ and
$(78,7)$, resp., is also divisible by $v^{7}$ for $A_{5}$ and by $v^{4}$ for
$B_{5}$.

######  Theorem 1.19.

The $E_{\infty}$-term of the ASS of $ku_{*}(K_{2})$ with exotic extensions
included contains exactly the following.

  * •

There is a trivial $ku_{*}$-module, which when $p=2$ has generators
corresponding to those enumerated at the end of Section 2 with gradings
decreased by 4, and similarly when $p$ is odd.

  * •

For every $S_{k,\ell}$ occurring in a summand of Theorem 1.11, there is a
chart of the same form as Figure 1.10 with $v$-towers of height $k+1$ on
generators in gradings $2p^{\ell+1}+2(p-1)(i-k_{0}-1)$ for $1\leq
i\leq\ell-k$. One must add to this the grading of the other factors
accompanying $S_{k,\ell}$ in Theorem 1.11.

  * •

For each occurrence of $B_{k}$ in Theorem 1.4, there is a summand

$\Sigma^{2(p^{k+1}+p^{k}+kp-k+1)}\widetilde{B}_{k}$

with gradings increased by those of other factors accompanying $B_{k}$ in 1.4.
Here $\widetilde{B}_{k}$ is as defined prior to (1.17).

  * •

For each summand $y_{k}^{e}A_{k}$ in Theorem 1.4, there is a variant of
$\Sigma^{2(p^{k+1}+p^{k}+kp-k+1)}\widetilde{B}_{k}$ with gradings increased by
$2ep^{k}$. In this variant, the initial $v$-towers are pushed up by $k$
filtrations and surrounded with a triangle of classes of the sort appearing in
the lower left corner of Figure 1.18. See Remark 1.20.

###### Proof.

Theorem 1.16 and our results for $ku^{*}(K_{2})$ give the $ku_{*}$-module
structure of $ku_{*}(K_{2})$, but that is not the same as the ASS picture.
Expanding on work done in [5] and [12] and using methods such as those in
Section 2, we were able to write the $E_{2}$-term of the ASS for
$ku_{*}(K_{2})$, and had conjectured the differentials (but not the
extensions) prior to embarking on our $ku$-cohomology project. We were unable
to prove the differentials, probably because we had not taken sufficient
advantage of the exact sequence with $k(1)_{*}(K_{2})$. Now that we know the
2-orders and $v$-heights of generators (by grading, at least, if not by name),
it is straightforward to see that the differentials must be as we expected.
The isomorphism (1.17) plays an important role here; the left hand side gives
the ASS form of the right hand side.

######  Remark 1.20.

Regarding the unusual portion of the ASS chart for part of $ku_{*}(K_{2})$ in
the lower left of Figure 1.18, this is obtained from [5, Fig. 4.2] with
$d_{6}$-differentials on all odd-graded towers. For $A_{k}$, it will be a
triangle going up to filtration $k$, with all but the first two dots on the
top row being part of $B_{k}$.

The structure of the rest of the paper is as follows. In Section 2, we compute
the $E_{2}$-term of the ASS for $ku^{*}(K_{2})$. In Section 3 we determine the
differentials in this ASS. In order to do so, we need to compare with
$k(1)^{*}(K_{2})$, where $k(1)$ is the spectrum for mod-$p$ connective
$KU$-theory, using the exact sequence

$\to k(1)^{*-1}(K_{2})\to ku^{*}(K_{2})\
\smash{\mathop{\longrightarrow}\limits^{p}}\ ku^{*}(K_{2})\to
k(1)^{*}(K_{2})\to ku^{*+1}(K_{2})\
\smash{\mathop{\longrightarrow}\limits^{p}}\ .$ (1.21)

In Section 3, we restate results about $k(1)^{*}(K_{2})$ from [7]. At the end
of Section 3, we show how the descriptions of $ku^{*}(K_{2})$ in Theorems 1.4
and 1.11 are obtained once we know the differentials. This exact sequence is
also used in determining the exotic extensions of (1.2), which is done in
Section 4. In Section 5, we propose complete formulas for the exact sequence
(1.21), and then in Section 6, we show that our proposed formulas account for
all elements of $k(1)^{*}(K_{2})$ exactly once.

The main point of Section 6 is to prove that there are no additional exotic
extensions in $ku^{*}(K_{2})$. An exotic extension $p\cdot A=B$ implies that
$A$ is not in the image from $k(1)^{*-1}(K_{2})$, and $B$ does not map
nontrivially to $k(1)^{*}(K_{2})$, so once we have shown that all elements are
accounted for, there can be no more extensions. Many of our formulas in
Section 5 are forced by naturality. However, many others occur in regular
families, but with surprising filtration jumps. We could probably prove that
the homomorphisms must be as we claim, by showing that there are no other
possibilities, but we prefer to forgo doing that. In the optional Section 7,
we discuss in more detail how the charts are obtained and provide an
explanation for the duality result (1.17).

##  2\. The $E_{2}$-term of the ASS for $ku^{*}(K_{2})$

We will need some notation. By $H^{*}K_{2}$, we understand
$H^{*}(K({\mathbb{Z}}_{p},2);{\mathbb{Z}}_{p})$. Let $E$ denote an exterior
algebra, $P$ a polynomial algebra, and $TP_{n}[x]=P[x]/(x^{n})$ the truncated
polynomial algebra. In all cases these will be over ${\mathbb{Z}}_{p}$, the
integers mod $p$. Let $\overline{E}$ denote the augmentation ideal of an
exterior algebra, and $E_{1}=E[Q_{0},Q_{1}]$, where $Q_{i}$ are the Milnor
primitives. Because $Q_{i}^{2}=0$ we have homology groups, $H_{*}(-;Q_{i})$,
defined for $E_{1}$-modules. We let $\langle y_{1},y_{2},\ldots\rangle$ denote
the ${\mathbb{Z}}_{p}$-span of classes $y_{i}$.

The Adams spectral sequence (ASS) for $ku^{*}(K_{2})$ has
$E_{2}^{s,t}=\operatorname{Ext}_{\mathcal{A}}^{s,t}(H^{*}(bu),H^{*}K_{2})$,
where $\mathcal{A}$ is the mod $p$ Steenrod algebra and
$H^{*}(bu)\approx\mathcal{A}/\mathcal{A}(Q_{0},Q_{1})$. Using a standard
change of rings theorem, [8], this is
$\operatorname{Ext}_{E_{1}}^{s,t}({\mathbb{Z}}_{p},H^{*}K_{2})$. This
converges to $ku^{-(t-s)}(K_{2})$. We depict this with $E_{2}^{s,t}$ in
position $(t-s,s)$ as usual, but label the axis with codegrees, the negative
of the homotopical degree, so the left side of the chart will have positive
gradings and refer to cohomological grading. In an attempt to avoid confusion,
we rewrite this as $G_{2}^{-(t-s),s}$. With this notation, the differentials
are $d_{r}:G_{r}^{a,b}\longrightarrow G_{r}^{a+1,b+r}$, multiplication by the
element $v\in ku^{-2(p-1)}$ (also considered in $G_{r}^{-2(p-1),1}$), is
$v:G_{r}^{a,b}\longrightarrow G_{r}^{a-2(p-1),b+1}$, and multiplication by the
element representing $p\in ku^{0}$, ($h_{0}\in G_{r}^{0,1}$), is
$h_{0}:G_{r}^{a,b}\longrightarrow G_{r}^{a,b+1}$.

In the paragraph preceding Remark 2.19, we will define elements $z_{j}\in
G_{2}^{2(p^{j+1}+1),0}$ for $j\geq 0$ and elements

$z_{i,j}\in G_{2}^{2(p^{j+1}+1+(p-1)(j-i)),0}$

as in (1.8) satisfying the properties in Definition 1.9.

######  Definition 2.1.

For $j\geq k_{0}$, we define $W_{j}=\langle
z_{j,j},z_{j-1,j},\ldots,z_{k_{0},j}\rangle$.

We also have $y_{i}\in G_{2}^{2p^{i},0}$ for $i\geq 0$, and

$q\in G^{9,0}_{2}\text{ if }p=2,\text{ and in }G^{4p-1,0}_{2}\text{ if $p$ is
odd.}$ (2.2)

One last definition, let $\Lambda_{j+1}=TP_{p}[z_{i}:\,i\geq j+1]$.

A picture of $P[v]\otimes W_{5}$ as a $P[v,h_{0}]$-module with $p=2$ appears
in Figure 2.3.

######  Figure 2.3.

A depiction of $P[v]\otimes W_{5}$

$136$$134$$132$$130$$z_{2,5}$$z_{3,5}$$z_{4,5}$$z_{5,5}$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$

The remainder of this section is devoted to the proof of the following result.

######  Theorem 2.4.

The $E_{2}$ term of the Adams spectral sequence for the reduced
$ku^{*}(K_{2})$ is isomorphic as a $P[h_{0},v]$-module to

$P[v,y_{1}]\otimes E[q]\otimes\bigl{(}\bigoplus_{j\geq k_{0}}(W_{j}\otimes
TP_{p-1}[z_{j}]\otimes\Lambda_{j+1})\bigr{)}$
$\oplus\bigl{(}P[h_{0},v,y_{1}]\otimes
E[v^{k_{0}}q]\bigr{)}\oplus\biggl{(}P[y_{1}]\otimes\begin{cases}\langle
y_{0}^{p-1}z_{0}\rangle&p\text{ odd}\\\ \langle
y_{0}z_{0},z_{1},h_{0}y_{0}z_{0}=vz_{1}\rangle&p=2.\end{cases}\biggr{)}$

plus a trivial $P[h_{0},v]$-module.

Some of the algebra structure of this $E_{2}$ will be useful later. For
example, the product structure among the $z_{j}$’s will be clear, and also the
formula

$(v^{2}q)^{2}=v^{4}z_{2},$ (2.5)

holds when $p=2$ since, as we shall see, in $H^{*}(K_{2})$,
$x_{9}^{2}-Q_{0}x_{17}\in\operatorname{im}(Q_{1})$.

We will give a detailed proof when $p=2$, and then sketch the minor changes
for odd $p$. There are two parts to proving this theorem. First, we must give
a complete description of the $E_{1}$-module structure of $H^{*}K_{2}$.
Second, we have to compute
$\operatorname{Ext}_{E_{1}}^{*,*}({\mathbb{Z}}_{2},-)$ of this. We begin the
first part.

Serre ([9]) showed that $H^{*}K_{2}$ is a polynomial algebra on classes
$u_{2^{j}+1}$ in degree $2^{j}+1$ for $j\geq 0$ defined by $u_{2}=\iota_{2}$
and $u_{2^{j+1}+1}=\operatorname{Sq}^{2^{j}}u_{2^{j}+1}$ for $j\geq 0$. We
easily have

$Q_{0}(u_{2})=u_{3},\ Q_{0}(u_{3})=0,\
Q_{0}(u_{2^{j}+1})=u_{2^{j-1}+1}^{2}\text{ for }j\geq 2,$

and

$Q_{1}(u_{2})=u_{5},\ Q_{1}(u_{3})=u_{3}^{2},\ Q_{1}(u_{5})=0,\
Q_{1}(u_{2^{j}+1})=u_{2^{j-2}+1}^{4}\text{ for }j\geq 3.$

Let $x_{5}=u_{5}+u_{2}u_{3}$ and write $H^{*}K_{2}$ as an associated graded
object:

$P[u_{2}^{2}]\otimes E[x_{5}]\otimes\bigl{(}E[u_{2}]\otimes
P[u_{3}]\bigr{)}\otimes_{j\geq 2}\left(E[u_{2^{j+1}+1}]\otimes
P[(u_{2^{j}+1})^{2}]\right)$

From this, we can read off

######  Lemma 2.6.

$H_{*}(H^{*}K_{2};Q_{0})=P[u_{2}^{2}]\otimes E[x_{5}]$

Letting $x_{9}=u_{9}+u_{3}^{3}$ and $x_{17}=u_{17}+u_{2}u_{5}^{3}$, we rewrite
again as

$\displaystyle P[u_{2}^{2}]\otimes TP_{4}[x_{9}]\otimes
TP_{4}[x_{17}]\otimes_{j>4}E[(u_{2^{j}+1})^{2}]$
$\displaystyle\otimes\bigl{(}E[u_{2}]\otimes
P[u_{5}]\bigr{)}\otimes\bigl{(}E[u_{3}]\otimes
P[u_{3}^{2}]\bigr{)}\otimes_{j>4}\bigl{(}E[u_{2^{j}+1}]\otimes
P[(u_{2^{j-2}+1})^{4}]\bigr{)}.$

Again we read off

######  Lemma 2.7.

$H_{*}(H^{*}K_{2};Q_{1})=P[u_{2}^{2}]\otimes TP_{4}[x_{9}]\otimes
TP_{4}[x_{17}]\otimes_{j>4}E[(u_{2^{j}+1})^{2}]$

An associated graded version of this is

######  Lemma 2.8.

$H_{*}(H^{*}K_{2};Q_{1})=P[u_{2}^{2}]\otimes E[x_{9}]\otimes
E[x_{17}]\otimes_{j>2}E[(u_{2^{j}+1})^{2}]$

The bulk of the work here is finding a nice splitting of $H^{*}K_{2}$ as an
$E_{1}$-module.

Let $N$ be the $E_{1}$-submodule with single nonzero elements in gradings 5,
7, 8, 9, and 10 with generators $x_{5}=u_{5}+u_{2}u_{3}$, $x_{7}=u_{2}u_{5}$,
and $x_{9}=u_{9}+u_{3}^{3}$, satisfying $Q_{0}x_{7}=Q_{1}x_{5}$ and
$Q_{0}x_{9}=Q_{1}x_{7}=x_{10}$. It has a $Q_{0}$-homology class $x_{5}$ and a
$Q_{1}$-homology class $x_{9}$. This class $x_{9}$ is called $q$ in Theorem
2.4 and in all other sections. A picture of $N$ is in Figure 2.9. In pictures
such as this, straight lines indicate $Q_{0}=\operatorname{Sq}^{1}$ and curved
lines $Q_{1}$.

######  Figure 2.9.

An $E_{1}$-module $N$.

$5$$9$$7$$10$

The $E_{1}$-submodule $P[u_{2}^{2}]\oplus P[u_{2}^{2}]\otimes N$ carries the
$Q_{0}$-homology of $H^{*}K_{2}$, while the remaining $Q_{1}$-homology is,
written in our usual way as an associated graded version,

$P[u_{2}^{2}]\otimes E[x_{9}]\otimes\overline{E}[x_{17},u_{2^{j}+1}^{2},\
j>2].$ (2.10)

We will exhibit a $Q_{0}$-free $E_{1}$-submodule $R$ whose $Q_{1}$-homology is
exactly the above $\overline{E}$. Moreover, $N\otimes R$ contains an
$E_{1}$-split summand $S$ which maps isomorphically to $\langle
x_{9}\rangle\otimes R$.

It is premature to state this because we haven’t defined $R$ and $S$ yet, but
for the record:

######  Proposition 2.11.

As an $E_{1}$ module, $\widetilde{H}^{*}K_{2}$ is isomorphic to $T\oplus F$
where $F$ is free over $E_{1}$ and $T$ is

$P[u_{2}^{2}]\otimes\bigl{(}\langle u_{2}^{2}\rangle\oplus N\oplus R\oplus
S\bigr{)}$

A start on $R$ and $S$.

For this to make sense, we need to find $R$ and $S$. The module $R$ is a
direct sum of shifted versions of modules $L_{k}$, $k\geq 0$, which have
generators $g_{2i}$, $0\leq i\leq k$, with $Q_{1}g_{2i}=Q_{0}g_{2i+2}$ for
$0\leq i<k$, $Q_{0}g_{0}\neq 0$, and $Q_{1}g_{2k}=0$. For example, $L_{3}$ is
depicted in Figure 2.12.

######  Figure 2.12.

The $E_{1}$-module $L_{3}$.

$g_{0}$$g_{2}$$g_{4}$$g_{6}$

A splitting map, $\langle x_{9}\rangle\otimes L_{k}\longrightarrow N\otimes
L_{k}$, for the epimorphism $N\otimes L_{k}\to\langle x_{9}\rangle\otimes
L_{k}$ is defined by

$x_{9}g_{2i}\mapsto x_{9}\otimes g_{2i}+x_{7}\otimes g_{2i+2}+x_{5}\otimes
g_{2i+4}\text{ for }0\leq i\leq k-2,$

$x_{9}g_{2k-2}\mapsto x_{9}\otimes g_{2k-2}+x_{7}\otimes g_{2k}$, and
$x_{9}\otimes g_{2k}\mapsto x_{9}\otimes g_{2k}$.

The $E_{1}$-module $M_{j}$

Let

$x_{2^{j}+1}=u_{2^{j}+1}+\begin{cases}u_{2}u_{5}^{3}&j=4\\\
u_{2}u_{3}u_{5}^{2}u_{9}^{2}&j=5\\\ u_{3}u_{5}^{2}u_{9}^{2}u_{17}^{2}&j=6\\\
0&j>6\end{cases}\text{ and
}w_{2^{j}-1}=\begin{cases}u_{2}u_{3}u_{5}^{2}&j=4\\\
u_{3}u_{5}^{2}u_{9}^{2}&j=5\\\ 0&j>5.\end{cases}$

Then $Q_{0}x_{2^{j}+1}=u_{2^{j-1}+1}^{2}+Q_{1}w_{2^{j}-1}$, so
$Q_{0}x_{2^{j}+1}$ and $u_{2^{j-1}+1}^{2}$ represent the same $Q_{1}$-homology
class. Define $E_{1}$-modules $M_{j}$ inductively by $M_{3}=0$, and for $j\geq
4$ there is a short exact sequence of $E_{1}$-modules

$0\to u_{2^{j-2}+1}^{2}M_{j-1}\to M_{j}\to M_{j}^{\prime}\to 0,$ (2.13)

where $M_{j}^{\prime}=\langle x_{2^{j}+1},Q_{0}x_{2^{j}+1}\rangle$ and
$Q_{1}x_{2^{j}+1}=u_{2^{j-2}+1}^{2}Q_{0}x_{2^{j-1}+1}$. The above definitions
of the $x_{2^{j}+1}$ are necessary to get this formula to work right.

There is an isomorphism of $E_{1}$-modules
$M_{j}\approx\Sigma^{2^{j}+1}L_{j-4}$ given by

$\Sigma^{2^{j}+1}g_{2i}\mapsto\begin{cases}x_{2^{j}+1}&i=0\\\
u_{2^{j-2}+1}^{2}x_{2^{j-1}+1}&i=1\\\
u_{2^{j-2}+1}^{2}u_{2^{j-3}+1}^{2}x_{2^{j-2}+1}&i=2\\\
u_{2^{j-2}+1}^{2}u_{2^{j-3}+1}^{2}\cdots
u_{2^{j-i-1}+1}^{2}x_{2^{j-i}+1}&2<i\leq j-4\\\ \end{cases}$ (2.14)

And we have

$H_{*}(M_{j};Q_{1})=\begin{cases}\langle u_{9}^{2},u_{17}\rangle&j=4\\\
\langle u_{17}^{2},u_{9}^{2}u_{17}\rangle&j=5\\\ \langle
u_{33}^{2},u_{17}^{2}u_{9}^{2}u_{17}\rangle&j=6\\\ \langle
u_{2^{j-1}+1}^{2},u_{2^{j-2}+1}^{2}\cdots u_{9}^{2}x_{17}\rangle&j>6\\\
\end{cases}$ (2.15)

The $E_{1}$-module $R$

Let

$R=\bigoplus_{j\geq 4}M_{j}\otimes
E[u_{2^{j}+1}^{2},u_{2^{j+1}+1}^{2},\ldots].$ (2.16)

Then $H_{*}(R;Q_{1})=\overline{E}[x_{17},u_{9}^{2},u_{17}^{2},\ldots]$, since
monomials in $\overline{E}$ without $x_{17}$ appear from a first term (of the
two in (2.15)) in $H_{*}(M_{j}\otimes E;Q_{1})$, where $j$ is minimal such
that $u_{2^{j-1}+1}^{2}$ appears in the monomial, while those with $x_{17}$,
and also containing a product $u_{9}^{2}\cdots u_{2^{j-2}+1}^{2}$ of maximal
length, occur as a second term in $H_{*}(M_{j}\otimes E;Q_{1})$.

###### Proof of Proposition 2.11.

We have the $E_{1}$-submodule $T$ given in Proposition 2.11. Because this
contains all of the $Q_{0}$ and $Q_{1}$ homology, what remains must be free
over $E_{1}$ by [11].

###### Proof of Theorem 2.4.

We compute $\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{2},T)$ with $T$ as in
Proposition 2.11. We will not be concerned with the free $E_{1}$-module $F$
but later we will give the Poincaré series for it. Each copy of $E_{1}$ in $F$
gives a ${\mathbb{Z}}_{2}$ in $G^{*,0}$ that corresponds to $Q_{0}Q_{1}$.

That

$\operatorname{Ext}_{E_{1}}^{*,*}({\mathbb{Z}}_{2},P[u_{2}^{2}])=P[v,h_{0},y_{1}]$

with $y_{1}\in G_{2}^{4,0}$ should be clear, given our labeling conventions.
We normally work with the reduced cohomologies, so the $y_{1}^{0}$ generator
above would be ignored. The $y_{1}$ notation is particularly useful when we
consider all primes $p$. It is $y_{0}^{p^{1}}$ where $y_{0}\in G_{2}^{2,0}$.
So $|y_{1}|=2p$.

We compute $\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{2},N)$ in two ways using
two different filtrations of $N$. From this we see that the generator of the
towers can be thought of either as $v^{2}x_{9}$ or $h_{0}^{2}x_{5}$.

Using Figure 2.9 as our guide, our first filtration is $\langle
x_{5},x_{8}\rangle$, $\langle x_{7},x_{10}\rangle$, and $\langle
x_{9}\rangle$. The $\operatorname{Ext}$ on $x_{9}\in G^{9,0}$ is just
$P[v,h_{0}]$. For the other two, we get $h_{0}$-towers on $x_{10}\in G^{10,0}$
and $x_{8}\in G^{8,0}$. The extensions in $N$ show these two $h_{0}$-towers
are connected by multiplication by $v$. In addition, a $d_{1}$ is forced on us
by the extensions. Figure 2.17 describes this completely.

######  Figure 2.17.

The first computation of $\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{2},N)$

$10$$8$$5$$3$$10$$8$$5$$3$$\Rightarrow$$v^{2}x_{9}$

Again referring to Figure 2.9, our second filtration is $\langle
x_{9},x_{10}\rangle$, $\langle x_{7},x_{8}\rangle$, and $\langle
x_{5}\rangle$. Now our $\operatorname{Ext}$ groups are $P[v,h_{0}]$ on
$x_{5}\in G^{5,0}$, $P[v]$ on $x_{8}\in G^{8,0}$ and $x_{10}\in G^{10,0}$.
Again, the $d_{1}$ is forced by the extensions in $N$. Figure 2.18 describes
the result.

######  Figure 2.18.

The second computation of $\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{2},N)$

$\Rightarrow$$10$$8$$5$$3$$=$$h_{0}^{2}x_{5}$$10$$8$$5$$3$

This concludes the computation of $\operatorname{Ext}$ for
$P[u_{2}^{2}]\otimes(\langle u_{2}^{2}\rangle\oplus N)$ of Proposition 2.11.
The result is the second line of Theorem 2.4.

We need to compute $\operatorname{Ext}$ for $P[u_{2}^{2}]\otimes(R\oplus S)$
and show it is the same as the top line in Theorem 2.4. Since $S\approx\langle
x_{9}\rangle\otimes R$, all we need to do is $P[u_{2}^{2}]\otimes R$ and
ignore the $E[x_{9}]$. Similarly we can ignore the $P[u_{2}^{2}]$ and the
$P[y_{1}]$ because for every power of $u_{2}^{2}$ we will have a copy of the
answer indexed by powers of $y_{1}$. All we have left now is $R$, but $R$ is
just many copies of the various $M_{j}$ and the indexing for the number of
copies is given by the $\Lambda_{j+1}$.

All that remains is to show that
$\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{2},M_{j})\approx P[v]\otimes
W_{j-2}$ with $W_{j-2}$ as in Definition 2.1.111The reason for this awkward
shift is that the gradings for $z_{j}$ which give the elegant statements in
Definition 1.1 and elsewhere are not particularly convenient in developing the
$E_{2}$ statement. Recall that $M_{j}=\Sigma^{2^{j}+1}L_{j-4}$. We can filter
$L_{j-4}$ into pairs of elements $g_{2i},Q_{0}g_{2i}$, for $0\leq i\leq j-4$.
Then $\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{2},M_{j})$ has a $P[v]$ on each
element $\Sigma^{2^{j}+1}Q_{0}g_{2i}$ which we denote by $z_{j-i-2,j-2}\in
G^{2^{j}+2+2i,0}$. The element $z_{j-2,j-2}$ is often called $z_{j-2}$. There
is no $d_{1}$, but undoing the filtration does solve the extension problem and
gives us $h_{0}z_{k,j-2}=vz_{k-1,j-2}$. This completes our computation and
thus our proof.

######  Remark 2.19.

To illustrate the last computation in the proof, consider the generators of
the $v$-towers for $\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{2},M_{7})$. They
are $z_{5}$, $z_{4}^{2}$, $z_{3}^{2}z_{4}$, and $z_{2}^{2}z_{3}z_{4}$, which
is what we have called $z_{5,5}$, $z_{4,5}$, $z_{3,5}$, and $z_{2,5}$, as
pictured in Figure 2.3. For future reference, we note that (with $\sim$
meaning homologous)

$z_{j}=Q_{0}x_{2^{j+2}+1}\sim
u_{2^{j+1}+1}^{2}=Q_{0}u_{2^{j+2}+1}=Q_{0}Q_{j+2}\iota_{2}=Q_{j+2}Q_{0}\iota_{2}.$
(2.20)

We now describe briefly the changes required when $p$ is odd. We have

$H^{*}(K_{2})=P[y_{0}]\otimes P[g_{1},g_{2},\ldots]\otimes
E[u_{0},u_{1},\ldots],$

with $|y_{0}|=2$, $|g_{j}|=2(p^{j}+1)$, $|u_{i}|=2p^{i}+1$,
$Q_{0}y_{0}=u_{0}$, $Q_{0}u_{i}=g_{i}$, $Q_{1}y_{0}=u_{1}$,
$Q_{1}u_{0}=g_{1}$, $Q_{1}u_{i}=g_{i-1}^{p}$, $i\geq 2$. Let
$y_{1}=y_{0}^{p}$. Then, similarly to the case $p=2$,

$H_{*}(H^{*}K_{2},Q_{0})=P[y_{1}]\otimes E[y_{0}^{p-1}u_{0}].$

Let $N=\langle
y_{0}^{p-1}u_{0},q=y_{0}^{p-1}u_{1},Q_{0}q=Q_{1}(y_{0}^{p-1}u_{0})\rangle$.
Then $P[y_{1}]\oplus P[y_{1}]\otimes N$ carries the $Q_{0}$-homology and part
of the $Q_{1}$-homology. Similarly to (2.10), the rest of the $Q_{1}$-homology
is

$P[y_{1}]\otimes E[q]\otimes\overline{E[w_{1}]\otimes
TP_{p}[g_{2},g_{3},\ldots]},$

where $w_{1}=u_{2}+u_{0}g_{1}^{p-1}$. There are $E_{1}$-submodules $M_{j}$ for
$j\geq 2$, defined inductively by $M_{2}=\langle
w_{1},g_{2}=Q_{0}w_{1}\rangle$, $M_{j}^{\prime}=\langle
u_{j},g_{j}=Q_{0}u_{j}\rangle$ for $j\geq 3$, and for $j\geq 3$, there exists
a short exact sequence of $E_{1}$-modules

$0\to g_{j-1}^{p-1}M_{j-1}\to M_{j}\to M_{j}^{\prime}\to 0,$

with $Q_{1}u_{j}=g_{j-1}^{p}$. There is an isomorphism of $E_{1}$-modules
$M_{j}\approx\Sigma^{2p^{j}+1}L_{j-2}$, where $L_{j}$ is similar to Figure
2.12, but with $i$th generator ($i\geq 0$) in grading $2(p-1)i$ rather than
$2i$.

Let

$R=\bigoplus_{j\geq 2}M_{j}\otimes TP_{p-1}[g_{j}]\otimes
TP_{p}[g_{j+1},\ldots].$

Then $H_{*}(R;Q_{1})=\overline{E[w_{1}]\otimes TP_{p}[g_{2},g_{3},\ldots]}$,
and so, similarly to Proposition 2.11, up to free $E_{1}$-modules

$H^{*}K_{2}\approx P[y_{1}]\otimes(\langle y_{1}\rangle\oplus N\oplus R\oplus
qR).$ (2.21)

Similarly to Figure 2.18, $\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{p},N)$ can
be read off from Figure 2.22. This gives the third summand and $vq$ part of
the second summand in Theorem 2.4, while the $\langle y_{1}\rangle$ part of
(2.21) gives the non-$vq$ part of the second summand. For the first summand in
Theorem 2.4, we replace $g_{j}$ by $z_{j-1}$, and then note that
$\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{p},M_{j})\approx P[v]\otimes
W_{j-1}$, similar to Figure 2.3. For example, $M_{3}$ has $v$-towers on
$g_{3}$ and $g_{2}^{p}$, which are renamed $z_{2}=z_{2,2}$ and
$z_{1}^{p}=z_{1,2}$, the generators of the $v$-towers of $W_{2}$. This
completes our sketch of proof of Theorem 2.4 when $p$ is odd.

######  Figure 2.22.

Computation of $\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{p},N)$

$4p$$y_{0}^{p-1}g_{1}$$vq$$2p+1$$3$$\cdot$$\cdot$$\cdot$

We explain here the reason for the $k_{0}$ in Definition 1.1. In Theorem 2.4,
$y_{0}^{p-1}z_{0}$ and $z_{1}$ are in the part that is not multiplied by
higher $z$’s when $p=2$, but when $p$ is odd, they form the module $M_{2}$,
whose Ext is $P[v]\otimes W_{1}$, which is multiplied by higher $z$’s. Since
$B_{k}$’s are multiplied by higher $z$’s, but $A_{k}$’s are not, this explains
why $z_{1}$ is in $B_{1}$ when $p$ is odd, but not when $p=2$. The reason for
the split in Theorem 2.4 is the difference in the submodules $N$. Its second
class is $y_{0}^{p-1}Q_{1}y_{0}$ in each. Applying $Q_{1}$ yields
$y_{0}^{p-2}(Q_{1}y_{0})^{2}$. This is 0 when $p$ is odd, but not when $p=2$.
The reason that the portion of Ext corresponding to $N$ is not multiplied by
higher $z$’s is that it gives part of the $Q_{0}$-homology, and this is not
multiplied by higher $z$’s.

We close this section with enumeration of the unimportant
${\mathbb{Z}}_{2}$-classes in $ku^{*}(K_{2})$ when $p=2$.

More on the $E_{1}$-free part when $p=2$

If we compute the $\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{2},F)$ for the
$E_{1}$ free part of $H^{*}K_{2}$, we just get a ${\mathbb{Z}}_{2}$
corresponding to the top element for each copy of $E_{1}$. If we find the
Poincaré series (PS) for the free part, all we have to do to get the PS for
these elements is multiply by $\frac{x^{4}}{(1+x)(1+x^{3})}$. The Poincaré
series for free part is obtained by subtracting the PS for the non-free part
of Proposition 2.11 from that of $H^{*}K_{2}$. This is:

$\prod_{k\geq
0}\frac{1}{(1-x^{2^{k}+1})}-\frac{1}{(1-x^{4})}\bigl{(}1+x^{5}+x^{7}+x^{8}+x^{9}+x^{10}\bigr{)}$
$-\frac{1}{(1-x^{2})(1-x^{4})}\bigl{(}\bigoplus_{j\geq
4}\bigl{(}x^{2^{j}+1}(1+x^{9})(1+x)(1-x^{2j-6})\prod_{k\geq
j}(1+x^{2^{k+1}+2})\bigr{)}\bigr{)}$

The first term is the PS for $H^{*}K_{2}$. The second is the PS for
$P[u_{2}^{2}]\otimes(\langle 1\rangle\oplus N)$. The last term is more
complicated but does the $S$ and $R$ terms. The $(1-x^{4})$ in the denominator
is for the $P[u_{2}^{2}]$. The $x^{9}$ is the shift that takes $R$ to $S$. The
$(1+x)$ is because they are $Q_{0}$ free. The
$x^{2^{j}+1}(1-x^{2j-6})/(1-x^{2})$ is for the odd part of $M_{j}$ and the
remainder is for $\Lambda$.

This is easy to put into a computer and calculate. For example, the number of
free generators in degree 79 is 245.

##  3\. Differentials in the ASS of $ku^{*}(K_{2})$

The main theorem of this section determines the differentials in the ASS for
$ku^{*}(K_{2})$.

######  Theorem 3.1.

The differentials in the spectral sequence whose $E_{2}$-term was given in
Theorem 2.4 are as follows. All $v$-towers are involved, either as source or
target, in exactly one of these. Here $M$ refers to any monomial (possibly
$=1$) in the specified algebra. Recall that $\Lambda_{j}=TP_{p}[z_{i}:\,i\geq
j]$, which is an exterior algebra if $p=2$. Also, recall
$y_{t}=y_{1}^{p^{t-1}}$. We give reference numbers to the differentials when
$p$ is odd, but references to these also apply to the corresponding
differential when $p=2$, as the proofs are extremely similar.

First with $p=2$.

$\displaystyle d_{\nu(i)+2}(y_{1}^{i})$ $\displaystyle=$ $\displaystyle
h_{0}^{\nu(i)}v^{2}qy_{1}^{i-1},\ i\geq 1;$ $\displaystyle
d_{\nu(i)+2}(y_{1}^{i}z_{j}M)$ $\displaystyle=$ $\displaystyle
v^{\nu(i)+2}qy_{1}^{i-1}z_{j-\nu(i),j}M,$ $\displaystyle j\geq\nu(i)+2,\
M\in\Lambda_{j};$ $\displaystyle
d_{2^{t}-t}(h_{0}^{t-2}v^{2}qy_{1}^{2^{t-1}-1}M)$ $\displaystyle=$
$\displaystyle v^{2^{t}}z_{t}M,$ $\displaystyle t\geq 2,\ M\in P[y_{t}];$
$\displaystyle d_{2^{t}-t}(qy_{1}^{2^{t-1}-1}z_{j-(t-2),j}M)$ $\displaystyle=$
$\displaystyle v^{2^{t}-t}z_{t}z_{j}M,$ $\displaystyle j\geq t\geq 2,\ M\in
P[y_{t}]\otimes\Lambda_{j+1}.$

Now with $p$ odd.

$\displaystyle d_{\nu(i)+2}(y_{1}^{i})$ $\displaystyle=$ $\displaystyle
h_{0}^{\nu(i)+1}vqy_{1}^{i-1},\ i\geq 1;$ (3.2) $\displaystyle
d_{\nu(i)+2}(y_{1}^{i}z_{j}M)$ $\displaystyle=$ $\displaystyle
v^{\nu(i)+2}qy_{1}^{i-1}z_{j-\nu(i)-1,j}M,$ $\displaystyle j\geq\nu(i)+2,\
M\in\Lambda_{j};$ $\displaystyle d_{p^{t}-t}(h_{0}^{t-1}vqy_{1}^{p^{t-1}-1}M)$
$\displaystyle=$ $\displaystyle v^{p^{t}}z_{t}M,$ $\displaystyle t\geq 1,\
M\in P[y_{t}];$ $\displaystyle d_{p^{t}-t}(qy_{1}^{p^{t-1}-1}z_{j-(t-1),j}M)$
$\displaystyle=$ $\displaystyle v^{p^{t}-t}z_{t}z_{j}M,$ $\displaystyle j\geq
t\geq 1,\ M\in P[y_{t}]\otimes TP_{p-1}[z_{j}]\otimes\Lambda_{j+1}.$

The proof occupies the rest of this section, except that at the end of the
section we explain briefly how this leads to our description of
$ku^{*}(K_{2})$ in Section 1, except for the exotic extensions.

By [10, Theorem A], $Q_{j}Q_{0}\iota_{2}$ is in the image from
$BP^{*}(K_{2})$, and hence must be a permanent cycle in our ASS. Thus by
(2.20), $z_{j}$ is a permanent cycle, and so (3.1) follows from (3.2), and
(3.1) follows from (3.1), using $pz_{i,\ell}=vz_{i-1,\ell}$, as noted in 1.9.

The differentials (3.2) follow from the result of [2] that
$H^{2pi+1}(K_{2};{\mathbb{Z}})\approx{\mathbb{Z}}/p^{\nu(i)+2}\oplus\bigoplus{\mathbb{Z}}_{p}$.
See also [3, Proposition 1.3.5] when $p=2$. The ASS converging to
$H^{*}(K_{2};{\mathbb{Z}})$ has
$E_{2}=\operatorname{Ext}_{A_{0}}({\mathbb{Z}}_{2},H^{*}K_{2})$, where
$A_{0}=\langle 1,Q_{0}\rangle$. We depict this $E_{2}$ similarly to our ASS
for $ku^{*}(K_{2})$. It has an $h_{0}$-tower for each element of
$H_{*}(H^{*}K_{2},Q_{0})$, which was described in Lemma 2.6. These come in
pairs in grading $2pi$ and $2pi+1$ corresponding to $y_{1}^{i}$ and
$y_{1}^{i-1}y_{0}^{p-1}u_{0}$. In order to get the
${\mathbb{Z}}/p^{\nu(i)+2}$, there must be a $d_{\nu(i)+2}$-differential, as
pictured on the right hand side of Figure 3.6.

Similarly to Figures 2.17 and 2.18, we have, for $p=2$ and $i\geq 1$, an
$h_{0}$-tower in the ASS for $ku^{*}(K_{2})$ arising from $G^{4i+1,2}$, called
either $h_{0}^{2}y_{1}^{i-1}x_{5}$ or $v^{2}y_{1}^{i-1}q$. There is also an
$h_{0}$-tower arising from $y_{1}^{i}\in G^{4i,0}$. The classes $y_{1}$ and
$x_{5}$ correspond to cohomology classes $u_{2}^{2}$ and $u_{5}+u_{2}u_{3}$.
Under the morphism $ku^{*}(K_{2})\to H^{*}(K_{2};{\mathbb{Z}})$, these towers
map across, as suggested in Figure 3.6. We deduce the
$d_{\nu(i)+2}$-differential claimed in (3.2), promulgated by the action of
$v$. Note that $x_{9}=q$.

######  Figure 3.6.

$ku^{*}(K_{2})\to H^{*}(K_{2};{\mathbb{Z}})$

$2pi$$2pi+1$$2pi$$2pi+1$$ku^{*}(K_{2})$$H^{*}(K_{2};{\mathbb{Z}})$$\longrightarrow$

The situation when $p$ is odd is extremely similar, using Figure 2.22. The
difference is that the $h_{0}$-tower in $2pi+1$ in the $ku^{*}$ ASS starts in
filtration 1 rather than 2. Its generator can be called $vy_{1}^{i-1}q$.

In Figure 3.7, we depict many of the differentials asserted in Theorem 3.1 in
grading $\leq 36$ when $p=2$. Regarding the third (final) summand in Theorem
2.4, which is $P[y_{1}]\otimes A_{1}$ when $p=2$, we have included
$y_{1}A_{1}$, $y_{1}^{3}A_{1}$, and $y_{1}^{5}A_{1}$. Not included are the
portions involving (3.2) and (3.1) when $i$ is odd, as this portion self-
annihilates. What is shown is (3.2) for $i=2$, 4, and 6, (3.1) for
$(t,k)=(1,0)$, $(1,1)$, $(1,2)$, and $(2,0)$, and (3.1) with $t=1$, $k=0$, and
$j=4$.

######  Figure 3.7.

Some differentials with $p=2$

$36$$32$$28$$24$$20$$y_{1}^{4}$$16$$12$$8$
$y_{1}z_{1}$$z_{2}^{2}$$z_{3}$$y_{1}^{4}z_{2}$$y_{1}qz_{2}$$z_{2}$$v^{2}y_{1}q$$y_{1}^{2}z_{2}$$v^{2}y_{1}^{3}q$$y_{1}^{2}$
$y_{1}^{6}$ $v^{2}y_{1}^{5}q$

In order to establish the remaining differentials, we will need the following
description of $k(1)^{*}(K_{2})$, which is proved in [7]. We shift by 1 the
subscripts of the classes $z_{j}$ and $w_{j}$ used there. The formulas for
$r(j)$ and $r^{\prime}(j)$ are as in [7]. We recapitulate some of their
properties. Those stated here but not there are easily proved by induction.

######  Proposition 3.8.

[7] For $j\geq 0$, $z_{j}$ is the reduction of the class in $ku^{*}(K_{2})$
and satisfies $|z_{j}|=2(p^{j+1}+1)$. The classes $w_{j}$ satisfy
$|w_{1}|=2p^{2}+1$, $|w_{2}|=2p^{3}-2p^{2}+6p-3$, and
$w_{j+2}=y_{j}^{p-1}w_{j}z_{j+1}^{p-1}$. The integers $r(j)$ and
$r^{\prime}(j)$ satisfy the following properties.

$\displaystyle r(0)=1,\ r(1)=p,\ r(j+2)=r(j)+p^{j+1}(p-1)+1;$ (3.9)
$\displaystyle r^{\prime}(0)=p-1,\ r^{\prime}(1)=p^{2}-p,$ $\displaystyle
r^{\prime}(j+2)=r^{\prime}(j)+p^{j+2}(p-1)-1,$ (3.10) $\displaystyle
r(j)-r^{\prime}(j-1)=j,$ (3.11) $\displaystyle r(j)+r^{\prime}(j)=p^{j+1},$
(3.12) $\displaystyle r(j+2)+r^{\prime}(j)=p^{j+2}+1,$ (3.13)
$\displaystyle(p-1)(r(j-1)+j-1)<p^{j},$ (3.14) $\displaystyle
p^{j+1}-p^{j}\leq r^{\prime}(j)<p^{j+1}-p^{j-1}.$ (3.15)

######  Theorem 3.16.

[7] For any $p$, $k(1)^{*}(K_{2})$ is a trivial $k(1)^{*}$-module plus

$\displaystyle\bigoplus_{j>0}TP_{r(j)}[v]\otimes P[y_{j+1}]\otimes
TP_{p-1}[y_{j}]\otimes\overline{E}[w_{j}]\otimes
E[w_{j+1}]\otimes\Lambda_{j+1}$ $\displaystyle\oplus$
$\displaystyle\bigoplus_{j\geq 1}TP_{r^{\prime}(j-1)}[v]\otimes
P[y_{j}]\otimes E[w_{j}]\otimes\overline{TP}_{p}[z_{j}]\otimes\Lambda_{j+1}$
$\displaystyle\oplus$ $\displaystyle
P[y_{1}]\otimes\biggl{(}\overline{E}[y_{0}^{p-1}z_{0}]\oplus\begin{cases}\overline{E}[z_{1}]&p=2\\\
0&p\text{ odd}\end{cases}\biggr{)}\oplus\bigoplus_{j\geq 1}P[y_{1}]\otimes
E[q]\otimes\overline{E}[z_{j}^{p}]\otimes\Lambda_{j+1}.$

The last line was not discussed in [7]; it is from free $E[Q_{1}]$ summands
which are not part of free $E_{1}$ summands, and plays a very important role.

Now we continue the proof of Theorem 3.1. We have already proved (3.2) and
(3.1). As already noted, the $z_{j}$’s are infinite cycles by [10], and so the
differentials in (3.1) are implied as soon as the corresponding differential
in (3.1) is proved.

As a warmup, we consider the cases $t=2$ and 3 of (3.1) when $p=2$. We make
extensive use of the exact sequence (1.21). Referring to Figure 3.7 is useful.

In even gradings $\leq 14$, $k(1)^{*}(K_{2})=0$ in positive filtration, by
Theorem 3.16. Thus the map $ku^{*}(K_{2})\to k(1)^{*}(K_{2})$ implies that in
the ASS for $ku^{*}(K_{2})$, $v^{s}z_{2}$ must be hit by a differential or
divisible by 2 for $s\geq 2$. In grading $<8$, there is nothing that can
divide it, and the only odd-grading $v$-tower in that range is on
$v^{2}y_{1}q$. Thus $d_{2}(v^{2}y_{1}q)=v^{4}z_{2}$, the case $t=2$, $M=1$ of
(3.1). Since $d_{2}(y_{1}^{2k})=0$ by (3.2), the case $t=2$ of (3.1) follows
for any $M$ by the derivation property. An analogous argument does not work at
the odd primes.

Similarly $v^{s}z_{3}$ must be hit or divisible for $s\geq 4$, and examination
of options in Figure 3.7 shows that we must have
$d_{5}(h_{0}v^{2}y_{1}^{3}q)=v^{8}z_{3}$, preceded by extensions. Since
$d_{5}(y_{1}^{8})=h_{0}^{3}v^{2}y_{1}^{7}q$, we deduce the case $t=3$, $M\in
P[y_{1}^{8}]$ of (3.1) using the derivation property, (2.5) and
$h_{0}z_{2}=0$. We do not have a priori knowledge that $y_{1}^{4}z_{3}$ is a
permanent cycle in the ASS of $ku^{*}(K_{2})$. However, if it supported a
nonzero differential, then the tower of $v$-height 4 on $y_{1}^{4}z_{3}$ in
the ASS of $k(1)^{*}(K_{2})$ would have to map to $v^{t}C$ for $0\leq t\leq 3$
for some $C$ in positive filtration in grading 51 in the ASS of
$ku^{*}(K_{2})$. Then $v^{4}C$ must be $d_{r}(B)$ with $r\geq 5$ and $B$ in
filtration 0 in grading 42. ($B$ cannot have higher filtration since
everything is $v$-towers, and $v^{3}C$ cannot be hit.) But the only possible
$B$ is $y_{1}^{6}z_{2}$, and we already know that
$v^{4}y_{1}^{6}z_{2}\in\operatorname{im}(d_{4})$. (Ordinarily this would not
preclude the possibility of $B$ supporting a differential, but it does since
everything is $v$-towers.) Thus $y_{1}^{4}z_{3}$ is a permanent cycle, and
consideration of its image in $k(1)^{*}(K_{2})$ implies that
$v^{s}y_{1}^{4}z_{3}$ is hit by a differential for some $s\geq 4$. The only
element in odd grading $<42$ not yet accounted for is $h_{0}v^{2}y_{1}^{7}q$
in grading 33, and so this must be the source of the differential. This is the
case $t=3$, $M=y_{1}^{4}$ of (3.1). The validity for all $M=y_{1}^{8i+4}$ (and
$t=3$) now follows similarly to what we did for $M=y_{1}^{8i}$ at the
beginning of this paragraph.

Now we switch our attention to the odd primes. The situation when $p=2$ is
extremely similar. We want to prove the following version of (3.1).

$d_{p^{t}-t}(h_{0}^{t-1}vqy_{1}^{(i+1)p^{t-1}-1})=v^{p^{t}}y_{1}^{ip^{t-1}}z_{t}.$
(3.17)

Now we work toward proving this. We illustrate with $p=5$, but it should be
clear how it generalizes to an arbitrary prime. One new thing is the
Divisibility Criterion as invoked in [7]. Each mod $(p-1)$ value of $i$ can be
considered separately. We will consider (3.17) with $p=5$ and $i=4\ell$; other
congruences follow similarly. We index the differential (3.17) by $(\ell,t)$.
We write $T$ (for vertical Tower) for the class
$h_{0}^{t-1}vqy_{1}^{(4\ell+1)5^{t-1}-1}$, and $M$ (for Monomial) is
$y_{1}^{4\ell 5^{t-1}}z_{t}$. We will often afflict $T$ and $M$ with the
parameters $(\ell,t)$. We write $|T|$ for $\frac{1}{2}(|T|+1)$. The
$\frac{1}{2}$ avoids extraneous factors of 2 that always cancel out. The $+1$
is so that this indicates the grading (times $\frac{1}{2}$) of the class that
it hits. $|M|$ denotes $\frac{1}{2}$ times the grading of $M$, and
$M^{\prime}$ equals $\frac{1}{2}$ times the grading of $v^{h}M$, where $h$ is
the $v$-height of $M$ in $k(1)^{*}(K_{2})$. We wish to show that the
differentials must be as claimed.

There are three types of constraints on the differentials involving these
classes. Constraint C1 is that if $T\to M$ (by which we mean that a certain
$T$ class supports a differential hitting $v^{i}M$ for some $i$ and a certain
monomial $M$), then $|T|\leq M^{\prime}$. (This says that the $v$-tower on $M$
cannot be hit while its image in $k(1)^{*}(K_{2})$ is nonzero.) Constraint C2
says that if $T(5\ell+1,t-1)\to M_{1}$ and $T(\ell,t)\to M_{2}$, then
$|M_{2}|>|M_{1}|$. Since $|T(5\ell+1,t-1)|=|T(\ell,t)|$, this says that as you
move up an $h_{0}$ tower222Note that $h_{0}T(5\ell+1,t-1)=T(\ell,t)$.,
differentials must get longer (unless they are hitting into an $h_{0}$ tower,
which is not the case here.) Constraint C3 says that if $T_{2}\to M_{1}$, then
there exists $M_{3}$ such that $|M_{1}|\geq|M_{3}|\geq M_{1}^{\prime}$ and
either

$M_{3}^{\prime}\leq|T_{2}|$

or

$T_{3}\to M_{3}\text{ has already been proved, and }|T_{3}|\leq|T_{2}|.$

The reason for C3 is that there must be extensions into the $M_{1}$-tower from
grading $M_{1}^{\prime}$ to $|T_{2}|+4$. The nonzero classes on the $v$-tower
(on $M_{3}$) supporting the extensions must go to at least $|T_{2}|+4$, and it
has nonzero classes at least to $M_{3}^{\prime}+4$, and if $T_{3}\to M_{3}$
was already proved, it has nonzero classes to $|T_{3}|+4$. Note that we are
saying that the $v$-tower on $M_{1}$ maps to 0 in $k(1)^{*}(K_{2})$ once we
get to grading $M_{1}^{\prime}$ (and hence in gradings $\leq M_{1}^{\prime}$
it is either hit by differentials or is divisible by $p$). There might be
classes of higher filtration in $k(1)^{*}(K_{2})$ to which it could map, but,
if so, we can modify the generator of the $M_{1}$ tower by the class on the
tower sitting above it. Also note that it is possible that extensions from the
tower $M_{3}$ don’t start from the generator, if there are $h_{0}$-extensions
on the tower for awhile. See Figure 3.18. There is an exception to the C3
requirement for $T(\ell,1)\to M(\ell,1)$. Here the extension into
$v^{4}y_{1}^{4\ell}z_{1}$ is obtained from the special class
$y_{1}^{4\ell}y_{0}^{4}z_{0}$.

######  Figure 3.18.

The role of $M_{3}$

$|M_{1}|$$M_{3}$$|T_{2}|$$|T_{2}|+4$$M_{1}^{\prime}$

With the above conventions, we have $|T|=5^{t}(4\ell+1)+1$,
$|M|=5^{t}(4\ell+5)+1$, and $M^{\prime}=|M|-4r^{\prime}(t-1)$, where
$4r^{\prime}(t-1)$ has the values 16, 80, 412, and 2076 for $t=1$, 2, 3, and
4. Increasing from $t$ to $t+2$ increases this by $4^{2}\cdot 5^{t+1}-4$. We
consider the cases in order of increasing $|M|$ and, for equal values of
$|M|$, increasing $\ell$. We tabulate a representative sample in Table 1. We
omit listing values of $\ell\equiv 3,4$ mod 5 because they behave similarly to
$\ell\equiv 2$.

Table 1. Cases in order $\ell$ | $t$ | $|T|$ | $|M|$ | $M^{\prime}$ |  | $\ell$ | $t$ | $|T|$ | $|M|$ | $M^{\prime}$
---|---|---|---|---|---|---|---|---|---|---
0 | 1 | 6 | 26 | 10 |  | 36 | 1 | 726 | 746 | 730
1 | 1 | 26 | 46 | 30 |  | 37 | 1 | 746 | 766 | 750
2 | 1 | 46 | 66 | 50 |  | 7 | 2 | 726 | 826 | 746
0 | 2 | 26 | 126 | 46 |  | 40 | 1 | 806 | 826 | 810
5 | 1 | 106 | 126 | 110 |  | 41 | 1 | 826 | 846 | 830
6 | 1 | 126 | 146 | 130 |  | 42 | 1 | 846 | 866 | 850
7 | 1 | 146 | 166 | 150 |  | 8 | 2 | 826 | 926 | 846
1 | 2 | 126 | 226 | 146 |  | 45 | 1 | 906 | 926 | 910
10 | 1 | 206 | 226 | 210 |  | 46 | 1 | 926 | 946 | 930
11 | 1 | 226 | 246 | 230 |  | 47 | 1 | 946 | 966 | 950
12 | 1 | 246 | 266 | 250 |  | 9 | 2 | 926 | 1026 | 946
2 | 2 | 226 | 326 | 246 |  | 50 | 1 | 1006 | 1026 | 1010
15 | 1 | 306 | 326 | 310 |  | 51 | 1 | 1026 | 1046 | 1030
16 | 1 | 326 | 346 | 330 |  | 52 | 1 | 1046 | 1066 | 1050
17 | 1 | 346 | 366 | 350 |  | 1 | 3 | 626 | 1126 | 714
3 | 2 | 326 | 426 | 346 |  | 10 | 2 | 1026 | 1126 | 1046
20 | 1 | 406 | 426 | 410 |  | 55 | 1 | 1106 | 1126 | 1110
21 | 1 | 426 | 446 | 430 |  | 56 | 1 | 1126 | 1146 | 1130
22 | 1 | 446 | 466 | 450 |  | 57 | 1 | 1146 | 1166 | 1150
4 | 2 | 426 | 526 | 446 |  | 11 | 2 | 1126 | 1226 | 1146
25 | 1 | 506 | 526 | 510 |  | 60 | 1 | 1206 | 1226 | 1210
26 | 1 | 526 | 546 | 530 |  | 61 | 1 | 1226 | 1246 | 1230
27 | 1 | 546 | 566 | 550 |  | 62 | 1 | 1246 | 1266 | 1250
0 | 3 | 126 | 626 | 214 |  |  |  | $\vdots$ |  |
5 | 2 | 526 | 626 | 546 |  | 154 | 1 | 3086 | 3106 | 3090
30 | 1 | 606 | 626 | 610 |  | 0 | 4 | 626 | 3126 | 1050
31 | 1 | 626 | 646 | 630 |  | 5 | 3 | 2626 | 3126 | 2714
32 | 1 | 646 | 666 | 650 |  | 30 | 2 | 3026 | 3126 | 3046
6 | 2 | 626 | 726 | 646 |  | 155 | 1 | 3106 | 3126 | 3110
35 | 1 | 706 | 726 | 710 |  | 156 | 1 | 3126 | 3146 | 3130

Before presenting a general argument, we illustrate with an example, starting
with $M_{1}=M(1,3)$. We will see that it builds a chart which is $y_{1}^{100}$
times Figure 1.7. In Table 1, we have $|M_{1}|=1126$. Its $v$-tower is
truncated at height $p^{3}=125$ by a differential on $T(1,3)$, with
$|T(1,3)|=626$, using our grading conventions. Playing the role of $M_{3}$ is
$M(6,2)$ with $|M_{3}|=726$. We have $M_{1}^{\prime}=714$. It is $v^{3}M_{3}$
which supports the extension in ”grading“ 714. Note that for $0\leq i\leq 2$,
$h_{0}v^{i}M_{3}\neq 0$, and so $p\cdot v^{i}M_{3}$ is not a $v$-multiple of
$M_{1}$. (In Figure 1.7, the class $y_{2}^{p-1}z_{2}$ corresponds to $M_{3}$.)
From Table 1, we see that $M_{3}^{\prime}=646$, which means that in “grading”
$\leq 646$, the $v$-tower on $M_{3}$ is either hit by a differential or
divisible by $p$. Table 1 says it is hit by a differential in 626. In
“gradings” from 646 to 630, it is divisible by $p$. It has its own, distinct,
$M_{3}$ class, namely $M(31,1)$. In Figure 1.7, this latter class corresponds
to $y_{1}^{p-1}y_{2}^{p-1}z_{1}$.

Now we start the proof. We begin with a lemma.

######  Lemma 3.19.

For $M=M(\ell^{\prime},t^{\prime})$ with $|M(5\ell+1,t-1)|<|M|<|M(\ell,t)|$,
we have $t^{\prime}<t$, $|T(\ell,t)|<|T(\ell^{\prime},t^{\prime})|$, and
$|M(5\ell+1,t-1)|<M^{\prime}$.

###### Proof.

The given inequalities quickly force $t^{\prime}<t$. The inequality
$|T(\ell,t)|<|T(\ell^{\prime},t^{\prime})|$ follows immediately. Finally, the
given inequalities prevent $M^{\prime}\leq|M(5\ell+1,t-1)|$.

To prove the differentials, we use induction on our ordering of the $M$’s. If
the differentials are not as posed, consider the smallest $|M|$ such that
$T(\ell,t)\to M$ with $M\neq M(\ell,t)$.

We cannot have $|M|>|M(\ell,t)|$, because $|M(\ell,t)|$ would contradict the
minimality of $|M|$. We cannot have $|M|\leq|M(5\ell+1,t-1)|$ by constraint
C2.

If $|M(5\ell+1,t-1)|<|M|<|M(\ell,t)|$, by constraint C3 and the lemma, we must
have $M_{3}$ with $|M(5\ell+1,t-1)|<M^{\prime}\leq|M_{3}|<|M|$. Because
$|M_{3}|<|M|$, we know $T_{3}\to M_{3}$ by induction. From the lemma, we get
$|T(\ell,t)|<|T_{3}|$, but that contradicts constraint C3.

We must have $T(\ell,t)\to M(\ell,t)$, and $M(5\ell+1,t-1)$ is eligible for
our $M_{3}$. This completes most of the proof of (3.17) and hence of Theorem
3.1.

Underlying the above analysis has been an assumption that the $M$-classes are
always hit by $T$-classes. We show now that it could not have occurred that an
$M$-class supported a differential. Assume that $M=y_{1}^{ip^{t-1}}z_{t}$ is
the $M$-class of lowest grading which supports a differential. We now revert
to letting $|x|$ denote the actual grading of a class $x$, not divided by 2.

In $k(1)^{*}(K_{2})$, $M$ supports a $v$-tower of $v$-height $r^{\prime}(t-1)$
by 3.16. We will show at the end of the proof that there is a number
$\Delta\leq t$ such that $v^{i}M$ maps nontrivially to $ku^{*+1}(K_{2})$ if
and only if $i\leq r^{\prime}(t-1)-\Delta$. (Usually $\Delta=1$.) The image of
$M$ in $ku^{|M|+1}(K_{2})$ is a class $C$ of positive filtration such that
$v^{r^{\prime}(t-1)-\Delta}C\neq 0$ and $v^{r^{\prime}(t-1)-\Delta+1}C=0\in
ku^{*}(K_{2})$, so there must be a differential in the ASS of $ku^{*}(K_{2})$
from a filtration-0 class hitting a class of filtration $\geq
r^{\prime}(t-1)-\Delta+2$ in grading $|M|+1-2(p-1)(r^{\prime}(t-1)-\Delta+1)$.
(The reason that the differential must start from filtration 0 is that in even
gradings, $E_{2}$ consists entirely of $v$-towers starting in filtration 0.)
This differential cannot come from another such $M$ because of our lowest-
grading assumption. It cannot come from a product of one or more $z$’s times
one of these $M$’s because $z$’s are infinite cycles. We must rule out the
possibility that this differential is one of type (3.1). They are
distinguished by having the smallest $z$-subscript at least 2 greater than the
$p$-exponent of the exponent of $y_{1}$.

The differential to $C$ has subscript $\geq r^{\prime}(t-1)-\Delta+2$, and so
the class in (3.1) would be $y_{1}^{\ell p^{r^{\prime}(t-1)-\Delta}}Z$ for
some positive integer $\ell$, where $Z$ is a product of $z_{j}$’s with $j\geq
r^{\prime}(t-1)-\Delta+2$, and each $j$ appears at most $p-1$ times, except
that the smallest $j$ might appear $p$ times. Equating this grading with
$|M|-2(p-1)(r^{\prime}(t-1)-\Delta+1)$, and cancelling a common factor 2 from
all terms yields

$\ell
p^{r^{\prime}(t-1)-\Delta+1}+\sum_{j}(p^{j+1}+1)=ip^{t}+p^{t+1}+1-(p-1)(r^{\prime}(t-1)-\Delta+1).$
(3.20)

Using (3.12) and (3.14) and $\Delta\leq t$, the right hand side of (3.20)
equals $p^{t}(i+1)+(p-1)(r(t-1)+\Delta-1)+1\equiv(p-1)(r(t-1)+\Delta-1)+1$ mod
$p^{t}$, with $(p-1)(r(t-1)+\Delta-1)+1\leq p^{t}$ (strict if $t>2$). Since
$r^{\prime}(t-1)-\Delta>t$, this implies that the $\displaystyle\sum_{j}$ on
the left hand side of (3.20) must contain at least $(p-1)(r(t-1)+\Delta-1)+1$
summands. We obtain

$\displaystyle\sum p^{j}\geq p\cdot
p^{r^{\prime}(t-1)-\Delta+2}+(p-1)(p^{r^{\prime}(t-1)-\Delta+3}+\cdots+p^{r^{\prime}(t-1)+r(t-1)})$
$\displaystyle=$ $\displaystyle p^{r^{\prime}(t-1)+r(t-1)+1}=p^{p^{t}+1},$

so $\sum p^{j+1}\geq p^{p^{t}+2}$, and hence $p^{t}(i+1)>p^{p^{t}+2}$. Thus
$i\geq p^{p^{t}-t+2}>p^{p^{t}-2t}$.

Since $d_{p^{t}-t+1}(y_{1}^{p^{p^{t}-t-1}})$ is defined,

$d_{r}(y_{1}^{p^{p^{t}-t-1}})=0\text{ for }r\leq p^{t}-t,$ (3.21)

and by the lowest-grading assumption,
$d_{p^{t}-t}(h_{0}^{t-1}vqy_{1}^{(i-p^{p^{t}-2t}+1)p^{t-1}-1})=v^{p^{t}}y_{1}^{(i-p^{p^{t}-2t})p^{t-1}}z_{t}$
and $y_{1}^{(i-p^{p^{t}-2t})p^{t-1}}z_{t}$ is a permanent cycle. Since

$y_{1}^{ip^{t-1}}z_{t}=y_{1}^{(i-p^{p^{t}-2t})p^{t-1}}z_{t}\cdot
y_{1}^{p^{p^{t}-t-1}},$

we deduce that $y_{1}^{ip^{t-1}}z_{t}$ survives to $E_{p^{t}-t}$ and (3.17),
using the derivation property of differentials.

Now we consider the need for $\Delta$ in the above argument. The worry is that
maybe part of the $v$-tower on $M$ in $k(1)^{*}(K_{2})$ might be in the image
from $ku^{*}(K_{2})$, due to a filtration jump from a lower tower, as sketched
in Figure 3.22, so that only a smaller part of the $M$-tower in
$k(1)^{*}(K_{2})$ maps to $ku^{*+1}(K_{2})$.

######  Figure 3.22.

An unwanted possibility

$c$ $c^{\prime}$ $c$
$c^{\prime}$$|M_{1}|$$|M_{2}|$$|M_{1}|$$|M_{2}|$$|M_{1}|+1$$ku^{*}(K_{2})$$k(1)^{*}(K_{2})$$ku^{*+1}(K_{2})$

The monomials
$M_{\varepsilon}=y_{t_{\varepsilon}}^{i_{\varepsilon}}z_{t_{\varepsilon}}$
($\varepsilon=1,2$) have
$|M_{\varepsilon}|=2(p^{t_{\varepsilon}}(i_{\varepsilon}+p)+1)$ and are
truncated in $k(1)^{*}(K_{2})$ in grading
$M_{\varepsilon}^{\prime}=|M_{\varepsilon}|-2(p-1)r^{\prime}(t_{\varepsilon}-1)$.
In $ku^{*}(K_{2})$, $M_{2}$ is truncated in grading
$|T_{2}|=|v^{p^{t_{2}}}M_{2}|=2(p^{t_{2}}(i_{2}+1)+1)$. In Figure 3.22,
elements $c$ are in grading $M_{2}^{\prime}$, and $c^{\prime}$ is in grading
$M_{1}^{\prime}+2(p-1)$. The necessary condition for nontrivial image in
$k(1)^{*}(K_{2})$ (and hence $\Delta>1$) is

$|T_{2}|+2(p-1)\leq M_{1}^{\prime}+2(p-1)\leq M_{2}^{\prime}.$ (3.23)

If this occurs, then we might have $\Delta$ as large as
$\dfrac{M_{2}^{\prime}-M_{1}^{\prime}}{2(p-1)}+1$. We now show in Lemma 3.24
that if (3.23) holds, then $(M_{2}^{\prime}-M_{1}^{\prime})/(2(p-1))<t$,
establishing the claim made earlier about $\Delta\leq t$.

We restrict to $p=5$, $i=4\ell$ for simplicity, and so that the reader can
refer to Table 1 as an aid. The argument easily generalizes to any prime and
any congruence. We divide everything by 2 as was done above, and also subtract
off the $+1$ which occurs in formulas for $|M|$ and $|T|$, so the numbers will
be 1 smaller than those in the table.

######  Lemma 3.24.

If $t_{1}>t_{2}$ and

$5^{t_{2}}(4\ell_{2}+1)+4\leq
5^{t_{1}}(4\ell_{1}+5)-4r^{\prime}(t_{1}-1)+4\leq
5^{t_{2}}(4\ell_{2}+5)-4r^{\prime}(t_{2}-1),$

then

$\textstyle\frac{1}{4}\bigl{(}5^{t_{2}}(4\ell_{2}+5)-4r^{\prime}(t_{2}-1)-(5^{t_{1}}(4\ell_{1}+5)-4r^{\prime}(t_{1}-1))\bigr{)}<t_{1}-1.$

###### Proof.

If there is a counterexample to this, then there is one with $\ell_{1}=0$,
since $\ell_{2}$ could be decreased by $5^{t_{1}-t_{2}}\ell_{1}$, so it
suffices to use $\ell_{1}=0$. Let $Q(k)=(5^{2k}-1)/24$ (called $q(k)$ in [7,
Lemma 5.3]). Then, using [7, Lemma 5.5], for $t=2k+\delta$ with $\delta=1$ or
2,

$5^{t+1}-4r^{\prime}(t-1)=5^{2k+\delta}+16\cdot 5^{\delta}Q(k)+4k+4\cdot
5^{\delta-1}.$

Since $16\cdot 5^{\delta}Q(k)+4k+4\cdot 5^{\delta-1}<3\cdot 5^{2k+\delta}$,
the hypothesis of the lemma says that $5^{t_{1}+1}-4r^{\prime}(t_{1}-1)$ mod
$4\cdot 5^{t_{2}}$ lies in the mod-$(4\cdot 5^{t_{2}})$ interval
$[5^{t_{2}},5^{t_{2}+1}-4r^{\prime}(t_{2}-1)-4]$.

Let $t_{1}=2k_{1}+\delta_{1}$ and $t_{2}=2k_{2}+\delta_{2}$. The condition is
restated as

$5^{2k_{1}+\delta_{1}}+16\cdot 5^{\delta_{1}}Q(k_{1})+4k_{1}+4\cdot
5^{\delta_{1}-1}$ (3.25)

lies in the mod-$(4\cdot 5^{t_{2}})$ interval

$[5^{t_{2}},5^{t_{2}}+16\cdot 5^{\delta_{2}}Q(k_{2})+4k_{2}+4\cdot
5^{\delta_{2}-1}-4].$ (3.26)

Let $\delta_{2}=1$. The reduction mod $4\cdot 5^{t_{2}}$ of (3.25) is

$5^{t_{2}}+16\cdot 5^{\delta_{1}}Q(k_{2})+4k_{1}+4\cdot 5^{\delta_{1}-1}.$
(3.27)

Let $\delta_{1}=2$. Then $5^{t_{2}}+16\cdot 5^{\delta_{1}}Q(k_{2})>4\cdot
5^{t_{2}}$ and equals $5^{2k_{2}+2}-(2000Q(k_{2}-1)+100)$, so (3.27) will
first be in the interval (3.26) when $4k_{1}+20=2000Q(k_{2}-1)+100$, hence
$k_{1}=500Q(k_{2}-1)+20$, so $t_{1}=1000Q(k_{2}-1)+42$. The left hand side of
the conclusion of the lemma is $\frac{1}{8}(M_{2}^{\prime}-M_{1}^{\prime})$
with $M_{1}^{\prime}$ and $M_{2}^{\prime}$ as in (3.23). For
$k_{1}=500Q(k_{2}-1)+20$, the value of $M_{1}^{\prime}$ is at the left end of
the interval (3.26), and so $\frac{1}{8}(M_{2}^{\prime}-M_{1}^{\prime})$
equals $\frac{1}{4}$ times the length plus 4 of (3.26), which is

$20Q(k_{2})+k_{2}+1=500Q(k_{2}-1)+k_{2}+21=\textstyle\frac{1}{2}t_{1}+k_{2}.$

Since $k_{2}<<t_{1}$, this is less than $t_{1}-1$, as claimed. If $k_{1}$ is
increased from the value $500Q(k_{2}-1)+20$, the value of $t_{1}$ increases,
while $M_{2}^{\prime}-M_{1}^{\prime}$ decreases, since $M_{1}^{\prime}$ is
moving through the interval, so the inequality asserted in the lemma is
satisfied more strongly.

Now, with $\delta_{2}=1$ continuing, let $\delta_{1}=1$. Since $k_{1}>k_{2}$,
(3.27) lies outside the interval (3.26) until $80Q(k_{2})+4k_{1}+4=4\cdot
5^{t_{2}}$, so

$k_{1}=5^{2k_{2}+1}-20Q(k_{2})-1=100Q(k_{2})+4$

and $t_{1}=200Q(k_{2})+9$. Again
$\frac{1}{8}(M_{2}^{\prime}-M_{1}^{\prime})=20Q(k_{2})+k_{2}+1\approx\frac{1}{10}t_{2}+k_{2}$,
so the conclusion of the lemma is satisfied more strongly.

A similar analysis works when $\delta_{2}=2$. In this case
$\frac{1}{8}(M_{2}^{\prime}-M_{1}^{\prime})\approx\frac{1}{2}t_{1}+k_{2}$ if
$\delta_{1}=1$, and
$\frac{1}{8}(M_{2}^{\prime}-M_{1}^{\prime})\approx\frac{1}{10}t_{1}+k_{2}$ if
$\delta_{1}=2$.

We close this section by explaining how Theorems 2.4 and 3.1 lead to the
descriptions of $ku^{*}(K_{2})$ given in Theorems 1.4 and 1.11, modulo exotic
extensions. We begin with the portion in even gradings and restrict our
attention to odd $p$. All elements in the $P[h_{0},v,y_{1}]$ part of Theorem
2.4 support differentials of type (3.2). Note that
$y_{0}^{p^{k}-1}=y_{0}^{p-1}y_{1}^{p^{k-1}-1}=\prod_{j=0}^{k-1}y_{j}^{p-1}$.
The first is easiest to write, the second occurs in Theorem 2.4, and the third
in 1.1 and Figure 1.7. From 1.1, $y_{0}^{p^{k}-1}z_{0}$ is in $A_{k}$ for
$k\geq 1$, the bottom right element in Figure 1.7. Then

$P[y_{1}]y_{0}^{p-1}z_{0}=\bigoplus\mathcal{M}_{k}^{A}\cdot
y_{0}^{p^{k}-1}z_{0}\subset\bigoplus\mathcal{M}_{k}^{A}A_{k}.$ (3.28)

The first part occurs in Theorem 2.4 and the last part in Theorem 1.4.

Now we consider $P[y_{1}]\otimes\bigoplus_{j\geq 1}W_{j}\otimes
TP_{p-1}[z_{j}]\otimes\Lambda_{j+1}$ in Theorem 2.4. The $\bigoplus$ part is
all monomials $z_{\ell}M$ with $\ell\geq 1$ and $M\in\Lambda_{\ell}$. From
Theorem 3.1, $y_{1}^{i}z_{\ell}M$ supports a differential (3.1) if
$\ell\geq\nu(i)+2$, while those with $\nu(i)\geq\ell-1$ are hit by
differentials (3.1) and (3.1), yielding $v$-towers with heights as given in
1.1. These are all monomials in $\bigoplus_{\ell\geq
1}P[y_{\ell},y_{\ell+1},\ldots]z_{\ell}\Lambda_{\ell}$. From 1.1 or (1.3), the
generators of the $v$-towers in $B_{k}$ are all

$z_{j}\prod_{i=j}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\},\ 1\leq j\leq k.$

Let $(z_{\ell}M)_{i}$ be the $y_{i}^{e}z_{i}^{e^{\prime}}$ factors of $M$.
Then $\mathcal{M}_{k}B_{k}$ consists of all monomials $z_{\ell}M$ such that
$(z_{\ell}M)_{i}$ equals $y_{i}^{p-1}$ or $z_{i}^{p-1}$ for $\ell\leq i<k$,
but not for $i=k$, and so every monomial $z_{\ell}M$ is in a unique
$\mathcal{M}_{k}B_{k}$. From Theorem 3.1, $z_{\ell}M$ has $v$-height
$p^{\ell}$ if and only if $M$ contains no $z$-factors, which explains the
split into $\mathcal{M}_{k}^{A}$ and $\mathcal{M}_{k}^{B}$ in Theorem 1.4.

Now we address the odd gradings. The $P[h_{0},v,y_{1}]vq$ part of Theorem 2.4
is totally removed either as sources (3.1) or targets (3.2) of differentials.
See grading 17 in Figure 3.7 for a nice illustration. The
$qy_{1}^{i-1}S_{\nu(i)+1,\ell}$ part of Theorem 1.11 is formed from
$TP_{\nu(i)+2}[v]qy_{1}^{i-1}W_{\ell}$ in 2.4 using (3.1). The generators of
$S_{\nu(i)+1,\ell}$ are $z_{1,\ell},\ldots,z_{\ell-\nu(i)-1,\ell}$, but to see
the differential from (3.1), one should write
$z_{t,\ell}=z_{t,t+\nu(i)+1}Z_{t+\nu(i)+1}^{\ell}$, where

$Z_{i}^{j}=(z_{i}\cdots z_{j-1})^{p-1}\text{ for }j>i,\text{ with
}Z_{i}^{i}=1.$ (3.29)

The remaining generators of $qy_{1}^{i-1}W_{\ell}$, namely
$qy_{1}^{i-1}z_{j,\ell}$ with $\ell-\nu(i)\leq j\leq\ell$, support
differentials (3.1). There can be no unexpected exotic extensions among these
summands for the reason noted at the end of Section 1. The
$\operatorname{ker}(p)$ elements in the $S$ summands play a very important
role in the exact sequence.

To see that there are no unexpected exotic extensions between these summands

##  4\. Exotic extensions

In this section, we prove the following expansion of (1.2).

######  Theorem 4.1.

If $i\geq 0$ and $k\geq k_{0}$,

$py_{k}^{i}y_{k-1}^{p-1}z_{k-1}=v^{p^{k-1}(p-1)}y_{k}^{i}z_{k}$

with an additional term $vy_{k}^{i}y_{k-1}^{p-1}z_{k-2}^{p}$ if $k\geq
k_{0}+2$.

The additional term is seen in Ext, and will be ignored in the rest of this
section. We have included the factor $y_{k}^{i}$, which is not automatic since
$y_{k}^{i}$ is not a permanent cycle. Since, for example, $y_{k+1}=y_{k}^{p}$,
we need not consider $y_{i}$ for $i>k$. It is automatic that this formula can
be multiplied by $z_{j}$’s, since they do survive the spectral sequence.

The extension is deduced from the exact sequence

$ku^{*}(K_{2})\ \smash{\mathop{\longrightarrow}\limits^{\cdot p}}\
ku^{*}(K_{2})\longrightarrow k(1)^{*}(K_{2})$

and the fact that $v^{r^{\prime}(k-1)}y_{k}^{i}z_{k}=0$ in $k(1)^{*}(K_{2})$
with $r^{\prime}(k-1)\geq p^{k}(p-1)$. Thus
$v^{r^{\prime}(k-1)}y_{k}^{i}z_{k}$ must be divisible by $p$ in
$ku^{*}(K_{2})$, and, as we will show, the $v$-tower on
$y_{k}^{i}y_{k-1}^{p-1}z_{k-1}$ provides the only classes that can do the
dividing. Once we know the division formula toward the end of the $v$-tower,
we can deduce that it holds earlier in the tower, as well. For example,
$r^{\prime}(2)=p^{3}-p^{2}+p-2$, which is the height in the top $v$-tower in
Figure 1.7 where the extensions into it do not also involve an
$h_{0}$-extension. We deduce the extensions from the earlier part of the
$v$-tower on $y_{2}^{p-1}z_{2}$ by naturality.

We illustrate in Figure 4.2, using the notation of the preceding section. Thus
$T_{i}$ is the class satisfying $d_{r}(T_{i})=v^{r}M_{i}$, Here the portion of
the top tower to the right of $M_{1}^{\prime}$ must be divisible by $p$. The
tower providing the extension must have $M_{1}^{\prime}\leq|M_{2}|<|M_{1}|$
and $|T_{2}|\leq|T_{1}|$.

######  Figure 4.2.

Conditions for extension

$\bullet$$M_{1}$$M_{2}$$M_{1}^{\prime}$$|T_{1}|$$|T_{2}|$

As we did for the differentials in the previous section, we will perform the
argument for $p=5$. It will be clear that it generalizes to an arbitrary odd
prime, and with minor modification to $p=2$. Also, we use $i=4\ell$ in Theorem
4.1. If instead we used $i=4\ell+d$ for $1\leq d\leq 3$, it will just add the
same amount to the quantities $|M|$, $|T|$, and $M^{\prime}$ involved in the
argument. We can use Table 1 to envision the analysis, with the $t$ there
replaced by $k$. For a monomial $M(\ell,k)=y_{k}^{4\ell}z_{k}$, we have, after
dividing by 2, $|M|=5^{k}(4\ell+5)+1$, $|T|=5^{k}(4\ell+1)+1$, and
$5^{k}(4\ell+1.16)+1<M^{\prime}\leq 5^{k}(4\ell+1.8)+1$, using (3.15). With
$M_{1}$ and $M_{2}$ as in Figure 4.2, we will show that $M_{2}(5\ell+1,k-1)$
is the unique monomial satisfying the inequalities stated just before Figure
4.2 for $M_{1}(\ell,k)$. Note that
$M(5\ell+1,k-1)=y_{k}^{4\ell}y_{k-1}^{4}z_{k-1}$. We omit the $+1$ in all the
formulas.

The inequalities are satisfied by $M_{2}(5\ell+1,k-1)$ since

$5^{k}(4\ell+1.8)\leq 5^{k-1}(4(5\ell+1)+5)<5^{k}(4\ell+5)\text{ and
}5^{k-1}(4(5\ell+1)+1)\leq 5^{k}(5\ell+1).$

If $k_{2}\geq k$, then the first inequality, after dividing by $5^{k}$,
becomes

$4\ell+1.8\leq 5^{k_{2}-k}(4\ell_{2}+5)<4\ell+5,$

which cannot be satisfied since the middle term is $\equiv 1$ mod 4. If
$k_{2}<k-1$, then

$M_{1}^{\prime}-|T_{1}|>5^{k}\cdot.16\geq 4\cdot 5^{k_{2}}=|M_{2}|-|T_{2}|,$

which is inconsistent with two of the inequalities. Let $k_{2}=k-1$. If
$\ell_{2}<5\ell+1$, then

$|M_{2}|=5^{k-1}(4\ell_{2}+5)\leq 5^{k-1}(4\cdot
5\ell+5)<5^{k}(4\ell+1.16)<M_{1}^{\prime},$

contradicting one of the inequalities. If $k_{2}=k-1$ and $\ell_{2}>5\ell+1$,
then

$|T_{2}|\geq 5^{k-1}(4(5\ell+2)+1)>5^{k}(4\ell+1)=|T_{1}|,$

contradicting one of the inequalities.

We deduce that $M_{2}=y_{k}^{4\ell}y_{k-1}^{4}z_{k-1}$, as claimed. We should
perhaps have noted that the extensions could not have come from classes with
more than one $z_{j}$-factor, because these are $z_{j}$ times a class on which
the extensions have already been determined.

##  5\. Proposed formulas for the exact sequence (1.21)

In this section we propose what we feel must be the correct complete formulas
for the exact sequence (1.21). Some homomorphisms are forced by naturality,
but many others involve significant filtration jumps. However, they all occur
in several families with nice properties. The 10-term exact sequence (5.2)
shows how the $S_{k,\ell}$ portions and the exotic extensions yield
compatibility of the differing $v$-tower heights in $ku^{*}(K_{2})$ and
$k(1)^{*}(K_{2})$. In Section 6, we show that all elements of
$k(1)^{*}(K_{2})$ are accounted for exactly once in these homomorphisms, which
implies that there can be no more exotic extensions. This does not require us
to prove that our homomorphism formulas are actually correct, as discussed at
the end of Section 1. We will focus on the case when $p$ is odd. We could
incorporate all primes together at the expense of involving the parameter
$k_{0}$, but things are complicated enough without that. In an earlier version
of this paper ([6]), a thorough analysis when $p=2$ was performed.

We propose that (1.21) can be split into exact sequences of length 4 and 10
(not including 0’s at the end). There are subgroups of $k(1)^{*}(K_{2})$
called $G_{k}^{1}$ and $G_{k}^{2}$ for $k\geq 1$ and $G^{i}_{k,\ell}$ for
$3\leq i\leq 6$ and $1\leq k<\ell$ such that there are exact sequences

$0\to G_{k}^{1}\to A_{k}\ \smash{\mathop{\longrightarrow}\limits^{p}}\
A_{k}\to G_{k}^{2}\to 0$ (5.1)

for $k\geq 1$, and, for $1\leq k<\ell$,

$\displaystyle 0$ $\displaystyle\to$ $\displaystyle G^{3}_{k,\ell}\to
y_{k}B_{k}Z_{k}^{\ell}\ \smash{\mathop{\longrightarrow}\limits^{p}}\
y_{k}B_{k}Z_{k}^{\ell}\to G^{4}_{k,\ell}\to y_{1}^{p^{k-1}-1}qS_{k,\ell}$
(5.2) $\displaystyle\ \smash{\mathop{\longrightarrow}\limits^{p}}\ $
$\displaystyle y_{1}^{p^{k-1}-1}qS_{k,\ell}\to G^{5}_{k,\ell}\to
B_{k}z_{\ell}\ \smash{\mathop{\longrightarrow}\limits^{p}}\ B_{k}z_{\ell}\to
G^{6}_{k,\ell}\to 0,$

with $Z_{k}^{\ell}$ as defined in (3.29). The sequence (5.1) can be tensored
with $TP_{p-1}[y_{k}]\otimes P[y_{k+1}]$, while (5.2) can be tensored with
$TP_{p-1}[y_{k}]\otimes P[y_{k+1}]\otimes
TP_{p-1}[z_{\ell}]\otimes\Lambda_{\ell+1}$. If $p$ is odd, there are also
exact sequences

$0\to G^{7}_{k,e}\to B_{k}z_{k}^{e}\
\smash{\mathop{\longrightarrow}\limits^{p}}\ B_{k}z_{k}^{e}\to G^{8}_{k,e}\to
0$ (5.3)

for $k\geq 1$ and $1\leq e\leq p-2$. This can be tensored with
$P[y_{k}]\otimes\Lambda_{k+1}$.

One can verify that the totality of $A_{k}$ and $B_{k}$ groups in these exact
sequences agrees with that in Theorem 1.4. We will study these exact sequences
by breaking them up into short exact sequences and isomorphisms involving
kernels and cokernels of $\cdot p$.

Let $K_{k}^{A}=\operatorname{ker}(\cdot p|A_{k})$,
$K_{k}^{B}=\operatorname{ker}(\cdot p|B_{k})$,
$C_{k}^{A}=\operatorname{coker}(\cdot p|A_{k})$, and
$C_{k}^{B}=\operatorname{coker}(\cdot p|B_{k})$. There are important elements
$g_{k}\in K_{k}^{A}$ and $K_{k}^{B}$ defined (up to unit coefficients) by
$g_{1}=z_{1}$, $g_{2}=v^{p-2}z_{2}$, and, for $k\geq 1$,

$g_{k+2}=v^{r^{\prime}(k)-1}z_{k+2}+g_{k}y_{k}^{p-1}z_{k+1}^{p-1}.$ (5.4)

To see that this is in $\operatorname{ker}(\cdot p)$, we use (1.2) to see that
$p\cdot v^{r^{\prime}(k)-1}z_{k+2}=v^{r^{\prime}(k)}z_{k+1}^{p}$, and that the
$v^{r^{\prime}(k-2)-1}z_{k}$ term in $g_{k}$ yields
$v^{r^{\prime}(k-2)-1}v^{p^{k}(p-1)}z_{k+1}z_{k+1}^{p-1}$ in $p\cdot
g_{k}y_{k}^{p-1}z_{k+1}^{p-1}$. Using (3.10), these terms cancel. Other terms
in $p\cdot g_{k}y_{k}^{p-1}z_{k+1}^{p-1}$ yield 0 since
$g_{k}\in\operatorname{ker}(\cdot p)$.

The $v$-towers in $K_{k}^{A}$ are generated by

$g_{k}\text{ and
}g_{j}z_{j}^{p-1}\prod_{i=j+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\},\ 1\leq
j\leq k-1.$ (5.5)

For example, using Figure 1.7 when $k=3$, these are
$g_{3}=v^{p^{2}-p-1}z_{3}+y_{1}^{p-1}z_{1}z_{2}^{p-1}$,
$g_{2}z_{2}^{p-1}=v^{p-2}z_{2}^{p}$, $g_{1}z_{1}^{p-1}z_{2}^{p-1}$, and
$g_{1}z_{1}^{p-1}y_{2}^{p-1}$. The $v$-heights are $p^{k}-(r^{\prime}(k-2)-1)$
for $g_{k}$, and $p^{j}-j-(r^{\prime}(j-2)-1)$ for the others, since they are
determined by $v$-heights of $z_{j}$ in $B_{k}$. The map $G_{k}^{1}\to
K_{k}^{A}$ sends $w_{k}$ to $g_{k}$ and

$w_{j}P\mapsto g_{j}P\text{ for
}P=z_{j}^{p-1}\prod_{i=j+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\},$ (5.6)

with $w_{j}$ as in 3.8 and 3.16. The $v$-height of $w_{j}$ is $r(j)$ if it is
not accompanied by $z_{j}$, and $r^{\prime}(j-1)$ if it is. By (3.13) and
((3.12) and (3.11)) the $v$-heights agree, so (5.6) is an isomorphism on
$v$-towers.

For $L=K_{k}^{A}$ or $K_{k}^{B}$ or $C_{k}^{A}$ or $C_{k}^{B}$, we say that a
${\mathbb{Z}}_{p}$ in $L$ is a class of $v$-height 1 in $L$ which is not part
of a larger $v$-tower in $L$. There is one ${\mathbb{Z}}_{p}$ in $K_{3}^{A}$,
as can be seen in Figure 1.7. This is the element
$v^{p-2}y_{1}^{p-1}z_{1}z_{2}^{p-1}$. Note that for $i<p-1$,
$v^{i}y_{1}^{p-1}z_{1}z_{2}^{p-1}+v^{i+p^{2}-p-1}z_{3}$ is part of a $v$-tower
in $K_{3}^{A}$, which continues with the elements $v^{i}z_{3}$ for
$i>p^{2}-3$, but $v^{i}y_{1}^{p-1}z_{1}z_{2}^{p-1}$ itself is in $K_{3}^{A}$
only for $i=p-2$. Using 1.1, we find that the ${\mathbb{Z}}_{p}$’s in
$K_{k}^{A}$ are

$v^{p^{t}-t-1}(y_{t}\cdots
y_{j-1})^{p-1}z_{t}z_{j}^{p-1}\prod_{i=j+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}\text{
for }1\leq t<j<k.$ (5.7)

For example, the elements $v^{p-2}(y_{1}y_{2})^{p-1}z_{1}$ and
$v^{p^{2}-3}y_{2}^{p-1}z_{2}$ in Figure 1.7 yield elements in $K_{4}^{A}$
after being multiplied by $z_{3}^{p-1}$. The basic formula for the
homomorphism from part of $k(1)^{*}(K_{2})$ to ${\mathbb{Z}}_{p}$’s in various
$K_{k}^{A}$ and $K_{k}^{B}$, possibly tensored with other classes as in
Theorem 1.4, is

$\bigl{(}q(y_{1}\cdots y_{t})^{p-1}z_{j-t,j}\mapsto
v^{p^{t}-t-1}y_{t}^{p-1}z_{t}z_{j}\bigr{)}\otimes P[y_{j}]\otimes
TP_{p-1}[z_{j}]\otimes\Lambda_{j+1}\text{ for }j>t\geq 1.$ (5.8)

The domain elements are in the second half of the third line of Theorem 3.16.
The ones that are in $G^{1}_{k}$ in the isomorphism $G^{1}_{k}\to K_{k}^{A}$
can be extracted using (5.7).

The isomorphism $G^{3}_{k,\ell}\to y_{k}K_{k}^{B}Z_{k}^{\ell}$ in (5.2) is
given using formulas analogous to (5.6) and (5.8). There are several minor
differences. One is that the $v$-tower on $y_{k}g_{k}Z_{k}^{\ell}$ is
truncated due to $v^{p^{k}-k}z_{k}=0$ in $B_{k}$ (as opposed to
$v^{p^{k}}z_{k}=0$ in $A_{k}$). This is compatible with the fact that the
$v$-height of $w_{k}z_{k}$ in $k(1)^{*}(K_{2})$ is $k$ less than that of
$w_{k}$, using Theorem 3.16 and (3.11). The other is that $K_{k}^{B}$ has
additional ${\mathbb{Z}}_{p}$’s

$v^{p^{t}-t-1}(y_{t}\cdots y_{k-1})^{p-1}z_{t}\text{ for }1\leq t\leq k-1,$
(5.9)

as seen in Figure 1.7 when $k=3$, but these are always multiplied by higher
$z$’s, and so (5.8) applies.

The isomorphisms $C_{k}^{A}\to G_{k}^{2}$ and $C_{k}^{B}z_{\ell}\to
G_{k,\ell}^{6}$ are defined simply by sending an element to one with the same
name. Moreover $C_{k}^{A}=C_{k}^{B}$ except for $(y_{0}\cdots
y_{k-1})^{p-1}z_{0}\in C_{k}^{A}-C_{k}^{B}$. When $k=3$, we see that the
${\mathbb{Z}}_{p}$’s in $C_{k}^{B}$ are
$\\{z_{1}^{p}z_{2}^{p-1},\,z_{2}^{p},\,y_{2}^{p-1}z_{1}^{p}\\}$ in Figure
1.7.333The class $y_{2}^{p-1}z_{1}^{p-1}$ should really be called
$y_{2}^{p-1}z_{1}^{p-1}+v^{p^{2}(p-1)-1}z_{3}$ so that $v$ times it is
divisible by $p$, hence 0 in $C_{k}^{B}$, but we will ignore this fine-tuning.
For future reference,

$\text{${\mathbb{Z}}_{p}$'s in }C_{k}^{B}\text{
are}\bigl{\\{}z_{t}^{p}\prod_{i=t+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\bigr{\\}}:\,1\leq
t<k\\}.$ (5.10)

The corresponding elements in $k(1)^{*}(K_{2})$ are from the third line of
3.16.

The $v$-towers in $C_{k}^{A}=C_{k}^{B}$ are generated by

$z_{k}\text{ and
}y_{t}^{p-1}z_{t}\prod_{i=t+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\},\ 1\leq
t<k.$ (5.11)

We will show that the $v$-height of $z_{k}$ in $C_{k}^{B}$ is
$r^{\prime}(k-1)$, which equals its $v$-height in $k(1)^{*}(K_{2})$. It
follows from 1.1 that the $v$-height of
$y_{t}^{p-1}z_{t}\prod_{i=t+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}$ equals
$r^{\prime}(t-1)$, establishing the isomorphisms out of $C_{k}^{A}$ and
$C_{k}^{B}z_{\ell}$. In Figure 1.7, the $v$-height of $z_{3}$ equalling
$p^{3}-p^{2}+p-2=r^{\prime}(2)$ is apparent.

The proof of the claim about $v$-heights is by induction. By (3.10),
$r^{\prime}(k-1)-r^{\prime}(k-3)=p^{k-1}(p-1)-1$. Let
$D=(|z_{k}|-|y_{k-1}^{p-1}z_{k-1}|)/(2(p-1))=p^{k-1}(p-1)$. This is the
filtration on the $z_{k}$-tower above the element $y_{k-1}^{p-1}z_{k-1}$. We
show that $v^{i-1+D}z_{k}$ is divisible by $p$ if and only if $v^{i}z_{k-2}$
is divisible by $p$. Thus the difference of the $v$-heights in cokernels
equals the difference of the corresponding $r^{\prime}$ values. From Theorem
4.1, we have

$pv^{i-1}y_{k-1}^{p-1}z_{k-1}=v^{i-1+D}z_{k}+v^{i}y_{k-1}^{p-1}z_{k-2}^{p}.$

The claim follows, since $v^{i}y_{k-1}^{p-1}z_{k-2}^{p}$ is divisible by $p$
if and only if $v^{i}z_{k-2}$ is, by 1.1.

The analysis of (5.3) is extremely similar.

Now $S_{k,\ell}$ becomes involved. Let
$S_{k,\ell}^{K}=\operatorname{ker}(\cdot p|S_{k,\ell})$ and
$S_{k,\ell}^{C}=\operatorname{coker}(\cdot p|S_{k,\ell})$. Then
$S_{k,\ell}^{K}$ consists of $TP_{k+1}[v]\langle z_{1,\ell}\rangle$ plus
${\mathbb{Z}}_{p}$’s on $v^{k}z_{i,\ell}$ for $2\leq i\leq\ell-k$, while
$S_{k,\ell}^{C}$ has $TP_{k+1}[v]\langle z_{\ell-k,\ell}\rangle$ plus
${\mathbb{Z}}_{p}$’s on $z_{i,\ell}$ for $1\leq i<\ell-k$. Next we consider
the short exact sequence

$0\to y_{k}C_{k}^{B}Z_{k}^{\ell}\
\smash{\mathop{\longrightarrow}\limits^{\phi}}\ G_{k,\ell}^{4}\
\smash{\mathop{\longrightarrow}\limits^{\psi}}\
y_{1}^{p^{k-1}-1}qS_{k,\ell}^{K}\to 0.$ (5.12)

The map $\phi$ sends everything except the $v$-tower on
$y_{k}z_{k}Z_{k}^{\ell}$ to classes with the same name, and the heights of
these $v$-towers agree, as seen above. The class
$y_{k}z_{k}Z_{k}^{\ell}=y_{k}z_{k,\ell}$ maps to a ${\mathbb{Z}}_{p}$ with the
same name in $k(1)^{*}(K_{2})$. We have
$\psi(w_{k}w_{k+1}Z_{k+1}^{\ell})=qy_{1}^{p^{k-1}-1}z_{1,\ell}$. Then
$v^{k+1}w_{k}w_{k+1}Z_{k+1}^{\ell}\in\operatorname{ker}(\psi)$, and we have

$\phi(vy_{k}z_{k,\ell})=v^{k+1}w_{k}w_{k+1}Z_{k+1}^{\ell}.$

We illustrate this in the schematic Figure 5.13, in which $X$, $\circ$, and
$\bullet$ map to elements with the same symbol. The expressions at the end of
the $v$-towers are their $v$-heights. In particular,
$v^{r^{\prime}(k-1)}y_{k}z_{k,\ell}=0$ in $y_{k}C_{k}^{B}Z_{k}^{\ell}$. The
$v$-heights agree by (3.11), and the gradings match by an induction proof. The
${\mathbb{Z}}_{p}$’s in $y_{1}^{p^{k-1}-1}qS^{K}_{k,\ell}$ are hit by
$\psi(y_{k}z_{i+k-1,\ell})=y_{1}^{p^{k-1}-1}qv^{k}z_{i,\ell}$, $2\leq
i\leq\ell-k$, another interesting filtration jump.

######  Figure 5.13.

Towers in exact sequence.

$\circ$$r^{\prime}(k-1)$$\circ$ $w_{k}w_{k+1}Z_{k+1}^{\ell}$
$y_{1}^{p^{k-1}-1}qz_{1,\ell}$$k+1$$k+1$$r(k)$$X$$X$$y_{k}C_{k}^{B}Z_{k}^{\ell}$$G_{k,\ell}^{4}$$y_{1}^{p^{k-1}-1}qS_{k,\ell}^{K}$$\
\smash{\mathop{\longrightarrow}\limits^{\phi}}\ $$\
\smash{\mathop{\longrightarrow}\limits^{\psi}}\ $$y_{k}z_{k,\ell}$

Finally we consider the short exact sequence

$0\to y_{1}^{p^{k-1}-1}qS^{C}_{k,\ell}\
\smash{\mathop{\longrightarrow}\limits^{\phi^{\prime}}}\ G^{5}_{k,\ell}\
\smash{\mathop{\longrightarrow}\limits^{\psi^{\prime}}}\ K_{k}^{B}z_{\ell}\to
0.$ (5.14)

Similarly to (5.5), the generators of $v$-towers in $K_{k}^{B}$ are $g_{k}$
and, for $1\leq j<k$, elements of the form
$g_{j}z_{j}^{p-1}\prod_{j+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}$. The morphism
$\psi^{\prime}$ is determined by $w_{j}\mapsto g_{j}$. The $v$-heights of the
corresponding elements in $k(1)^{*}(K_{2})$ and $K_{k}^{B}$ both equal
$r^{\prime}(j-1)$ for $j<k$. However, the $v$-height of $w_{k}z_{\ell}$ is
$r(k)$, which is $k$ greater than $r^{\prime}(k-1)$. We have
$\phi^{\prime}(vy_{1}^{p^{k-1}-1}qz_{\ell-k,\ell})=v^{r^{\prime}(k-1)}w_{k}z_{\ell}$.
The class $y_{1}^{p^{k-1}-1}qz_{\ell-k,\ell}$ at the base of the $v$-tower
maps to a ${\mathbb{Z}}_{p}$ with the same name. The picture is quite similar
to Figure 5.13 with $k+1$ and $r^{\prime}(k-1)$ interchanged.

The ${\mathbb{Z}}_{p}$ classes $y_{1}^{p^{k-1}-1}qz_{i,\ell}$ for $1\leq
i<\ell-k$ are mapped by $\phi^{\prime}$ to classes with the same name in
$G_{k,\ell}^{5}\subset k(1)^{*}(K_{2})$. The ${\mathbb{Z}}_{p}$’s in
$K_{k}^{B}z_{\ell}$ are of the same form as in (5.7), and are hit by analogues
of (5.8).

##  6\. All accounted for

In this section, we show that all elements of $k(1)^{*}(K_{2})$ are involved
in exactly one of the homomorphisms involving some $G$-group described in the
preceding section. As discussed earlier, this implies that there can be no
exotic extensions in $ku^{*}(K_{2})$ other than those in (1.2), because an
additional extension would decrease the number of elements in
$\operatorname{ker}(\cdot p|ku^{*}(K_{2}))$ and $\operatorname{coker}(\cdot
p|ku^{*}(K_{2}))$, and these must correspond to elements of $k(1)^{*}(K_{2})$.
It also provides an excellent check on our analysis.

Let $p$ be odd, and

$G^{i}=\begin{cases}\displaystyle\bigoplus_{k\geq 1}G_{k}^{i}\otimes
TP_{p-1}[y_{k}]\otimes P[y_{k+1}]&1\leq i\leq 2\\\
\displaystyle\bigoplus_{1\leq k<\ell}G_{k,\ell}^{i}\otimes
TP_{p-1}[y_{k}]\otimes P[y_{k+1}]\otimes
TP_{p-1}[z_{\ell}]\otimes\Lambda_{\ell+1}&3\leq i\leq 6\\\
\displaystyle\bigoplus_{k\geq 1}\bigoplus_{e=1}^{p-2}G_{k,e}^{i}\otimes
P[y_{k}]\otimes\Lambda_{k+1}&7\leq i\leq 8.\end{cases}$

######  Theorem 6.1.

$G^{1}\oplus\cdots\oplus G^{8}$ equals $k(1)^{*}(K_{2})$, as described in
Theorem 3.16.

As throughout the paper, ${\mathbb{Z}}_{p}$’s coming from $E_{1}$-free
submodules of $H^{*}(K_{2})$ are ignored here. The remainder of this section
is devoted to the proof of Theorem 6.1. There are four parts of Theorem 3.16.
We deal with them one-at-a-time.

Case 1\. $P[y_{1}]y_{0}^{p-1}z_{0}$. In (3.28), it is shown that these classes
form a subset of $\bigoplus\mathcal{M}_{k}^{A}A_{k}$, and they map to classes
with the same name in $G^{2}$.

Case 2\. $\bigoplus_{j>0}TP_{r(j)}[v]\otimes P[y_{j+1}]\otimes
TP_{p-1}[y_{j}]\otimes\overline{E}[w_{j}]\otimes
E[w_{j+1}]\otimes\Lambda_{j+1}$. The generators of $v$-towers of height $r(j)$
occur in $G^{1}$, $G^{4}$, and $G^{5}$. From (5.6), only $w_{j}$ is in
$G_{j}^{1}$. So $G^{1}$ has $TP_{p-1}[y_{j}]\otimes P[y_{j+1}]w_{j}$. From
Figure 5.13, $G^{4}_{j,\ell}$ has $w_{j}w_{j+1}Z_{j+1}^{\ell}$. Note that
$\bigoplus_{\ell}Z_{j+1}^{\ell}TP_{p-1}[z_{\ell}]\otimes\Lambda_{\ell+1}=\Lambda_{j+1}$,
since the $\ell$-component gives the monomials whose smallest
non-$(p-1)$-power is a power of $z_{\ell}$, so $G^{4}$ contains
$P[y_{j+1}]\otimes TP_{p-1}[y_{j}]w_{j}w_{j+1}\otimes\Lambda_{j+1}$. From the
analysis following (5.14), $G^{5}_{j,\ell}$ has only $w_{j}z_{\ell}$ of
$v$-height $r(j)$, so $G^{5}$ will have $P[y_{j+1}]\otimes
TP_{p-1}[y_{j}]w_{j}\otimes\overline{\Lambda}_{j+1}$. Thus $G^{1}\oplus G^{5}$
contains the part without $w_{j+1}$, while $G^{4}$ contains the part with
$w_{j+1}$.

Case 3\. $\bigoplus_{j\geq 1}TP_{r^{\prime}(j-1)}[v]\otimes P[y_{j}]\otimes
E[w_{j}]\otimes\overline{TP}_{p}[z_{j}]\otimes\Lambda_{j+1}$. The generators
of $v$-towers of height $r^{\prime}(j-1)$ occur in each $G^{i}$ as follows.

  * $G^{1}$:

$\displaystyle{w_{j}z_{j}^{p-1}\bigoplus_{k\geq j+1}TP_{p-1}[y_{k}]\otimes
P[y_{k+1}]\otimes\bigoplus_{i=j+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}}$. This
can be deduced from (5.6).

  * $G^{2}$:

From (5.11),

$z_{j}TP_{p-1}[y_{j}]\otimes P[y_{j+1}]\oplus y_{j}^{p-1}z_{j}\bigoplus_{k\geq
j+1}TP_{p-1}[y_{k}]\otimes
P[y_{k+1}]\otimes\prod_{i=j+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}.$

  * $G^{3}$:

We use (5.5) and (5.6) and adapt some arguments used in Case 2 to obtain

$w_{j}z_{j}^{p-1}\biggl{(}\overline{TP}_{p}[y_{j}]\otimes
P[y_{j+1}]\otimes\Lambda_{j+1}\oplus\bigoplus_{k\geq
j+1}\overline{TP}_{p}[y_{k}]P[y_{k+1}]z_{k}^{p-1}\Lambda_{k+1}\prod_{i=j+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}\biggr{)}.$

  * $G^{4}$:

We use (5.11) and (5.12) to obtain

$y_{j}^{p-1}z_{j}\bigoplus_{k\geq j+1}\overline{TP}_{p}[y_{k}]\otimes
P[y_{k+1}]z_{k}^{p-1}\Lambda_{k+1}\prod_{i=j+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}.$

  * $G^{5}$:

We use (5.14) and
$\bigoplus_{\ell>k}z_{\ell}TP_{p-1}[z_{\ell}]\otimes\Lambda_{\ell+1}\approx\overline{\Lambda}_{k+1}$
to obtain

$w_{j}z_{j}^{p-1}\bigoplus_{k\geq j+1}TP_{p-1}[y_{k}]\otimes
P[y_{k+1}]\otimes\overline{\Lambda}_{k+1}\otimes\prod_{i=j+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}.$

  * $G^{6}$:

We combine the analysis for $G^{2}$ and the observation used for $G^{5}$ to
obtain

$\displaystyle z_{j}TP_{p-1}[y_{j}]\otimes
P[y_{j+1}]\otimes\overline{\Lambda}_{j+1}$ $\displaystyle\oplus$
$\displaystyle y_{j}^{p-1}z_{j}\bigoplus_{k\geq j+1}TP_{p-1}[y_{k}]\otimes
P[y_{k+1}]\otimes\overline{\Lambda}_{k+1}\otimes\prod_{i=j+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}$

  * $G^{7}$:

Similarly to $G^{3}$, we have

$\bigoplus_{e=1}^{p-2}\biggl{(}w_{j}z_{j}^{e}\otimes
P[y_{j}]\otimes\Lambda_{j+1}\oplus w_{j}z_{j}^{p-1}\bigoplus_{k\geq
j+1}z_{k}^{e}\otimes
P[y_{k}]\otimes\Lambda_{k+1}\otimes\prod_{i=j+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}\biggr{)}.$

  * $G^{8}$:

Using (5.11), we get

$\bigoplus_{e=1}^{p-2}\biggl{(}z_{j}^{e}\otimes
P[y_{j}]\otimes\Lambda_{j+1}\oplus y_{j}^{p-1}z_{j}\bigoplus_{k\geq
j+1}z_{k}^{e}\otimes
P[y_{k}]\otimes\Lambda_{k+1}\otimes\prod_{i=j+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}\biggr{)}.$

We begin by analyzing the portion including the factor $w_{j}$. We will show
that

$G^{1}\oplus G^{3}\oplus G^{5}\oplus
G^{7}=P[y_{j}]w_{j}\otimes\overline{TP}_{p}[z_{j}]\otimes\Lambda_{j+1}.$

Here, and in the remainder of our analysis of Case 3, $G^{i}$ refers just to
the relevant portion of $G^{i}$, here the part with
$TP_{r^{\prime}(j-1)}[v]w_{j}$. The first part of $G^{7}$ gives all terms with
$z_{j}^{e}$ for $1\leq e\leq p-2$. The remaining part has factors
$w_{j}z_{j}^{p-1}$, which we will omit writing. Combining $G^{1}$ and $G^{5}$
removes the bar in $G^{5}$. The first part of $G^{3}$ gives the part with
positive exponent of $y_{j}$, which we now omit.

Let $E_{\ell}=P[y_{\ell}]\otimes\Lambda_{\ell}$, thought of as monomials in
$y_{i}$ and $z_{i}$ for $i\geq\ell$ with exponents $\leq p-1$. The remaining
parts of the $G^{i}$’s under consideration combine to

$\bigoplus_{k\geq j+1}\biggl{(}TP_{p-1}[y_{k}]\oplus
y_{k}z_{k}^{p-1}TP_{p-1}[y_{k}]\oplus\bigoplus_{e=1}^{p-2}z_{k}^{e}TP_{p}[y_{k}]\biggr{)}\otimes
E_{k+1}\otimes\prod_{i=j+1}^{k-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}.$ (6.2)

We wish to show this equals $E_{j+1}$. The portion in parentheses is all
monomials in $TP_{p}[y_{k},z_{k}]$ except $y_{k}^{p-1}$ and $z_{k}^{p-1}$. For
a monomial $M$ in $E_{j+1}$, let $M_{i}$ denote its $y_{i}^{s}z_{i}^{t}$
factor. The $k$-summand in (6.2) is all monomials $M$ in $E_{j+1}$ for which
$k$ is the smallest $i$ such that $M_{i}$ is neither $y_{i}^{p-1}$ nor
$z_{i}^{p-1}$. Thus the sum over all $k$ yields all of $E_{j+1}$, as claimed.

A very similar argument shows that the $G^{2}\oplus G^{4}\oplus G^{6}\oplus
G^{8}$ part for Case 3 equals the portion which includes just the 1 in
$E[w_{j}]$; i.e.,
$P[y_{j}]\otimes\overline{TP}_{p}[z_{j}]\otimes\Lambda_{j+1}$.

Case 4\. $\bigoplus_{j\geq 1}P[y_{1}]\otimes
E[q]\otimes\overline{E}[z_{j}^{p}]\otimes\Lambda_{j+1}$. We first consider the
part without the $q$, and fix $j$ and omit writing the $z_{j}^{p}$. The
desired answer is $P[y_{1}]\otimes\Lambda_{j+1}$. These come from the
${\mathbb{Z}}_{p}$’s in $G^{2}\oplus G^{4}\oplus G^{6}\oplus G^{8}$. Similarly
to Case 3, $G^{2}$ and $G^{6}$ combine to give

$\bigoplus_{k\geq j+1}TP_{p-1}[y_{k}]\otimes
P[y_{k+1}]\otimes\Lambda_{k+1}\otimes\prod_{i=k+1}^{j-1}\\{z_{i}^{p-1},y_{i}^{p-1}\\}.$

This, together with the portion of $G^{4}$ from $\operatorname{im}(\phi)$ in
(5.12) obtained using (5.10), and the ${\mathbb{Z}}_{p}$’s in $G^{8}$ obtained
using (5.10) give exactly (6.2), which we showed equals
$P[y_{j+1}]\otimes\Lambda_{j+1}$.444Here the classes in (6.2) are
${\mathbb{Z}}_{p}$’s and are multiplied by $z_{j}^{p}$, whereas in Case 3 they
were multiplied by $w_{j}z_{j}^{p-1}$ and were generators of $v$-towers of
height $r^{\prime}(j-1)$. The element $X$ in Figure 5.13 with $k$ replaced by
$j$ yields, from $G^{4}$,

$\displaystyle y_{j}TP_{p-1}[y_{j}]\otimes
P[y_{j+1}]\otimes\bigoplus_{\ell>j}Z_{j+1}^{\ell}TP_{p-1}[z_{\ell}]\otimes\Lambda_{\ell+1}$
$\displaystyle=$ $\displaystyle y_{j}TP_{p-1}[y_{j}]\otimes
P[y_{j+1}]\otimes\Lambda_{j+1},$

which combines with the portion just obtained to yield
$P[y_{j}]\otimes\Lambda_{j+1}$.

The last line of the $G^{4}_{k,\ell}$ discussion in Section 5 describes
${\mathbb{Z}}_{p}$’s in $G^{4}$ mapped by $\psi$ in (5.12). Those with a
$z_{j}^{p}$ factor yield

$\displaystyle\bigoplus_{k=1}^{j-1}y_{k}TP_{p-1}[y_{k}]P[y_{k+1}]\bigoplus_{\ell>j}Z_{j+1}^{\ell}TP_{p-1}[z_{\ell}]\Lambda_{\ell+1}$
$\displaystyle=$
$\displaystyle\bigoplus_{k=1}^{j-1}(P[y_{k}]-P[y_{k+1}])\otimes\Lambda_{j+1}$
$\displaystyle=$ $\displaystyle(P[y_{1}]-P[y_{j}])\otimes\Lambda_{j+1}.$

Combining this with the result of the preceding paragraph yields the desired
$P[y_{1}]\otimes\Lambda_{j+1}$.

We finish this section by showing that the ${\mathbb{Z}}_{p}$’s including a
factor $q$ are obtained exactly once. We omit writing the $q$. The classes
which we must obtain are $P[y_{1}]\bigoplus_{j\geq 1}z_{j}^{p}\Lambda_{j+1}$.
There are eight ways these appear in $G^{i}$-sets.

  1. (1)

In $G^{1}$, using (5.7) and (5.8), for $1\leq i<j<k$,

$y_{1}^{p^{j-1}-1}z_{i,j}z_{j}^{p-2}\prod_{s=j+1}^{k-1}\\{z_{s}^{p-1},y_{s}^{p-1}\\}\otimes
TP_{p-1}[y_{k}]\otimes P[y_{k+1}].$

  2. (2)

In $G^{3}$, using (5.9) and (5.8), for $1\leq i<k<\ell$,

$y_{1}^{p^{k-1}-1}y_{k}z_{i,k}z_{k}^{p-2}Z_{k+1}^{\ell}\otimes
TP_{p-1}[y_{k}]\otimes P[y_{k+1}]\otimes
TP_{p-1}[z_{\ell}]\otimes\Lambda_{\ell+1}.$

  3. (3)

In $G^{3}$, using (5.7) and (5.8), for $1\leq i<j<k<\ell$,

$y_{1}^{p^{j-1}-1}y_{k}z_{i,j}z_{j}^{p-2}\prod_{s=j+1}^{k-1}\\{z_{s}^{p-1},y_{s}^{p-1}\\}Z_{k}^{\ell}\otimes
TP_{p-1}[y_{k}]\otimes P[y_{k+1}]\otimes
TP_{p-1}[z_{\ell}]\otimes\Lambda_{\ell+1}.$

  4. (4)

From $\operatorname{im}(\phi^{\prime})$ in (5.14), for $1\leq k<\ell$ and
$1\leq i\leq\ell-k$,

$y_{1}^{p^{k-1}-1}z_{i,\ell}\otimes TP_{p-1}[y_{k}]\otimes P[y_{k+1}]\otimes
TP_{p-1}[z_{\ell}]\otimes\Lambda_{\ell+1}.$

  5. (5)

From $\psi^{\prime}$ in (5.14), using (5.9) and (5.8), for $k<\ell$ and
$\ell-k<i<\ell$,

$y_{1}^{p^{k-1}-1}z_{i,\ell}\otimes TP_{p-1}[y_{k}]\otimes P[y_{k+1}]\otimes
TP_{p-1}[z_{\ell}]\otimes\Lambda_{\ell+1}.$

  6. (6)

From $\psi^{\prime}$ in (5.14), using (5.7) and (5.8), for $i<j<k<\ell$,

$y_{1}^{p^{j-1}-1}z_{i,j}z_{j}^{p-2}\prod_{s=j+1}^{k-1}\\{z_{s}^{p-1},y_{s}^{p-1}\\}\cdot
z_{\ell}\otimes TP_{p-1}[y_{k}]\otimes P[y_{k+1}]\otimes
TP_{p-1}[z_{\ell}]\otimes\Lambda_{\ell+1}.$

  7. (7)

From (5.3), using (5.9) and (5.8), for $i<k$ and $1\leq e\leq p-2$,

$y_{1}^{p^{k-1}-1}z_{i,k}z_{k}^{e-1}P[y_{k}]\otimes\Lambda_{k+1}.$

  8. (8)

From (5.3), using (5.7) and (5.8), for $i<j<k$ and $1\leq e\leq p-2$,

$y_{1}^{p^{j-1}-1}z_{i,j}z_{j}^{p-2}\prod_{s=j+1}^{k-1}\\{z_{s}^{p-1},y_{s}^{p-1}\\}\cdot
z_{k}^{e}P[y_{k}]\otimes\Lambda_{k+1}.$

First combine (1)+(6) to put a $\otimes\Lambda_{k+1}$ at the end of (1), and
then, similarly to the simplification of (6.2), combine with (3)+(8) to get

$\bigoplus_{i<j}y_{1}^{p^{j-1}-1}P[y_{j+1}]z_{i,j}z_{j}^{p-2}\Lambda_{j+1}.$
(6.3)

We combine and relabel (4)+(5) to give

$\bigoplus_{i<j}y_{1}^{p^{j-1}-1}TP_{p-1}[y_{j}]P[y_{j+1}]z_{i,j+1}\Lambda_{j+1}$
(6.4)

together with

$\bigoplus_{i\geq j\geq
1}y_{1}^{p^{j-1}-1}TP_{p-1}[y_{j}]P[y_{j+1}]z_{i}^{p}\Lambda_{i+1}.$ (6.5)

Let $Y(s)=y_{1}^{p^{s}-1}TP_{p-1}[y_{s+1}]P[y_{s+2}]=\langle
y_{1}^{i}:\nu(i+1)=s\rangle$. Then (6.5) is

$\bigoplus_{i>s\geq 0}Y(s)z_{i}^{p}\Lambda_{i+1}.$ (6.6)

We simplify and relabel (2) to

$\bigoplus_{i<j}y_{1}^{p^{j-1}-1}y_{j}TP_{p-1}[y_{j}]P[y_{j+1}]z_{i,j}z_{j}^{p-2}\Lambda_{j+1}.$
(6.7)

(6.3), (6.7), and (7) combine to give

$\bigoplus_{i<j}y_{1}^{p^{j-1}-1}P[y_{j}]z_{i,j}TP_{p-1}[z_{j}]\Lambda_{j+1}=\bigoplus_{i\leq
j-1\leq t}Y(t)z_{i,j}TP_{p-1}[z_{j}]\Lambda_{j+1}.$

For any $t\geq i$, the coefficient of $Y(t)z_{i}^{p}$ in (6.4) plus this is

$Z_{i+1}^{t+2}\Lambda_{t+2}\oplus\bigoplus_{j=i+1}^{t+1}Z_{i+1}^{j}TP_{p-1}[z_{j}]\Lambda_{j+1}=\Lambda_{i+1},$

as the second part has all monomials not divisible by $Z_{i+1}^{t+2}$.
Combining this with (6.6) yields the desired result,

$\bigoplus_{s\geq 0}Y(s)\bigoplus_{i\geq 1}z_{i}^{p}\Lambda_{i+1}.$

##  7\. An explanation of self-duality of $B_{k}$

In this optional section, we discuss some observations about the ASS of
$ku^{*}(K_{2})$ and $ku_{*}(K_{2})$ which, among other things, provide an
explanation of the self-dual nature of the $B_{k}$ summands which occur in
both $ku^{*}(K_{2})$ and $ku_{*}(K_{2})$. We restrict to $p=2$.

We first observe that, for $k\geq 1$, there is an $E_{1}$-submodule,
$\mathcal{M}_{k}$, of $H^{*}(K_{2})$ such that
$\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{2},\mathcal{M}_{k})$ (resp.
$\operatorname{Ext}_{E_{1}}(\mathcal{M}_{k},{\mathbb{Z}}_{2})$) is closed
under the differentials in the ASS converging to $ku^{*}(K_{2})$ (resp.
$ku_{*}(K_{2})$), yielding the chart $A_{k}$ (resp. the $ku$-homology analogue
of $A_{k}$ discussed in Theorem 1.19). For example, with $M_{j}$ as in (2.13)
and $N$ as in Figure 2.9, $\mathcal{M}_{3}$ is as depicted in Figure 7.1.

######  Figure 7.1.

The $E_{1}$-module $\mathcal{M}_{3}$.

$17$$26$$33$$30$$36$$22$
$y_{1}^{4}$$y_{1}^{3}N$$y_{1}^{2}M_{4}$$y_{1}x_{9}M_{4}$$M_{5}$

The two ASSs for $\mathcal{M}_{3}$ will yield the charts for $A_{3}$ and its
homology analogue pictured in [4].

The situation for $B_{k}$ is slightly more complicated. There is no
$E_{1}$-submodule of $H^{*}(K_{2})$ which, by itself, can give a chart
$B_{k}z_{\ell}$. Some of the differentials that truncate $v$-towers in
$B_{k}z_{\ell}$ come from classes that are part of a summand that includes
$y_{1}^{2^{k-1}-1}qS_{k,\ell}$. We find that, for $2\leq k<\ell$, there is an
$E_{1}$-submodule $\mathcal{M}_{k,\ell}$ of $H^{*}K_{2}$ such that
$\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{2},\mathcal{M}_{k,\ell})$ is closed
under the differentials in the ASS converging to $ku^{*}(K_{2})$ and yields
the chart

$B_{k}z_{\ell}\oplus y_{1}^{2^{k-1}-1}qS_{k,\ell}\oplus
y_{k}B_{k}Z_{k}^{\ell}.$

Note that these three subsets of $ku^{*}(K_{2})$ appeared together in the
10-term exact sequence (5.2).

This $\mathcal{M}_{k,\ell}$ is symmetric; i.e., there is an integer $D$ such
that $\Sigma^{D}\mathcal{M}_{k,\ell}^{*}$ and $\mathcal{M}_{k,\ell}$ are
isomorphic $E_{1}$-modules, where $\mathcal{M}_{k,\ell}^{*}$ is obtained from
$\mathcal{M}_{k,\ell}$ by negating gradings and dualizing $Q_{0}$ and $Q_{1}$.
This implies that the $v$-towers in
$\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{2},\mathcal{M}_{k,\ell})$ and
$\operatorname{Ext}_{E_{1}}(\mathcal{M}_{k,\ell},{\mathbb{Z}}_{2})$ correspond
nicely. Moreover, the differentials in the two ASSs correspond, too, obtaining
isomorphic charts, although the gradings in one decrease from left to right,
while in the other they increase.

We illustrate with an example, $\mathcal{M}_{3,4}$, and then discuss the
implication for self-duality of $B_{k}$. In Figure 7.2, we depict
$\mathcal{M}_{3,4}$.

######  Figure 7.2.

The $E_{1}$-module $\mathcal{M}_{3,4}$.

$70$$75$$80$$96$$102$$91$ $86$
$y_{1}^{7}x_{9}M_{5}$$y_{1}^{6}z_{3}M_{4}$$y_{1}^{5}x_{9}z_{3}M_{4}$$y_{1}^{4}M_{6}$$y_{1}^{3}x_{9}M_{6}$$y_{1}^{2}z_{4}M_{4}$$y_{1}x_{9}z_{4}M_{4}$$z_{4}M_{5}$

In Figure 7.3, we depict the ASS chart for both
$\operatorname{Ext}_{E_{1}}({\mathbb{Z}}_{2},\mathcal{M}_{3,4})$ and
$\operatorname{Ext}_{E_{1}}(\mathcal{M}_{3,4},{\mathbb{Z}}_{2})$. They are
isomorphic except that, from left to right, the gradings start with 102 for
the first and 70 for the second. We label the portions of the chart
corresponding to the eight summands of $\mathcal{M}_{3,4}$ just by the
$M$-factor, since accompanying factors differ for the two versions. For
example, the $M_{5}$ on the left-hand side is $z_{4}M_{5}$ for the first
spectral sequence, and is $y_{1}^{7}x_{9}M_{5}$ for the second.

######  Figure 7.3.

Two ASSs for $\mathcal{M}_{2,3}$.

$102$$70$$M_{5}$$M_{4}$$92$$80$$M_{4}$$M_{6}$$86$$86$$M_{6}$$M_{4}$$76$$96$$M_{4}$$M_{5}$

For the $ku^{*}(K_{2})$ version, $B_{3}z_{4}$ is on the left hand side of
Figure 7.3, and $y_{3}B_{3}z_{3}$ on the right hand side, with
$y_{1}^{3}qS_{3,4}$ separating them. The duality isomorphism in Theorem 1.16
says that the Pontryagin dual of $B_{3}z_{4}$ is isomorphic as a
$ku_{*}$-module to $\Sigma^{4}$ of the right hand side of the $ku_{*}(K_{2})$
version of Figure 7.3, and we see that this is isomorphic to a shifted version
of $B_{3}$ with indices negated. This is the self-duality statement, that the
Pontryagin dual of $B_{k}$ is isomorphic as a $ku_{*}$-module to a shifted
version of $B_{k}$ with indices negated.

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