Dependencies

From Table 9 in the Appendix, we have

Therefore, the element \(x_{126,6} \in \mathrm{Ext}_A^{6, 126+6}\) cannot kill \(h_1h_4x_{109,12}\).

From Axiom 7.6(2) and the rigidity Axioms 0.119 and 0.120 for the synthetic Adams spectral sequence for \(S^{0,0}\), the right side of the equation in statement \((5')\) is nonzero if and only if statement \((3)\) is true.

The statement in Axiom 7.3(1) was originally stated as

in [ . Upon inspection, \(\pi _{125,125+5} S^{0,0}\) doesn’t contain any \(\lambda \)-torsion classes, so this is equivalent to the version we stated, which is more consistent with the statement in part (2).

Let \(\theta _5 = [h_5^2]\) represent any synthetic homotopy class in \(\pi _{62,62+2}S^{0,0}\) detected by \(h_5^2\) in the Adams \(E_2\)-page. For convenience, we use the same notation, \(\theta _5\), to denote its image in \(\pi _{62,62+2}S^{0,0}/\lambda ^r\) via the map \(S^{0,0} \rightarrow S^{0,0}/\lambda ^r\) for all \(r \ge 1\). Similarly, let \(\eta = [h_1] \in \pi _{1,1+1}S^{0,0}\).

Assuming that both statements \((3)\) and \((5')\) in Proposition 7.8 are true, we have a relation: For any homotopy class \([\lambda ^4 h_0^2x_{125,9,2}]\),

for some \([h_1h_4x_{109,12}]\).

For \(x,y,\delta \) in Proposition 4.5 we always have

if \(0\le a\le n\) and \(0\le a+r-n-1{\lt}m-n\) (the differential is trivial if \(a\) exceeds this range).

The element \(h_0^2x_{125,9,2}\) survives to the Adams \(E_5\)-page, and is not killed by any classical differential.

The element \(h_1x_{121,7}\) survives to the Adams \(E_6\)-page, and is not killed by any classical differential.

2. The element \(h_6 M d_0\) survives to the Adams \(E_\infty \)-page.

3. The element \(h_5x_{91,11}\) survives to the Adams \(E_\infty \)-page.

1. \(x_{126,8,4} + x_{126,8}\) survives to the \(E_6\)-page.

2. \(h_1h_4x_{109,12}\) is a permanent cycle, and can only be killed by

   \[ d_6(x_{126,8,4} + x_{126,8}) \ \text{or} \ d_{12}(h_6^2). \]

3. \(h_0^2x_{124,8}\) survives to the \(E_\infty \)-page.

4. In \(\mathrm{Ext}_A^{25,125+25}\), the element \(g^4\Delta h_1g\) is the only one that survives to the classical \(E_5\)-page.

1. \(x_{123,9}+h_0x_{123,8}\) survives to the Adams \(E_{12}\)-page, and is not killed by any classical differential.

2. We have a classical nonzero differential

   \[ d_2 (x_{125,8}) = h_1 (x_{123,9}+h_0x_{123,8}) + h_0^2x_{124,8}. \]

The statement \((4)\) in Proposition 7.8 is equivalent to the following statement \((4')\):

1. For every \(\theta _5\), we have \(\theta _5^2\) is detected by \(\lambda ^6 h_0^2x_{124,8}\). In particular,

   \[ \theta _5^2 = \lambda ^6 [h_0^2x_{124,8}] \neq 0 \in \pi _{124,124+4}S^{0,0} \]

   for some \([h_0^2x_{124,8}]\).

The statement \((5)\) in Proposition 7.8 is equivalent to the following statement \((5')\):

1. For every homotopy class \([h_0^2x_{124,8}]\), we have \(\lambda ^3 \eta [h_0^2x_{124,8}]\) is detected by \(\lambda ^6 h_1h_4x_{109,12}\). In particular, we have

   \[ \lambda ^3 \eta [h_0^2x_{124,8}] = \lambda ^6 [h_1h_4x_{109,12}] \in \pi _{125,125+8} S^{0,0} \]

   for some \([h_1h_4x_{109,12}]\).

There exists a homotopy class \([\lambda ^4 h_1 x_{121,7}] \in \pi _{122,122+4}S^{0,0}/\lambda ^9\), such that

Assuming that both statements \((3)\) and \((5')\) in Proposition 7.8 are true, the synthetic Toda bracket

does not contain zero, and is detected by \(\lambda ^4 h_0^2x_{125,9,2}\). Here \(\alpha _1 = [x_{123,9}+h_0x_{123,8}]\) refers to the homotopy class described in Lemma 7.14.

There exists a homotopy class

with the following properties:

1. For any homotopy class \([h_0^2 x_{124,8}]\), there exist homotopy classes

   \[ \alpha _2 \in \pi _{124,124+13} S^{0,0}/\lambda ^9, \ \alpha _3 \in \pi _{125,125+15} S^{0,0}/\lambda ^9, \]

   such that

   \begin{align*} \lambda ^3 \eta \cdot \alpha _1 & = \lambda ^3 [h_0^2 x_{124,8}] + \lambda ^6 \alpha _2 & & \in \pi _{124,124+7} S^{0,0}/\lambda ^9,\\ \eta \cdot \alpha _2 & = \lambda \cdot \alpha _3 & & \in \pi _{125,125+14} S^{0,0}/\lambda ^9, \end{align*}
2. \[ \lambda ^3 \cdot \alpha _1 \cdot [h_0] = 0 \in \pi _{123,123+7} S^{0,0}/\lambda ^9. \]

Let \(X \in \mathcal{S}^{\mathrm{fin}}\) be a finite spectrum. The synthetic Adams spectral sequence for \(\nu X\) is isomorphic to the \(\lambda \)-Bockstein spectral sequence:

1. \(E_2^{s,t}(X) \cong \pi _{t-s,t}(\nu X / \lambda )\).

2. If there is a classical Adams differential \(d_r x = y\) for \(x \in E_2^{s,t}(X) \cong \pi _{t-s,t}(\nu X / \lambda )\), then \(x\) admits a lift to \(\pi _{t-s,t}(\nu X / \lambda ^{r-1})\) whose image under the Bockstein \(\nu X / \lambda ^{r-1} \to \Sigma ^{1,-r+1} \nu X / \lambda \) equals \(d_r(x)\).

Let \(X \in \mathcal{S}^{\mathrm{fin}}\) be a finite spectrum. The synthetic Adams spectral sequence for \(\nu X\) has \(E_2\)-page

where an element in \(E_2^{s,t}(X)\) has tridegree \((s,t,t)\) and \(\lambda \) has tridegree \((0,0,-1)\). Given a classical Adams differential \(d_r^{\mathrm{cl}}(x) = y\), the corresponding synthetic differential is \(d_r(x) = \lambda ^{r-1}y\), which is \(\lambda \)-linear, and all synthetic Adams differentials arise in this way.

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

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

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

Exactly one of the following two statements is true:

1. The element \(h_6^2\) survives to the \(E_\infty \)-page.

2. There is a nonzero classical Adams differential

   \[ d_{12}(h_6^2) = h_1h_4x_{109,12}. \]

Furthermore, statement \((2)\) is true if and only if the following three statements are all true:

1. \[ d_6(x_{126,8,4} + x_{126,8}) = 0. \]

2. There exists a \(\theta _5\) such that \(\theta _5^2\) is detected by \(\lambda ^6 h_0^2x_{124,8}\). In particular,

   \[ \theta _5^2 = \lambda ^6 [h_0^2x_{124,8}] \neq 0 \in \pi _{124,124+4}S^{0,0} \]

   for some \([h_0^2x_{124,8}]\).

3. There exists a homotopy class \([h_0^2x_{124,8}]\) such that \(\lambda ^3 \eta [h_0^2x_{124,8}]\) is detected by \(\lambda ^6 h_1h_4x_{109,12}\). In particular, we have

   \[ \lambda ^3 \eta [h_0^2x_{124,8}] = \lambda ^6 [h_1h_4x_{109,12}] \in \pi _{125,125+8} S^{0,0} \]

   for some \([h_1h_4x_{109,12}]\).

If statement \((3)\) is true, then statement \((5)\) in Proposition 7.8 must be false.

According to Axiom 7.15, it is not yet known whether the element \(h_0^2x_{125,9,2}\) supports a nonzero \(d_5\)-differential. Assuming that both statements \((3)\) and \((5')\) in Proposition 7.8 are true, Lemma 7.16 specifically implies that \(\lambda ^4 h_0^2x_{125,9,2}\) detects a nonzero homotopy class in \(\pi _{125,125+7} S^{0,0}/\lambda ^9\). Therefore, under these assumptions, we would have \(d_5(h_0^2x_{125,9,2}) = 0\).

Axiom 7.3 is proved using the quadratic construction on a map from the mod 2 Moore spectrum to the sphere spectrum, where the restriction on the bottom cell is \(\theta _5\). Therefore, it is necessary to use a \(\theta _5\) of order 2.

Notably, [ confirms that all classical \(\theta _5\)’s indeed have order 2. Furthermore, from Proposition 0.121 and an analysis of the differentials in the classical Adams spectral sequence, we find that \(\pi _{62,62+2}S^{0,0}\) doesn’t contain any \(\lambda \)-torsion. Consequently, all synthetic \(\theta _5\)’s also have order 2, making it valid to apply Axiom 7.3 to any \(\theta _5\).

Additionally, since the proof of Axiom 7.3 shows that the expression \(\lambda \eta \theta _5^2\) corresponds to the total differential \(\delta _1: S^{0,0}/\lambda \rightarrow S^{1,-1}\) on \(h_6^2\), the value of the expression \(\lambda \eta \theta _5^2\) is consistent for every choice of \(\theta _5\). (Note that our grading for the triangulation translation functor is smashing with \(S^{1,0}\), which is consistent with [ but is different from [ , so the target of \(\delta _1\) is \(S^{1,-1}\). )

The element \(h_6^2\) survives to the \(E_\infty \)-page in the Adams spectral sequence.

2. The element \(h_6^2\) survives to the \(E_{r+3}\)-page of the classical Adams spectral sequence if and only if for some \(\theta _5\),

   \[ \lambda \eta \theta _5^2 = 0 \ \text{in} \ \pi _{125,125+4} S^{0,0}/\lambda ^{r+1}. \]

3. In particular, \(h_6^2\) is a permanent cycle in the classical Adams spectral sequence if and only if for some \(\theta _5\),

   \[ \lambda \eta \theta _5^2 = 0 \ \text{in} \ \pi _{125,125+4} S^{0,0}. \]

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

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.)