2. The $E_2$ -term of the ASS for $kup^*(K_2)$

We will need some notation. By $H^*K_2$ , we understand $H^*\left(K\!\left({\mathbb{Z}}_p,2\right);\,{\mathbb{Z}}_p\right)$ . 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 ASS for $kup^*(K_2)$ has $E_2^{s,t} = \operatorname{Ext}_{\mathcal{A}}^{s,t}(H^*(bup),H^*K_2)$ , where $\mathcal{A}$ is the mod $p$ Steenrod algebra and $H^*(bup) \approx \mathcal{A}/\mathcal{A}(Q_0,Q_1)$ . Using a standard change of rings theorem [Reference Liulevicius10], this is $\operatorname{Ext}_{E_1}^{s,t}({\mathbb{Z}}_p,H^*K_2)$ . This converges to $kup^{-(t-s)}(K_2)$ . 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 $v \in kup^{-2(p-1)}$ (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 $p \in kup^0$ , $\big(h_0 \in G_r^{0,1}\big)$ , is $h_0 \,:\, G_r^{a,b} \longrightarrow G_r^{a,b+1}$ .

In the paragraph preceding Remark 2.17, we will define elements $z_j \in G_2^{2\!\left(p^{j+1}+1\right),0}$ for $j \ge 0$ and elements:

\begin{equation*}z_{i,j} \in G_2^{2(p^{j+1}+1+(p-1)(j-i)),0}\end{equation*}

as in (1.12) satisfying the properties in Definition 1.13.

We also have $y_i \in G_2^{2p^i,0}$ for $i\ge 0$ , and

(2.2) \begin{equation} q\in G^{9,0}_2\text{ if }p=2,\text{ and in }G^{4p-1,0}_2\text{ if $p$ is odd.} \end{equation}

Cf. (1.3), (1.2), and (2.2). One last definition, let $\Lambda _{j+1}=TP_p[z_i\,:\,i\ge j+1]$ .

A picture of $P[v]\otimes W_5$ as a $P[v,h_0]$ -module with $p=2$ appears in Figure 5.

Figure 5.

A depiction of $P[v]\otimes W_5$ .

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 $kup^*(K_2)$ is isomorphic as a $P[h_0,v]$ -module to

\begin{equation*} P[v,y_1]\otimes E[q] \otimes \left( \bigoplus _{j\ge k_0}\left(W_j\otimes TP_{p-1}[z_j]\otimes \Lambda _{j+1}\right)\right) \end{equation*}

\begin{equation*} \oplus \bigl (P[h_0,v,y_1]\otimes E\big[ v^{k_0}q\big]\bigr ) \oplus \biggl ( P[y_1]\otimes \begin {cases}\left\langle y_0^{\,p-1}z_0\right\rangle &p\text { odd}\\[5pt] \big\langle y_0z_0,z_1,h_0y_0z_0=vz_1\big\rangle &p=2.\end {cases} \biggr ) \end{equation*}

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

(2.5) \begin{equation} (v^2q)^2 = v^4z_2, \end{equation}

holds when $p=2$ since, as we shall see, in $H^*(K_2)$ , $x_9^2-Q_0x_{17}\in \operatorname{im}(Q_1)$ .

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 ([Reference Serre11]) showed that $H^*K_2$ is a polynomial algebra on classes $u_{2^j+1}$ in degree $2^j+1$ for $j\ge 0$ defined by $u_2=\iota _2$ and $u_{2^{j+1}+1}=\operatorname{Sq}^{2^j}u_{2^j+1}$ for $j\ge 0$ . We easily have

\begin{equation*} 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\ge 2, \end{equation*}

and

\begin{equation*} 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\ge 3. \end{equation*}

Let $x_5=u_5+u_2u_3$ and write $H^*K_2$ as an associated graded object:

\begin{equation*} P\big[u_2^2\big]\otimes E[x_5] \otimes \bigl ( E[u_2] \otimes P[u_3] \bigr ) \otimes _{j\ge 2} \left ( E[u_{2^{j+1}+1}]\otimes P\big[\big(u_{2^j+1}\big)^2\big] \right ) \end{equation*}

From this, we can read off

Lemma 2.6.

\begin{equation*} H_*\left(H^*K_2;\,Q_0\right)=P\big[u_2^2\big]\otimes E[x_5] \end{equation*}

Letting $x_9=u_9+u_3^3$ and $x_{17}=u_{17}+u_2u_5^3$ , we rewrite again as:

\begin{gather*} P\big[u_2^2\big] \otimes TP_4[x_9]\otimes TP_4[x_{17}]\otimes _{j\gt 4} E\!\left[\left(u_{2^j+1}\right)^2\right]\\ \otimes \bigl ( E[u_2]\otimes P[u_5]\bigr ) \otimes \left( E[u_3] \otimes P\!\left[u_3^2\right]\right) \otimes _{j\gt 4} \left(E\!\left[u_{2^j+1}\right] \otimes P\!\left[\left(u_{2^{j-2}+1}\right)^4\right]\right). \end{gather*}

Again we read off

Lemma 2.7.

\begin{equation*} H_*(H^*K_2;\,Q_1) = P\big[u_2^2\big] \otimes TP_4[x_9] \otimes TP_4[x_{17} ] \otimes _{j\gt 4} E\!\left[\left(u_{2^j+1}\right)^2\right] \end{equation*}

An associated graded version of this is

Lemma 2.8.

\begin{equation*} H_*\left(H^*K_2;\,Q_1\right) = P\big[u_2^2\big] \otimes E[x_9] \otimes E[x_{17} ] \otimes _{j\gt 2} E\!\left[\left(u_{2^j+1}\right)^2\right] \end{equation*}

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 $x_5=u_5 + u_2 u_3$ , $x_7=u_2u_5$ , and $x_9= u_9 + u_3^3$ , satisfying $Q_0x_7=Q_1x_5$ and $Q_0x_9=Q_1x_7=x_{10}$ . 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 6. In pictures such as this, straight lines indicate $Q_0=\operatorname{Sq}^1$ and curved lines $Q_1$ .

Figure 6.

An $E_1$ -module $N$ .

The $E_1$ -submodule $P\big[u_2^2\big]\oplus P\big[u_2^2\big]\otimes N$ 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,

(2.10) \begin{equation} P\big[u_2^2\big]\otimes E[x_9]\otimes \overline{E}\!\left[x_{17},u_{2^j+1}^2,\ j \gt 2\right]. \end{equation}

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

\begin{equation*} P\big[u_2^2\big]\otimes \bigl (\langle u_2^2\rangle \oplus N\oplus R\oplus S\bigr ) \end{equation*}

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 $L_k$ , $k \ge 0$ , which have generators $g_{2i}$ , $0\le i\le k$ , with $Q_1g_{2i}=Q_0g_{2i+2}$ for $0\le i\lt k$ , $Q_0g_0\ne 0$ , and $Q_1g_{2k}=0$ . For example, $L_3$ is depicted in Figure 7.

Figure 7.

The $E_1$ -module $L_3$ .

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:

\begin{equation*} 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\le i\le k-2,\end{equation*}

$x_9 g_{2k-2} \mapsto x_9\otimes g_{2k-2}+x_7\otimes g_{2k}$ , and $x_9 \otimes g_{2k} \mapsto x_9\otimes g_{2k}$ .

The $E_1$ -module $M_j$ . Let

\begin{equation*} x_{2^j+1}=u_{2^j+1}+ \begin {cases}u_2u_5^3&j=4\\[4pt] u_2u_3u_5^2u_9^2&j=5\\[4pt] u_3u_5^2u_9^2u_{17}^2&j=6\\[4pt] 0&j\gt 6 \end {cases}\quad \text{and}\ w_{2^j-1}= \begin {cases}u_2u_3u_5^2&j=4\\[4pt] u_3u_5^2u_9^2&j=5\\[4pt] 0&j\gt 5. \end {cases}\end{equation*}

Then $Q_0x_{2^j+1}=u_{2^{j-1}+1}^2+Q_1w_{2^j-1}$ , so $Q_0x_{2^j+1}$ 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 $j\ge 4$ there is a short exact sequence of $E_1$ -modules:

(2.13) \begin{equation} 0\to u_{2^{j-2}+1}^2M_{j-1}\to M_j\to M^{\prime}_j\to 0, \end{equation}

where $M^{\prime}_j=\langle x_{2^j+1},Q_0x_{2^j+1}\rangle$ and $Q_1x_{2^j+1}=u_{2^{j-2}+1}^2Q_0x_{2^{j-1}+1}$ . 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:

(2.14) \begin{equation} \Sigma ^{2^j+1} g_{2i} \mapsto \begin{cases} x_{2^j+1} & i = 0 \\[4pt] u_{2^{j-2}+1}^2 x_{2^{j-1}+1} & i = 1 \\[4pt] u_{2^{j-2}+1}^2 u_{2^{j-3}+1}^2 x_{2^{j-2}+1} & i = 2 \\[4pt] 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 \lt i \le j-4 \\[4pt] \end{cases} \end{equation}

And we have

(2.15) \begin{equation}H_*(M_j;\,Q_1)=\begin{cases}\big\langle u_9^2, u_{17} \big\rangle & j = 4 \\\big\langle u_{17}^2, u_9^2 u_{17} \big\rangle & j = 5 \\\big\langle u_{33}^2, u_{17}^2 u_9^2 u_{17} \big\rangle & j = 6 \\\big\langle u_{2^{j-1}+1}^2,u_{2^{j-2}+1}^2\cdots u_9^2x_{17}\big\rangle & j > 6 \\\end{cases}\end{equation}

The $E_1$ -module $R$ . Let

(2.16) \begin{equation} R=\bigoplus _{j\ge 4}M_j\otimes E\!\left[u_{2^j+1}^2,u_{2^{j+1}+1}^2,\ldots \right]. \end{equation}

Then $H_*(R;\,Q_1)=\overline{E}\!\left[x_{17},u_9^2,u_{17}^2,\ldots \right]$ , 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 [Reference Wall13].

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 $Q_0Q_1$ .

That

\begin{equation*} \operatorname {Ext}_{E_1}^{*,*}\left({\mathbb{Z}}_2,P\!\left[u_2^2\right]\right)=P[v,h_0,y_1] \end{equation*}

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 $v^2x_9$ or $h_0^2x_5$ .

Using Figure 6 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 8 describes this completely.

Figure 8.

The first computation of $\operatorname{Ext}_{E_1}({\mathbb{Z}}_2,N)$ .

Again referring to Figure 6, 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 $x_5 \in G^{5,0}$ and $P[v]$ 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 9 describes the result.

Figure 9.

The second computation of $\operatorname{Ext}_{E_1}({\mathbb{Z}}_2,N)$ .

This concludes the computation of $\operatorname{Ext}$ for $P\big[u_2^2\big]\otimes (\langle u_2^2 \rangle \oplus N)$ of Proposition 2.11. The result is the second line of Theorem 2.4.

We need to compute $\operatorname{Ext}$ for $P\big[u_2^2\big] \otimes (R \oplus S)$ 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 $P\big[u_2^2\big] \otimes R$ and ignore the $E[x_9]$ . Similarly, we can ignore the $P\big[u_2^2\big]$ 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.Footnote 2 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 $0 \le i \le j-4$ . 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 $h_0 z_{k,j-2} = v z_{k-1,j-2}$ . 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 $z_5$ , $z_4^2$ , $z_3^2z_4$ , and $z_2^2z_3z_4$ , which is what we have called $z_{5,5}$ , $z_{4,5}$ , $z_{3,5}$ , and $z_{2,5}$ , as pictured in Figure 5. For future reference, we note that (with $\sim$ meaning homologous)

(2.20) \begin{equation} z_j=Q_0x_{2^{j+2}+1}\sim u_{2^{j+1}+1}^2=Q_0u_{2^{j+2}+1}=Q_0Q_{j+2}\iota _2=Q_{j+2}Q_0\iota _2. \end{equation}

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

\begin{equation*}H^*(K_2)=P[y_0]\otimes P[g_1,g_2,\ldots ]\otimes E[u_0,u_1,\ldots ],\end{equation*}

with $|y_0|=2$ , $|g_j|=2(p^j+1)$ , $|u_i|=2p^i+1$ , $Q_0y_0=u_0$ , $Q_0u_i=g_i$ , $Q_1y_0=u_1$ , $Q_1u_0=g_1$ , and $Q_1u_i=g_{i-1}^p$ , $i\ge 2$ . Let $y_1=y_0^p$ . Then, similarly to the case $p=2$ ,

\begin{equation*}H_*(H^*K_2,Q_0)=P[y_1]\otimes E\!\left[y_0^{\,p-1}u_0\right].\end{equation*}

Let $N=\left\langle y_0^{\,p-1}u_0, q=y_0^{\,p-1}u_1, Q_0q=Q_1\left(y_0^{\,p-1}u_0\right)\right\rangle$ . 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

\begin{equation*}P[y_1]\otimes E[q]\otimes \overline {E[w_1]\otimes TP_p[g_2,g_3,\ldots ]},\end{equation*}

where $w_1=u_2+u_0g_1^{p-1}$ . There are $E_1$ -submodules $M_j$ for $j\ge 2$ , defined inductively by $M_2=\langle w_1,g_2=Q_0w_1\rangle$ , $M^{\prime}_j=\langle u_j,g_j=Q_0u_j \rangle$ for $j\ge 3$ , and for $j\ge 3$ , there exists a short exact sequence of $E_1$ -modules:

\begin{equation*}0\to g_{j-1}^{p-1}M_{j-1}\to M_j\to M^{\prime}_j\to 0,\end{equation*}

with $Q_1u_j=g_{j-1}^p$ . 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 7, but with $i$ th generator ( $i\ge 0$ ) in grading $2(p-1)i$ rather than $2i$ .

Let

\begin{equation*}R=\bigoplus _{j\ge 2}M_j\otimes TP_{p-1}[g_j]\otimes TP_p[g_{j+1},\ldots ].\end{equation*}

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:

(2.21) \begin{equation} H^*K_2\approx P[y_1]\otimes (\langle y_1\rangle \oplus N\oplus R\oplus qR). \end{equation}

Similarly to Figure 9, $\operatorname{Ext}_{E_1}({\mathbb{Z}}_p,N)$ can be read off from Figure 10. 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 5. 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 10.

Computation of $\operatorname{Ext}_{E_1}({\mathbb{Z}}_p,N)$ .

We explain here the reason for the $k_0$ in Definition 1.5. 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 $y_0^{\,p-1}Q_1y_0$ in each. Applying $Q_1$ yields $y_0^{\,p-2}(Q_1y_0)^2$ . 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 $kup^*(K_2)$ 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 to 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

\begin{align*} \prod _{k\ge 0} \frac {1}{(1-x^{2^k+1})} & -\frac {1}{(1-x^4)} \left(1 + x^5+x^7+x^8+x^9+x^{10}\right)\\& -\frac {1}{(1-x^2)(1-x^4)} \left( \bigoplus _{j\ge 4} \left( x^{2^j+1}(1+x^9)(1+x)\left(1-x^{2j-6}\right) \prod _{k\ge j} \left(1+x^{2^{k+1}+2}\right) \right) \right) \end{align*}

The first term is the PS for $H^*K_2$ . The second is the PS for $P\big[u_2^2\big]\otimes (\langle 1 \rangle \oplus N)$ . The last term is more complicated but does the $S$ and $R$ terms. The $(1-x^4)$ in the denominator is for the $P\big[u_2^2\big]$ . The $x^9$ is the shift that takes $R$ to $S$ . The $(1+x)$ is because they are $Q_0$ free. The $x^{2^j+1} \left(1-x^{2j-6}\right)/(1-x^2)$ 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 $kup^*(K_2)$

The main theorem of this section determines the differentials in the ASS for $kup^*(K_2)$ .

Theorem 3.1. The differentials in the spectral sequence whose $E_2$ -term was given in Theorem 2.1 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 $\Lambda _j=TP_p[z_i\,:\,i\ge j]$ , 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 $p=2$ .

\begin{eqnarray*} d_{\nu (i)+2}\left(y_1^i\right)&=&h_0^{\nu (i)}v^2q y_1^{i-1},\ i\ge 1;\\[4pt] d_{\nu (i)+2}\left(y_1^iz_jM\right)&=&v^{\nu (i)+2}q y_1^{i-1}z_{j-\nu (i),j}M,\\[4pt] &&j\ge \nu (i)+2,\ M\in \Lambda _j;\\[4pt] d_{2^t-t}\left(h_0^{t-2}v^2q y_1^{2^{t-1}-1}M\right)&=&v^{2^t}z_tM,\\[4pt] &&t\ge 2,\ M\in P[y_t];\\[4pt] d_{2^t-t}\left(q y_1^{2^{t-1}-1}z_{j-(t-2),j}M\right)&=&v^{2^t-t}z_tz_jM,\\[4pt] &&j\ge t\ge 2,\ M\in P[y_t]\otimes \Lambda _{j+1}. \end{eqnarray*}

Now with $p$ odd.

(3.2) \begin{align} d_{\nu (i)+2}\left(y_1^i\right)=h_0^{\nu (i)+1}vq y_1^{i-1},\ i\ge 1; \qquad \end{align}

(3.3) \begin{align} \quad d_{\nu (i)+2}\left(y_1^iz_jM\right)&=v^{\nu (i)+2}q y_1^{i-1}z_{j-\nu (i)-1,j}M,\nonumber\\[4pt] &\quad j\ge \nu (i)+2,\ M\in \Lambda _j; \qquad\qquad\end{align}

(3.4) \begin{align} d_{p^t-t}\left(h_0^{t-1}vq y_1^{p^{t-1}-1}M\right)&=v^{p^t}z_tM,\nonumber\\[4pt] &\quad t\ge 1,\ M\in P[y_t];\qquad\qquad\qquad\qquad\end{align}

(3.5) \begin{align} d_{p^t-t}\left(q y_1^{p^{t-1}-1}z_{j-(t-1),j}M\right)& = v^{p^t-t}z_tz_jM,\nonumber\\[4pt]&\quad j\ge t\ge 1,\ M\in P[y_t]\otimes TP_{p-1}[z_j]\otimes \Lambda _{j+1}.\end{align}

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 $kup^*(K_2)$ in Section 1, except for the exotic extensions.

By [Reference Tamanoi12, Theorem A], $Q_jQ_0\iota _2$ 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 (3.3) follows from (3.2), and (3.5) follows from (3.4), using $pz_{i,\ell }=vz_{i-1,\ell }$ , as noted in 1.13.

The differentials (3.2) follow from the result of [Reference Browder3] that $H^{2pi+1}\big(K_2;{\mathbb{Z}}\big)\approx{\mathbb{Z}}/p^{\nu (i)+2}\oplus \bigoplus{\mathbb{Z}}_p$ . See also [Reference Clément4, Proposition 1.3.5] when $p=2$ . The ASS converging to $H^*\big(K_2;\,{\mathbb{Z}}\big)$ has $E_2=\operatorname{Ext}_{A_0}\big({\mathbb{Z}}_2,H^*K_2\big)$ , where $A_0=\langle 1,Q_0\rangle$ . We depict this $E_2$ similarly to our ASS for $kup^*(K_2)$ . 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 11.

Figure 11.

$ kup^*(K_2)\to H^*\big(K_2;\,{\mathbb{Z}}\big)$ .

Similarly to Figures 8 and 9, we have, for $p=2$ and $i\ge 1$ , an $h_0$ -tower in the ASS for $kup^*(K_2)$ arising from $G^{4i+1,2}$ , called either $h_0^2y_1^{i-1}x_5$ or $v^2y_1^{i-1}q$ . 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 $u_5+u_2u_3$ . Under the morphism $kup^*(K_2)\to H^*\big(K_2;\,{\mathbb{Z}}\big)$ , these towers map across, as suggested in Figure 11. We deduce the $d_{\nu (i)+2}$ -differential claimed in (3.2), promulgated by the action of $v$ . Note that $x_9=q$ .

The situation when $p$ is odd is extremely similar, using Figure 10. The difference is that the $h_0$ -tower in $2pi+1$ in the $kup^*$ ASS starts in filtration 1 rather than 2. Its generator can be called $vy_1^{i-1}q$ .

In Figure 12, we depict many of the differentials asserted in Theorem 3.1 in grading $\le 36$ 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 $y_1A_1$ , $y_1^3A_1$ , and $y_1^5A_1$ . Not included are the portions involving (3.2) and (3.3) when $i$ is odd, as this portion self-annihilates. What is shown is (3.2) for $i=2$ , 4, and 6, (3.4) for $(t,k)=(1,0)$ , $(1,1)$ , $(1,2)$ , and $(2,0)$ , and (3.5) with $t=1$ , $k=0$ , and $j=4$ .

Figure 12.

Some differentials with $p=2$ .

In order to establish the remaining differentials, we will need the following description of $k(1)^*(K_2)$ , which is proved in [Reference Davis, Ravenel and Wilson8]. 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 [Reference Davis, Ravenel and Wilson8]. We recapitulate some of their properties. Those stated here but not there are easily proved by induction.

Proposition 3.8. [Reference Davis and Wilson8] For $j\ge 0$ , $z_j$ is the reduction of the class in $kup^*(K_2)$ and satisfies $|z_j|=2\!\left(p^{j+1}+1\right)$ . The classes $w_j$ satisfy $|w_1|=2p^2+1$ , $|w_2|=2p^3-2p^2+6p-3$ , and $w_{j+2}=y_j^{p-1}w_jz_{j+1}^{p-1}$ . The integers $r(j)$ and $r^{\prime}(j)$ satisfy the following properties:

(3.9) \begin{align} &r(0)=1,\ r(1)=p,\ r(j+2)=r(j)+p^{j+1}(p-1)+1;\nonumber\\[8pt] &r^{\prime}(0)=p-1,\ r^{\prime}(1)=p^2-p,\end{align}

(3.10) \begin{align} r^{\prime}(j+2)=r^{\prime}(j)+p^{j+2}(p-1)-1, \end{align}

(3.11) \begin{align} r(j)-r^{\prime}(j-1)=j, \end{align}

(3.12) \begin{align} r(j)+r^{\prime}(j)=p^{j+1}, \end{align}

(3.13) \begin{align} r(j+2)+r^{\prime}(j)=p^{j+2}+1, \end{align}

(3.14) \begin{align} (p-1)(r(j-1)+j-1)\lt p^j, \end{align}

(3.15) \begin{align} p^{j+1}-p^j\le r^{\prime}(j)\lt p^{j+1}-p^{j-1}. \end{align}

Theorem 3.16. [Reference Davis and Wilson8] For any $p$ , $k(1)^*(K_2)$ is a trivial $k(1)^*$ -module plus

\begin{eqnarray*} &&\bigoplus _{j\gt 0}TP_{r(j)}[v]\otimes P\!\left[y_{j+1}\right]\otimes TP_{p-1}\big[y_j\big]\otimes \overline{E}\big[w_{j}\big]\otimes E\!\left[w_{j+1}\right]\otimes \Lambda _{j+1}\\[4pt] &\oplus &\bigoplus _{j\ge 1}TP_{r^{\prime}(j-1)}[v]\otimes P[y_{j}]\otimes E\big[w_{j}\big]\otimes \overline{TP}_p[z_{j}]\otimes \Lambda _{j+1}\\[4pt] &\oplus &P[y_1]\otimes \biggl (\overline{E}\big[y_0^{\,p-1}z_0\big]\oplus \begin{cases}\overline{E}[z_1]&p=2\\[4pt] 0&p\text{ odd}\end{cases}\biggr )\oplus \bigoplus _{j\ge 1} P[y_1]\otimes E[q ]\otimes \overline{E}\big[z_j^p\big]\otimes \Lambda _{j+1}. \end{eqnarray*}

The last line was not discussed in [Reference Davis, Ravenel and Wilson8]; 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 (3.3). As already noted, the $z_j$ ’s are infinite cycles by [Reference Tamanoi12], and so the differentials in (3.5) are implied as soon as the corresponding differential in (3.4) is proved.

As a warmup, we consider the cases $t=2$ and 3 of (3.4) when $p=2$ . We make extensive use of the exact sequence (1.25). Referring to Figure 12 is useful.

In even gradings $\le 14$ , $k(1)^*(K_2)=0$ in positive filtration, by Theorem 3.16. Thus, the map $kup^*(K_2)\to k(1)^*(K_2)$ implies that in the ASS for $kup^*(K_2)$ , $v^sz_2$ must be hit by a differential or divisible by 2 for $s\ge 2$ . In grading $\lt 8$ , there is nothing that can divide it, and the only odd-grading $v$ -tower in that range is on $v^2y_1q$ . Thus, $d_2\big(v^2y_1q\big)=v^4z_2$ , the case $t=2$ , $M=1$ of (3.4). Since $d_2\big(y_1^{2k}\big)=0$ by (3.2), the case $t=2$ of (3.4) follows for any $M$ by the derivation property. An analogous argument does not work at the odd primes.

Similarly $v^sz_3$ must be hit or divisible for $s\ge 4$ , and examination of options in Figure 12 shows that we must have $d_5\!\left(h_0v^2y_1^3q\right)=v^8z_3$ , preceded by extensions. Since $d_5\big(y_1^8\big)=h_0^3v^2y_1^7q$ , we deduce the case $t=3$ , $M\in P\big[y_1^8\big]$ of (3.4) using the derivation property (2.5) and $h_0z_2=0$ . We do not have a priori knowledge that $y_1^4z_3$ is a permanent cycle in the ASS of $kup^*(K_2)$ . However, if it supported a nonzero differential, then the tower of $v$ -height 4 on $y_1^4z_3$ in the ASS of $k(1)^*(K_2)$ would have to map to $v^tC$ for $0\le t\le 3$ for some $C$ in positive filtration in grading 51 in the ASS of $kup^*(K_2)$ . Then, $v^4C$ must be $d_r(B)$ with $r\ge 5$ and $B$ in filtration 0 in grading 42. ( $B$ cannot have higher filtration since everything is $v$ -towers, and $v^3C$ cannot be hit.) But the only possible $B$ is $y_1^6z_2$ , and we already know that $v^4y_1^6z_2\in \operatorname{im}(d_4)$ . (Ordinarily this would not preclude the possibility of $B$ supporting a differential, but it does since everything is $v$ -towers.) Thus, $y_1^4z_3$ is a permanent cycle, and consideration of its image in $k(1)^*(K_2)$ implies that $v^sy_1^4z_3$ is hit by a differential for some $s\ge 4$ . The only element in odd grading $\lt 42$ not yet accounted for is $h_0v^2y_1^7q$ in grading 33, and so this must be the source of the differential. This is the case $t=3$ , $M=y_1^4$ of (3.4). 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 (3.4):

(3.17) \begin{equation} d_{p^t-t}\left(h_0^{t-1}vqy_1^{(i+1)p^{t-1}-1}\right)=v^{p^t}y_1^{ip^{t-1}}z_t. \end{equation}

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 [Reference Davis, Ravenel and Wilson8]. 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 $h_0^{t-1}vq y_1^{(4\ell +1)5^{t-1}-1}$ , 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 $\frac 12(|T|+1)$ . The $\frac 12$ avoids extraneous factors of 2 that always cancel out. The $+1$ is so that this indicates the grading (times $\frac 12$ ) of the class that it hits. $|M|$ denotes $\frac 12$ times the grading of $M$ , and $M^{\prime}$ equals $\frac 12$ times the grading of $v^hM$ , 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 $v^iM$ for some $i$ and a certain monomial $M$ ), then $|T|\le M^{\prime}$ . (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 $|M_2|\gt |M_1|$ . Since $|T(5\ell +1,t-1)|=|T(\ell,t)|$ , this says that as you move up an $h_0$ tower,Footnote 3 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 $|M_1|\ge |M_3|\ge M^{\prime}_1$ and either

\begin{equation*}M^{\prime}_3\le |T_2|\end{equation*}

or

\begin{equation*}T_3\to M_3\text { has already been proved, and } |T_3|\le |T_2|.\end{equation*}

The reason for C3 is that there must be extensions into the $M_1$ -tower from grading $M^{\prime}_1$ 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 $M^{\prime}_3+4$ , 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 $M^{\prime}_1$ (and hence in gradings $\le M^{\prime}_1$ 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 13. There is an exception to the C3 requirement for $T(\ell,1)\to M(\ell,1)$ . Here, the extension into $v^4y_1^{4\ell }z_1$ is obtained from the special class $y_1^{4\ell }y_0^4z_0$ .

Figure 13.

The role of $M_3$ .

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.

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 2. 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 $M^{\prime}_1=714$ . It is $v^3M_3$ which supports the extension in “grading” 714. Note that for $0\le i\le 2$ , $h_0v^iM_3\ne 0$ , and so $p\cdot v^iM_3$ is not a $v$ -multiple of $M_1$ . (In Figure 2, the class $y_2^{p-1}z_2$ corresponds to $M_3$ .) From Table 1, we see that $M^{\prime}_3=646$ , which means that in “grading” $\le 646$ , 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 2, 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.

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

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 $M\ne M(\ell,t)$ .

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

If $|M(5\ell +1,t-1)|\lt |M|\lt |M(\ell,t)|$ , by constraint C3 and the lemma, we must have $M_3$ with $|M(5\ell +1,t-1)|\lt M^{\prime}\le |M_3|\lt |M|$ . Because $|M_3|\lt |M|$ , we know $T_3\to M_3$ by induction. From the lemma, we get $|T(\ell,t)|\lt |T_3|$ , 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 3.2. We will show at the end of the proof that there is a number $\Delta \le t$ such that $v^iM$ maps nontrivially to $kup^{*+1}(K_2)$ if and only if $i\le r^{\prime}(t-1)-\Delta$ . (Usually $\Delta =1$ .) The image of $M$ in $kup^{|M|+1}(K_2)$ is a class $C$ of positive filtration such that $v^{r^{\prime}(t-1)-\Delta }C\ne 0$ and $v^{r^{\prime}(t-1)-\Delta +1}C=0\in kup^*(K_2)$ , so there must be a differential in the ASS of $kup^*(K_2)$ from a filtration-0 class hitting a class of filtration $\ge r^{\prime}(t-1)-\Delta +2$ 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 (3.3). They are distinguished by having the smallest $z$ -subscript at least 2 greater than the $p$ -exponent of the exponent of $y_1$ .

Figure 14.

An unwanted possibility.

The differential to $C$ has subscript $\ge r^{\prime}(t-1)-\Delta +2$ , and so the class in (3.3) 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 $j\ge r^{\prime}(t-1)-\Delta +2$ , 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 canceling a common factor 2 from all terms yield

(3.20) \begin{equation} \ell p^{r^{\prime}(t-1)-\Delta +1} + \sum _j \left(p^{j+1}+1\right) =ip^t+p^{t+1}+1-(p-1)(r^{\prime}(t-1)-\Delta +1). \end{equation}

Using (3.12) and (3.14) and $\Delta \le t$ , 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 $(p-1)(r(t-1)+\Delta -1)+1\le p^t$ (strict if $t\gt 2$ ). Since $r^{\prime}(t-1)-\Delta \gt t$ , 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

\begin{eqnarray*} &&\sum p^{j}\ge p\cdot p^{r^{\prime}(t-1)-\Delta +2}+(p-1)\left(p^{r^{\prime}(t-1)-\Delta +3}+\cdots + p^{r^{\prime}(t-1)+r(t-1)}\right)\\[4pt] &=&p^{r^{\prime}(t-1)+r(t-1)+1}=p^{p^t+1}, \end{eqnarray*}

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

Since $d_{p^t-t+1}\left(y_1^{p^{p^t-t-1}}\right)$ is defined,

(3.21) \begin{equation} d_r\left(y_1^{p^{p^t-t-1}}\right)=0\text{ for }r\le p^t-t, \end{equation}

and by the lowest-grading assumption, $ d_{p^t-t}\left(h_0^{t-1}vqy_1^{\left(i-p^{p^t-2t}+1\right)p^{t-1}-1}\right)=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

\begin{equation*}y_1^{ip^{t-1}}z_t=y_1^{\left(i-p^{p^t-2t}\right)p^{t-1}}z_t\cdot y_1^{p^{p^t-t-1}},\end{equation*}

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 $kup^*(K_2)$ , due to a filtration jump from a lower tower, as sketched in Figure 14, so that only a smaller part of the $M$ -tower in $k(1)^*(K_2)$ maps to $kup^{*+1}(K_2)$ .

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 $M^{\prime}_\varepsilon =|M_\varepsilon |-2(p-1)r^{\prime}(t_\varepsilon -1)$ . In $kup^*(K_2)$ , $M_2$ is truncated in grading $|T_2|=|v^{p^{t_2}}M_2|=2(p^{t_2}(i_2+1)+1)$ . In Figure 14, elements $c$ are in grading $M^{\prime}_2$ , and $c^{\prime}$ is in grading $M^{\prime}_1+2(p-1)$ . The necessary condition for nontrivial image in $k(1)^*(K_2)$ (and hence $\Delta \gt 1$ ) is

(3.23) \begin{equation} |T_2|+2(p-1)\le M^{\prime}_1+2(p-1)\le M^{\prime}_2. \end{equation}

If this occurs, then we might have $\Delta$ as large as $\dfrac{M^{\prime}_2-M^{\prime}_1}{2(p-1)}+1$ . We now show in Lemma 3.24 that if (3.23) holds, then $\big(M^{\prime}_2-M^{\prime}_1\big)/(2(p-1))\lt t$ , establishing the claim made earlier about $\Delta \le t$ .

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 $t_1\gt t_2$ and

\begin{equation*}5^{t_2}(4\ell _2+1)+4\le 5^{t_1}(4\ell _1+5)-4r^{\prime}(t_1-1)+4\le 5^{t_2}(4\ell _2+5)-4r^{\prime}(t_2-1),\end{equation*}

then

\begin{equation*} \frac 14\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 )\lt t_1-1.\end{equation*}

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 [Reference Davis, Ravenel and Wilson8, Lemma 5.3]). Then, using [Reference Davis, Ravenel and Wilson8, Lemma 5.5] for $t=2k+\delta$ with $\delta =1$ or 2,

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

Since $16\cdot 5^\delta Q(k)+4k+4\cdot 5^{\delta -1}\lt 3\cdot 5^{2k+\delta }$ , 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:

(3.25) \begin{equation} 5^{2k_1+\delta _1}+16\cdot 5^{\delta _1}Q(k_1)+4k_1+4\cdot 5^{\delta _1-1} \end{equation}

lies in the mod- $(4\cdot 5^{t_2})$ interval:

(3.26) \begin{equation} [5^{t_2},5^{t_2}+16\cdot 5^{\delta _2}Q(k_2)+4k_2+4\cdot 5^{\delta _2-1}-4]. \end{equation}

Let $\delta _2=1$ . The reduction mod $4\cdot 5^{t_2}$ of (3.25) is

(3.27) \begin{equation} 5^{t_2}+16\cdot 5^{\delta _1}Q(k_2)+4k_1+4\cdot 5^{\delta _1-1}. \end{equation}

Let $\delta _1=2$ . Then, $5^{t_2}+16\cdot 5^{\delta _1}Q(k_2)\gt 4\cdot 5^{t_2}$ 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 $\frac 18\big(M^{\prime}_2-M^{\prime}_1\big)$ with $M^{\prime}_1$ and $M^{\prime}_2$ as in (3.23). For $k_1=500Q(k_2-1)+20$ , the value of $M^{\prime}_1$ is at the left end of the interval (3.26), and so $\frac 18\big(M^{\prime}_2-M^{\prime}_1\big)$ equals $\frac 14$ times the length plus 4 of (3.26), which is

\begin{equation*}20Q(k_2)+k_2+1=500Q(k_2-1)+k_2+21= \frac 12t_1+k_2.\end{equation*}

Since $k_2\ll t_1$ , 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 $M^{\prime}_2-M^{\prime}_1$ decreases, since $M^{\prime}_1$ 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 $k_1\gt k_2$ , (3.27) lies outside the interval (3.26) until $80Q(k_2)+4k_1+4=4\cdot 5^{t_2}$ , so

\begin{equation*}k_1=5^{2k_2+1}-20Q(k_2)-1=100Q(k_2)+4\end{equation*}

and $t_1=200Q(k_2)+9$ . Again $\frac 18\big(M^{\prime}_2-M^{\prime}_1\big)=20Q(k_2)+k_2+1\approx \frac 1{10}t_2+k_2$ , so the conclusion of the lemma is satisfied more strongly.

A similar analysis works when $\delta _2=2$ . In this case, $\frac 18\big(M^{\prime}_2-M^{\prime}_1\big)\approx \frac 12t_1+k_2$ if $\delta _1=1$ , and $\frac 18\big(M^{\prime}_2-M^{\prime}_1\big)\approx \frac 1{10}t_1+k_2$ if $\delta _1=2$ .

We close this section by explaining how Theorems 2.4 and 3.1 lead to the descriptions of $kup^*(K_2)$ given in Theorems 1.8 and 1.15, 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 1.5 and Figure 2. From 1.5, $y_0^{\,p^k-1}z_0$ is in $A_k$ for $k\ge 1$ , the bottom right element in Figure 2. Then,

(3.28) \begin{equation} 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^AA_k. \end{equation}

The first part occurs in Theorem 2.4 and the last part in Theorem 1.8.

Now we consider $P[y_1]\otimes \bigoplus _{j\ge 1} W_j\otimes TP_{p-1}[z_j]\otimes \Lambda _{j+1}$ in Theorem 2.4. The $\bigoplus$ part is all monomials $z_\ell M$ with $\ell \ge 1$ and $M\in \Lambda _\ell$ . From Theorem 3.1, $y_1^iz_\ell M$ supports a differential (3.3) if $\ell \ge \nu (i)+2$ , while those with $\nu (i)\ge \ell -1$ are hit by differentials (3.4) and (3.5), yielding $v$ -towers with heights as given in 1.5. These are all monomials in $\bigoplus _{\ell \ge 1}P[y_\ell,y_{\ell +1},\ldots ]z_\ell \Lambda _\ell$ . From 1.5 or (1.7), the generators of the $v$ -towers in $B_k$ are all

\begin{equation*}z_j\prod _{i=j}^{k-1}\left\{z_i^{p-1},y_i^{p-1}\right\},\ 1\le j\le k.\end{equation*}

Let $(z_\ell M)_i$ be the $y_i^ez_i^{e^{\prime}}$ factors of $ M$ . Then, $\mathcal{M}_kB_k$ consists of all monomials $z_\ell M$ such that $(z_\ell M)_i$ equals $y_i^{p-1}$ or $z_i^{p-1}$ for $\ell \le i\lt k$ , but not for $ i=k$ , and so every monomial $z_\ell M$ is in a unique $\mathcal{M}_kB_k$ . 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 1.8.

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 (3.4) or targets (3.2) of differentials. See grading 17 in Figure 12 for a nice illustration. The $qy_1^{i-1}S_{\nu (i)+1,\ell }$ part of Theorem 1.15 is formed from $TP_{\nu (i)+2}[v]qy_1^{i-1}W_\ell$ in 2.1 using (3.3). 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 (3.3), one should write $z_{t,\ell }=z_{t,t+\nu (i)+1}Z_{t+\nu (i)+1}^\ell$ , where

(3.29) \begin{equation} Z_i^j=(z_i\cdots z_{j-1})^{p-1}\text{ for }j\gt i, \text{ with }Z_i^i=1. \end{equation}

The remaining generators of $qy_1^{i-1}W_\ell$ , namely $qy_1^{i-1}z_{j,\ell }$ with $ \ell -\nu (i)\le j\le \ell$ , support differentials (3.5). 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.

4. Exotic extensions

In this section, we prove the following expansion of (1.6).

Theorem 4.1. If $i\ge 0$ and $k\ge k_0$ ,

\begin{equation*}py_{k}^iy_{k-1}^{p-1}z_{k-1}=v^{p^{k-1}(p-1)}y_{k}^iz_{k}\end{equation*}

with an additional term $vy_{k}^iy_{k-1}^{p-1}z_{k-2}^p$ if $k\ge k_0+2$ .

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 $i\gt k$ . 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:

\begin{equation*}kup^*(K_2)\ \smash {\mathop {\longrightarrow }\limits ^{\cdot p}}\ kup^*(K_2)\longrightarrow k(1)^*(K_2)\end{equation*}

and the fact that $v^{r^{\prime}(k-1)}y_{k}^iz_{k}=0$ in $k(1)^*(K_2)$ with $r^{\prime}(k-1)\ge p^k(p-1)$ . Thus, $v^{r^{\prime}(k-1)}y_{k}^iz_{k}$ must be divisible by $p$ in $kup^*(K_2)$ , and, as we will show, the $v$ -tower on $y_{k}^iy_{k-1}^{p-1}z_{k-1}$ 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 2 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 15, using the notation of the preceding section. Thus, $T_i$ is the class satisfying $d_r(T_i)=v^rM_i$ , Here, the portion of the top tower to the right of $M^{\prime}_1$ must be divisible by $p$ . The tower providing the extension must have $M^{\prime}_1\le |M_2|\lt |M_1|$ and $|T_2|\le |T_1|$ .

Figure 15.

Conditions for extension.

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 1.15. If instead we used $i=4\ell +d$ for $1\le d\le 3$ , 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 $5^k(4\ell +1.16)+1\lt M^{\prime}\le 5^k(4\ell +1.8)+1$ , using (3.15). With $M_1$ and $M_2$ as in Figure 15, we will show that $M_2(5\ell +1,k-1)$ is the unique monomial satisfying the inequalities stated just before Figure 15 for $M_1(\ell,k)$ . Note that $M(5\ell +1,k-1)=y_k^{4\ell }y_{k-1}^4z_{k-1}$ . We omit the $+1$ in all the formulas.

The inequalities are satisfied by $M_2(5\ell +1,k-1)$ since

\begin{equation*}5^k(4\ell +1.8)\le 5^{k-1}(4(5\ell +1)+5)\lt 5^k(4\ell +5)\text { and }5^{k-1}(4(5\ell +1)+1)\le 5^k(5\ell +1).\end{equation*}

If $k_2\ge k$ , then the first inequality, after dividing by $5^k$ , becomes

\begin{equation*}4\ell +1.8\le 5^{k_2-k}(4\ell _2+5)\lt 4\ell +5,\end{equation*}

which cannot be satisfied since the middle term is $\equiv 1$ mod 4. If $k_2\lt k-1$ , then

\begin{equation*}M^{\prime}_1-|T_1|\gt 5^k\cdot .16\ge 4\cdot 5^{k_2}=|M_2|-|T_2|,\end{equation*}

which is inconsistent with two of the inequalities. Let $k_2=k-1$ . If $\ell _2\lt 5\ell +1$ , then

\begin{equation*}|M_2|=5^{k-1}(4\ell _2+5)\le 5^{k-1}(4\cdot 5\ell +5)\lt 5^k(4\ell +1.16)\lt M^{\prime}_1,\end{equation*}

contradicting one of the inequalities. If $k_2=k-1$ and $\ell _2\gt 5\ell +1$ , then

\begin{equation*}|T_2|\ge 5^{k-1}(4(5\ell +2)+1)\gt 5^k(4\ell +1)=|T_1|,\end{equation*}

contradicting one of the inequalities.

We deduce that $M_2=y_k^{4\ell }y_{k-1}^4z_{k-1}$ , 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 (1.25)

In this section, we propose what we conjecture must be the correct complete formulas for the exact sequence (1.25). 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 $kup^*(K_2)$ 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 ([Reference Davis and Wilson7]), a thorough analysis when $p=2$ was performed.

We propose that (1.25) 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 $k\ge 1$ and $G^i_{k,\ell }$ for $3\le i\le 6$ and $1\le k\lt \ell$ such that there are exact sequences:

(5.1) \begin{equation} 0\to G_k^1\to A_k\ \smash{\mathop{\longrightarrow }\limits ^{p}}\ A_k\to G_k^2\to 0 \end{equation}

for $k\ge 1$ , and, for $1\le k\lt \ell$ ,

(5.2) \begin{eqnarray} \nonumber 0&\to & G^3_{k,\ell }\to y_kB_{k}Z_k^\ell \ \smash{\mathop{\longrightarrow }\limits ^{p}}\ y_kB_{k}Z_k^\ell \to G^4_{k,\ell }\to y_1^{p^{k-1}-1}q S_{k,\ell }\\[4pt] &\ \smash{\mathop{\longrightarrow }\limits ^{p}}\ &y_1^{p^{k-1}-1}q S_{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, \end{eqnarray}

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:

(5.3) \begin{equation} 0\to G^7_{k,e}\to B_kz_k^e\ \smash{\mathop{\longrightarrow }\limits ^{p}}\ B_kz_k^e\to G^8_{k,e}\to 0 \end{equation}

for $k\ge 1$ and $1\le e\le p-2$ . 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 1.8. 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 $k\ge 1$ ,

(5.4) \begin{equation} g_{k+2}=v^{r^{\prime}(k)-1}z_{k+2}+g_ky_k^{p-1}z_{k+1}^{p-1}. \end{equation}

To see that this is in $\operatorname{ker}({\cdot} p)$ , we use (1.6) 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 $p\cdot g_ky_k^{p-1}z_{k+1}^{p-1}$ . Using (3.10), these terms cancel. Other terms in $p\cdot g_ky_k^{p-1}z_{k+1}^{p-1}$ yield 0 since $g_k\in \operatorname{ker}({\cdot} p)$ .

The $v$ -towers in $K_k^A$ are generated by:

(5.5) \begin{equation} g_k\text{ and }g_jz_j^{p-1}\prod _{i=j+1}^{k-1}\left\{z_i^{p-1},y_i^{p-1}\right\},\ 1\le j\le k-1. \end{equation}

For example, using Figure 2 when $k=3$ , these are $g_3=v^{p^2-p-1}z_3+y_1^{p-1}z_1z_2^{p-1}$ , $g_2z_2^{p-1}=v^{p-2}z_2^p$ , $g_1z_1^{p-1}z_2^{p-1}$ , and $g_1z_1^{p-1}y_2^{p-1}$ . 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