5 Extensions on a classical \(E_r\)-page

To state the Generalized Leibniz Rule in terms of the classical Adams spectral sequence, we need to define \(f\)-extensions not only on homotopy groups but on the \(E_r\)-page as well.

Notation 5.1

For a map between classical spectra \(f: X\to Y\), consider the associated synthetic map from Notation 0.126

\[ \hat f: \Sigma ^{0,e(f)}\nu X\to \nu Y. \]

For any \(2\le r\le \infty \), we denote the following mod \(\lambda ^{r-1}\) reduction maps by \(\hat f_{r-1}\):

\[ \hat f_{r-1}: \Sigma ^{0,e(f)}\nu X/\lambda ^{r-1}\to \nu Y/\lambda ^{r-1}. \]

The \(E_0\)-page of the \(\hat f_{r-1}\)-ESS is isomorphic to

\[ \begin{aligned} \prescript {\hat f_{r-1}\! }{}{E}_0^{s,t,t-k+e(f)}\cong & E_\infty ^{s,t,t-k}(\nu X/\lambda ^{r-1})\oplus E_\infty ^{s,t,t-k+e(f)}(\nu Y/\lambda ^{r-1})\\ \cong & \big(Z_{r-1-k}^{s,t}(X)/B_{1+k}^{s,t}(X)\big)\oplus \big(Z_{r-1-k+e(f)}^{s,t}(Y)/B_{1+k-e(f)}^{s,t}(Y)\big). \end{aligned} \]

A nontrivial \(d^{\hat f_{r-1}}_n\) differential can be interpreted as a map from the subgroup

\begin{equation} \label{eq:fhat-ess-source} \prescript {\hat f_{r-1}\! }{}{Z}_{n-1}^{s,t,t-k}(X) \subset Z_{r-1-k}^{s,t}(X)/B_{1+k}^{s,t}(X) \end{equation}

5.1

to the quotient group

\begin{equation} \label{eq:fhat-ess-target} \big(Z_{r-1-k-n+e(f)}^{s+n,t+n}(Y)/B_{1+k+n-e(f)}^{s+n,t+n}(Y) \big)/\prescript {\hat f_{r-1}\! }{}{B}_{n-1}^{s+n,t+n,t-k}(Y). \end{equation}

5.2

The differential \(d^{\hat f_{r-1}}_n\) is trivial for degree reasons when

\[ n{\lt}\mathrm{AF}(f)\text{ or }n{\gt}r-2-k+e(f). \]

Definition 5.2

Let \(x\in Z^{s,t}_{r-1}(X)\) and \(y\in Z^{s+n,t+n}_{r-1-n+e(f)}(Y)\) for some

\[ e(f)\le n\le r-2+e(f). \]

We say that there is an \((f,E_r)\)-extension from \(x\) to \(y\), denoted by

\begin{equation} \label{eq:f-Er-ext-classical} d_{n}^{f,E_r}(x)=y \end{equation}

5.3

if there exists a synthetic \(\hat f_{r-1}\)-extension

\begin{equation} \label{eq:f-Er-ext-synthetic} d_{n}^{\hat f_{r-1}}(x)=\lambda ^{n-e(f)} y. \end{equation}

5.4

where \(x\) is viewed as an element of the subgroup (5.1) with \(k=0\), and \(\lambda ^{n-e(f)} y\) is viewed as an element of the quotient group (5.2) with \(k=0\).

We say that this \((f,E_r)\)-extension in (5.3) is essential if the corresponding synthetic \(\hat f_{r-1}\)-extension in (5.4) is an essential differential in the \(\hat f_{r-1}\)-ESS.

For \(r=\infty \), we similar define an \((f,E_\infty )\)-extension using the corresponding synthetic \(\hat f\)-extension.

tikz diagram — Figure 2 \((f,E_r)\)-extension

Remark 5.3

Consider an \((f,E_r)\)-extension \(d_{n}^{f,E_r}(x)=y\) in (5.3). The element \(y\) should be interpreted as a coset with indeterminacy given by \(B_{1+n-e(f)}^{s+n,t+n}(Y)\), plus the sum of images of \(d^{\hat f_{r-1}}_{{\lt}n}\) differentials. This \((f,E_r)\)-extension is essential if this coset does not contain 0.

Definition 5.4

Let \(x\in Z^{s,t}_{r-1}(X)\) and \(y\in Z^{s+n,t+n}_{r-1-n+e(f)}(Y)\) for some

\[ e(f)\le n\le r-2+e(f). \]

We say that there is an \((f,E_r)\)-extension of level \(l\) from \(x\) to \(y\), denoted by

\begin{equation} \label{eq:f-Er-level-ext-classical} d_{n}^{f,E_r,l}(x)=y \end{equation}

5.5

if there exists a synthetic \(\hat f_{r-1}\)-extension

\begin{equation} \label{eq:f-Er-level-ext-synthetic} d_{n}^{\hat f_{r-1}}(\lambda ^l x)=\lambda ^{n-e(f)+l} y. \end{equation}

5.6

where \(\lambda ^l x\) is viewed as an element of the subgroup (5.1) with certain \(k\), and \(\lambda ^{n-e(f)+l} y\) is viewed as an element of the quotient group (5.2).

We say that this \((f,E_r)\)-extension of level \(l\) in (5.5) is essential if the corresponding synthetic \(\hat f_{r-1}\)-extension in (5.6) is an essential differential in the \(\hat f_{r-1}\)-ESS.

For \(r=\infty \), we similarly define an \((f,E_\infty )\)-extension of level \(l\) using the corresponding synthetic \(\hat f\)-extension.

For \(l {\lt} l'\), an \((f,E_r)\)-extension of level \(l\) implies an extension of the same type of level \(l'\), but the latter might be inessential.

Proposition 5.5

Suppose we have an \((f,E_r)\)-extension of level \(l'\),

\[ d_n^{f,E_r,l'}(x) = y. \]

Then either

there is an essential \((f,E_r)\)-extension of level \(l\), \(d_m^{f,E_r,l}(x) = z\), for some \(m {\lt} n\), or
we have \(d_n^{f,E_r,l}(x) = y + z\) for some \(z\) lying in the indeterminacy of \(d_n^{f,E_r,l'}\).

Definition 5.6

A crossing of the \((f,E_r)\)-extension \(d_{n}^{f,E_r}(x)=y\) in (5.3) is defined as an essential \((f, E_{r-a})\)-extension from some \(x'\in Z_{r-1-a}^{s+a,t+a}(X)\) to

\[ y'\in Z_{r-1-n+b+e(f)}^{s+n-b,t+n-b}(Y)\backslash B_{1+n-b-e(f)}^{s+n-b,t+n-b}(Y) \]

(where \(\backslash \) denotes the difference of sets and it means that \(y'\) should survive to the classical Adams \(E_{r-n+b+e(f)}\) page while it should not be hit by an Adams differential of length at most \(1+n-b-e(f)\)) for \(0{\lt}a \le r-2\) and \(0\le b\le n-a-e(f)\). See Figure 3.

Proposition 5.7

An \((f,E_r)\)-extension \(d_{n}^{f,E_r}(x)=y\) (5.3) has a crossing if and only if the synthetic \(\hat f_{r-1}\)-extension \(d_{n}^{\hat f_{r-1}}(x)=\lambda ^{n-e(f)} y\) (5.4) has a crossing.

Proof ▶

By definition a crossing of the synthetic extension (5.4) has form

\begin{equation} \label{eq:syn-crossing-form} d_{n-a-b}^{\hat f_r}(\lambda ^ax')=\lambda ^{n-b-e(f)}y' \end{equation}

5.7

for some \(x'\in Z_{r-1-a}^{s+a,t+a}(X)\) and \(y'\in Z_{r-1-n+b+e(f)}^{s+n-b,t+n-b}(Y)\). Consider the following commutative diagram

\[ \begin{CD} \end{CD} \Sigma ^{0,e(f)}\nu X/\lambda ^{r-1-a} @{\gt}{\hat f_{r-1-a}}{\gt}{\gt} \nu Y/\lambda ^{r-1-a} \\ @V{\lambda ^a}VV @VV{\lambda ^a}V \\ \Sigma ^{0,e(f)+a}\nu X/\lambda ^{r-1} @{\gt}{\hat f_{r-1}}{\gt}{\gt} \Sigma ^{0,a}\nu Y/\lambda ^{r-1} \end{CD} \]

We see that (5.7) lifts to

\begin{equation} \label{eq:syn-crossing-lift} d_{n-a-b}^{\hat f_{r-1-a}}(x')= \lambda ^{n-a-b-e(f)}y'' \end{equation}

5.8

for some \(y''\in Z_{r-1-n+b+e(f)}^{s+n-b,t+n-b}(Y)\) such that

\[ y''\equiv y'\mod B_{1+n-b-e(f)}^{s+n-b,t+n-b}(Y) \]

where the indeterminacy \(B_{1+n-b-e(f)}^{s+n-b,t+n-b}(Y)\) is introduced by dividing \(\lambda ^a\). Therefore, the differential (5.7) is essential if and only if the differential (5.8) is essential and \(y''\notin B_{1+n-b-e(f)}^{s+n-b,t+n-b}(Y)\). This completes the proof.

The above Definition 5.6 and Proposition 5.7 extend to the case \(r=\infty \) as follows.

Definition 5.8

A crossing of the \((f,E_\infty )\)-extension \(d_{n}^{f,E_\infty }(x)=y\) is defined as an essential \((f, E_\infty )\)-extension from some \(x'\in Z_{\infty }^{s+a,t+a}(X)\) to

\[ y'\in Z_{\infty }^{s+n-b,t+n-b}(Y)\backslash B_{1+n-b-e(f)}^{s+n-b,t+n-b}(Y) \]

(where \(\backslash \) denotes the difference of sets, meaning \(y'\) survives to the classical Adams \(E_{\infty }\)-page while it is not hit by any Adams differential) for \(a{\gt}0\) and \(0\le b\le n-a-e(f)\).

Proposition 5.9

An \((f,E_\infty )\)-extension \(d_{n}^{f,E_\infty }(x)=y\) has a crossing if and only if the synthetic \(\hat f\)-extension \(d_{n}^{\hat f}(x)=\lambda ^{n-e(f)} y\) has a crossing.

Similarly, the crossing notion extends to extensions of level \(l\).

Definition 5.10

A crossing of the \((f,E_r)\)-extension of level \(l\) \(d_{n}^{f,E_r,l}(x)=y\) in (5.5) is defined as an essential \((f, E_{r-a})\)-extension of level \(l\) from some \(x'\in Z_{r-1-a}^{s+a,t+a}(X)\) to

\[ y'\in Z_{r-1-n+b+e(f)}^{s+n-b,t+n-b}(Y)\backslash B_{1+n-b-e(f)}^{s+n-b,t+n-b}(Y) \]

for \(0{\lt}a \le r-2\) and \(0\le b\le n-a-e(f)\).

For \(r=\infty \), we similarly define a crossing of an \((f,E_\infty )\)-extension of level \(l\) as an essential \((f, E_\infty )\)-extension of level \(l\) from some \(x'\in Z_{\infty }^{s+a,t+a}(X)\) to \(y'\in Z_{\infty }^{s+n-b,t+n-b}(Y)\backslash B_{1+n-b-e(f)}^{s+n-b,t+n-b}(Y)\) for \(a{\gt}0\) and \(0\le b\le n-a-e(f)\).

Proposition 5.11

An \((f,E_r)\)-extension of level \(l\) \(d_{n}^{f,E_r,l}(x)=y\) (5.5) has a crossing if and only if the corresponding synthetic \(\hat f_{r-1}\)-extension \(d_{n}^{\hat f_{r-1}}(\lambda ^l x)=\lambda ^{n-e(f)+l} y\) (5.6) has a crossing.

Proposition 5.12

If an \((f,E_r)\)-extension of level \(l\), \(d_n^{f,E_r,l}(x)=y\), has a crossing, then there is an essential \((f,E_r)\)-extension of level \(0\), \(d_m^{f,E_r,0}(x')=z'\), such that the implied extension at level \(a+l\) satisfies the degree conditions of a crossing.

Proof ▶

If there is a crossing, we apply the alternatives of Proposition 5.5.

Proposition 5.13

If an \((f,E_r)\)-extension of level \(l\), \(d_n^{f,E_r,l}(x)=y\), has no crossing, then the induced extension at any level \(l' {\gt} l\) also has no crossing.

Proposition 5.14

Consider \(f: X\to Y\), \(x\in Z_\infty ^{s,t}(X)\) and \(y\in Z_\infty ^{s+n,t+n}(Y)\). Then

\[ d^{f,E_\infty }_n(x)=y \]

implies that

\[ d^{f}_n(x+B_\infty ^{s,t}(X))=y+B_\infty ^{s,t}(Y). \]

(The implied differential could be inessential.)

Proof ▶

This follows directly from inverting \(\lambda \), which induces a map of spectral sequences from the \(\hat f\)-ESS to the \(f\)-ESS. The induced map on the \(E_0\)-pages are of the following form:

\[ \begin{CD} \end{CD} Z_\infty ^{s,t}(X)/B_{1+t-w}^{s,t}(X) @{\gt}{d^{\hat f}_0}{\gt}{\gt} Z_\infty ^{s,t}(Y)/B_{1+t-w-e(f)}^{s,t}(Y) \\ @V{\lambda ^{-1}}VV @VV{\lambda ^{-1}}V \\ E_\infty ^{s,t}(X) @{\gt}{d^{f}_0}{\gt}{\gt} E_\infty ^{s, t}(Y) \end{CD} \]

Remark 5.15

Since the synthetic Adams \(E_\infty \)-page contains more information than the classical Adams \(E_\infty \)-page, the \((f,E_\infty )\)-extensions similarly provide more information compared to the classical \(f\)-extensions.

Proposition 5.16

Consider \(f: X\to Y\), \(x\in Z_\infty ^{s,t}(X)\) and \(y\in Z_\infty ^{s+n,t+n}(Y)\). Then

\[ d^{f,E_\infty , l}_n(x)=y \]

implies that

\[ d^{f}_n(x+B_\infty ^{s,t}(X))=y+B_\infty ^{s,t}(Y). \]

Proof ▶

The same as Proposition 5.14, since we invert \(\lambda \).