Dependencies

We can further generalize Theorem 6.1 by using the conditions in Theorem 0.52 rather than those Corollary 0.53. We leave this generalization to the reader.

A synthetic generalization of the Mahowald Trick, again lacking any no-crossing conditions, is also presented in Chua’s work [ . This version is also incorrect for similar reasons.

A version of the Generalized Leibniz Rule without the no-crossing conditions is presented in the synthetic setting in Chua’s work [ . However, there is no doubt that this version is incorrect. For further details, see Remark 6.7.

The essential \((f, E_\infty )\)-extension:

is a crossing for both the \((f, E_\infty )\)-extension,

and the inessential \((f, E_\infty )\)-extension:

This indicates that if we were to disregard the no-crossing condition (5), and apply the Generalized Leibniz Rule to these two cases of the \((f, E_\infty )\)-extensions, we would arrive at two conflicting classical statements:

Several versions of classical generalizations of Mahowald’s original trick appear in the literature and are often referred to as geometric boundary theorems. Notable examples include the works of Behrens [ , and Ma [ .

There is also a version of the Generalized Mahowald Trick where the assumptions involve \((g, E_{m_1+2})\)-extensions and \((h, E_{r'})\)-extensions of level \(k\). We leave this generalization to the reader.

Suppose \(r\ge r'\) and there exists an \((f,E_{r'})\)-extension

for \(x\in Z_{r-1}^{s,t}(X)\) and \(y\in Z_{r'-1-n+e(f)}^{s+n,t+n}(Y)\). Assume that this extension has no crossing of the form \(d^{f,E_{r'-a}}_{n-a-b}(x')=y'\) for any \(b{\gt}0\), \(0{\lt}a\le n-e(f)\), and

Under these conditions, we also have an \((f,E_{r})\)-extension

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

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

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

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.

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

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

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

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.

In Example ??, we use the Generalized Mahowald Trick to establish the \((f, E_3)\)-extension:

for the map \(f=\nu :S^3 \rightarrow S^0\), as discussed in Example ??(2). However, to apply the Generalized Leibniz Rule in proving the classical Adams differential

in Example ??, we need the \(E_\infty \)-page version of the \((f, E_3)\)-extension:

We apply Corollary 6.14 to confirm this.

As checked in Example ??(2), the \((f, E_3)\)-extension has no crossing. Consequently, by Corollary 6.14, we obtain the required \((f, E_\infty )\)-extension:

Let \(X\to Y\to Z\to \Sigma X\) and \(X'\to Y'\to Z'\to \Sigma X'\) be distinguished triangles of (synthetic) spectra. By smashing these distinguished triangles together, we obtain the following commutative diagram of cofiber sequences:

If \(a\in \pi _n(X\wedge Z')\) and \(b\in \pi _n(Y\wedge Y')\) map to the same element in \(\pi _n(Y\wedge Z')\), then there exists \(c\in \pi _n(Z\wedge X')\) such that

1. \(b\) and \(c\) map to the same element in \(\pi _n(Z\wedge Y')\), and

2. \(a\) and \(c\) map to the same element via boundary maps in \(\pi _{n-1}(X\wedge X')\).

Consider a homotopy commutative diagram of converging spectral sequences

Assume \({}^{p}\! E_0 = {}^{p}\! E_r\) and \({}^{q}\! E_0 = {}^{q}\! E_r\) for some \(r \ge 0\) (i.e., the \(p\)-ESS and \(q\)-ESS have stationary pages up to \(r\)). Then \((d_r^p, d_r^q)\) induces a map from the \(f\)-ESS to the \(g\)-ESS.

In the setting of Theorem 0.52, if condition (4) is strengthened to “\(d_l^g(z) = w\) has no crossing” (and condition (5) is dropped), then \(d_{m+l-n}^q(y) = w\).

Let \(X, Y, Z \in \mathcal{S}^{\mathrm{fin}}\) be finite spectra. Suppose that \(X \xrightarrow {f} Y \xrightarrow {g} Z \xrightarrow {h} \Sigma X\) is a distinguished triangle with \(e(f) + e(g) + e(h) = 1\). Then there is a distinguished triangle of synthetic spectra

The construction uses Axiom 0.124 to establish the \(\lambda \)-relations between the classical and synthetic connecting maps.

Let \(X \in \mathcal{S}^{\mathrm{fin}}\) be a finite spectrum. For \(r \ge 2\), the \(E_\infty \)-page of the synthetic Adams spectral sequence for \(\nu X / \lambda ^r\) is

The Adams differential \(d_r(x)=y\) has a crossing on the \(E_{n+1}\)-page if and only if the corresponding \(\delta _n\)-extension

for

has a crossing.

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.

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.

Suppose in the classical Adams spectral sequence of \(X\) we have \(d_r(x)=y\), where \(x\in Z_{r-1}^{s,t}(X)\) and \(y\in Z_\infty ^{s+r,t+r-1}(X)\). Consider the map

1. If \(r\ge n+1\), then we view \(x\) as an element of

   \[ E_\infty ^{s,t,t}(\nu X/\lambda ^n)\cong Z_n^{s,t}(X), \]

   and \(\lambda ^{r-n-1} y\) as an element of

   \[ E_\infty ^{s+r,t+r-1, t+n}(\nu X/\lambda ^{m-n})\cong Z_{m-r+1}^{s+r,t+r-1}(X)/B_{r-n}^{s+r,t+r-1}(X). \]

   We then have

   \[ d^{\delta }_r(x)=\lambda ^{r-n-1}y, \]

   which is trivial if \(r{\gt}m\).

2. If \(r{\lt}n+1\), then we view \(\lambda ^{n+1-r}x\) as an element of

   \[ E_\infty ^{s,t,t-n-1+r}(\nu X/\lambda ^n)\cong Z^{s,t}_{r-1}(X)/B^{s,t}_{n+2-r}(X), \]

   and \(y\) as an element of

   \[ E_\infty ^{s+r,t+r-1, t+r-1}(\nu X)\cong Z_\infty ^{s+r,t+r-1}(X). \]

   In this case, we have

   \[ d^{\delta }_r(\lambda ^{n+1-r}x)=y. \]

Suppose that we have an \((f,E_r)\)-extension \(d^{f,E_r}_n(x)=y\), where \(x\in Z_{r-1}^{s,t}(X)\) and \(y\in Z_{r-1-n+e(f)}^{s+n,t+n}(Y)\). Then for all \(2\le r'{\lt}r\), we also have

Furthermore, if \(d^{f,E_r}_n(x)=y\) is essential and \(n\le r'-2+e(f)\), then (6.14) is inessential if and only if there exists some \(0{\lt}a'\le n-e(f)\) and an element

such that

for some \(b\ge 0\).

In Chua’s work [ , it is stated that for a synthetic map \(\alpha : X\rightarrow Y\), and an element \(x \in \pi _{*,*}X/\lambda \), there exists a differential from a maximal \(\alpha \)-extension of \(x\) to a maximal \(\alpha \)-extension of \(d_r(x)\). According to [ , a maximal \(\alpha \)-extension of \(x'\) is defined as \(\alpha [x']\), where \([x']\) represents a lift of \(x'\) to the \(E_{r'}\)-page, chosen such that \(\alpha [x']\) is the most \(\lambda \)-divisible among all such lifts.

In the context of the counter-example above, let \(r=2, r'=\infty \),

and consider \(x = h_4\) in Ext, with \(x' = d_2(x) = h_0h_3^2\).

The \((f, E_\infty )\)-extension:

is equivalent to the existence of a synthetic homotopy class \([h_0 h_3^2]\) in \(\pi _{14, 17} S^{0,0}\), such that

Similarly, the inessential \((f, E_\infty )\)-extension:

implies the existence of another synthetic homotopy class \([h_0 h_3^2]\) in \(\pi _{14, 17} S^{0,0}\), such that

Between these two lifts of \(h_0h_3^2\), the first \([h_0h_3^2]\) is clearly the maximal \([h_0]\)-extension according to Chua’s definition [ . Consequently, the incorrect version of the Generalized Leibniz Rule in [ , would lead to an incorrect conclusion:

Let \(f: X\to Y\) be a map between two classical spectra. Suppose that \(2\le n\le r\), \(e(f)\le m\le n-2+e(f)\), \(l\ge e(f)\), and we have

and the following conditions hold:

1. \(d_{r}(x)=x_\infty \),

2. \(d_{m}^{f,E_n}(x)=y\),

3. \(d_{l}^{f,E_\infty }(x_\infty )=y_\infty \),

4. the differential in (1) has no crossing on the \(E_n\)-page or (2) has no crossing.

5. the differential in (3) has no crossing.

Then we have an Adams differential

Let \(f: X\to Y\) be a map between two classical spectra. Suppose that \(2\le n\le r\), \(e(f)\le m\le n-2+e(f)\), \(l\ge e(f)\), \(k\ge 0\), and we have

and the following conditions hold:

1. \(d_{r}(x)=x_\infty \),

2. \(d_{m}^{f,E_n, k}(x)=y\),

3. \(d_{l}^{f,E_\infty , k}(x_\infty )=y_\infty \),

4. the differential in (1) has no crossing on the \(E_n\)-page or (2) has no crossing.

5. the differential in (3) has no crossing.

Then we have an Adams differential

Let \(f: X\to Y\) be a map between two classical spectra. Suppose that \(2\le n\le r\), \(e(f)\le m\le n-2+e(f)\), \(l\ge e(f)\), \(k\ge 0\), and we have

and the following conditions hold:

1. \(d_{r}(x)=x_\infty \),

2. \(d_{m}^{f,E_n, k}(x)=y\),

3. \(d_{r+l-m}(y)= y_\infty \),

4. the differential in (1) has no crossing on the \(E_n\)-page or (2) has no crossing,

5. the differential in (3) has no crossing.

Then we have an \((f,E_\infty )\)-extension

Consider a distinguished triangle of spectra

with \(e(f)+e(g)+e(h)=1\). Suppose that \(r=n+m+l\), \(n_1=n-e(f)\ge 1\), \(m_1=m-e(g)\ge 0\), \(l_1=l-e(h)\ge 0\), and

such that

1. \(d^{h,E_{r'}}_{l}\bar x=x\), where \(r'=r-m_1=n_1+l_1+1\),

2. \(d_r \bar x=\bar y\),

3. the \((h,E_{r'})\)-extension in (1) has no crossing, or the Adams differential (2) has no crossing on the \(E_{r'}\)-page,

4. \(d^{g,E_{m_1+2}}_{m}y=\bar y\),

5. for \(0 \le i \le n-1\), the Adams \(E_\infty \)-page of \(Y\) vanishes at the positions that are potential targets of \((f,E_r)\)-extensions of degree \(i\) from \(x\): \(E_\infty ^{s+l+i, t+l+i-1}(Y) = 0\).

Then we have \(x\in Z_{n+m+e(h)}^{s+l,t+l-1}(X)\) and

(See Figure 4.)