Existence of $b_{0}b_1g_{0}\widetilde{\gamma }_{s}$-element in the stable homotopy of spheres

Abstract

Let $p$ be a prime greater than five and $A$ be the mod $p$ Steenrod algebra. In this paper, we show that the composite map $\beta _1\varphi _s$ is nontrivial in the stable homotopy of spheres $\pi _{2(p-1)[(s+1)p^2+(s+1)p+s]-9}(S)$ , where $4\le s<p$ and $\varphi _s$ is represented by $b_1g_{0}\widetilde{\gamma }_{s}\in \mathrm{Ext}_A^{s+4,2(p-1)[(s+1)p^2+sp+s]+s-3}(\mathbb Z /p, \mathbb Z /p)$ in the Adams spectral sequence.

1 Introduction and statement of the main results

Let $A$ be the mod $p$ Steenrod algebra and $S$ be the sphere spectrum localized at an arbitrary odd prime $p$. To determine the stable homotopy groups of spheres $\pi _{*}(S)$ is one of the central problems in homotopy theory. One of the main tools to reach it is the Adams spectral sequence (see [1]):

$$\begin{aligned} E_2^{s,t}=\mathrm{Ext}_A^{s,t}(\mathbb Z /p, \mathbb Z /p)\Rightarrow \pi _{t-s}(S), \end{aligned}$$

where the $E_2$-term is the cohomology of $A$.

Throughout this paper, we fix $q=2(p-1)$. The known results on $\mathrm{Ext}_A^{*,*}(\mathbb Z /p, \mathbb Z /p)$ are as follows: $\mathrm{Ext}_A^{0,*}(\mathbb Z /p, \mathbb Z /p)=\mathbb Z /p$ by its definition. From [6], we have that for odd prime $p$ $\mathrm{Ext}_A^{1,*}(\mathbb Z /p, \mathbb Z /p)$ has $\mathbb Z /p$-basis consisting of $a_0\in \mathrm{Ext}_A^{1,1}(\mathbb Z /p, \mathbb Z /p)$ and $h_i\in \mathrm{Ext}_A^{1,p^iq}(\mathbb Z /p, \mathbb Z /p)$ for all $i\ge 0$ and $\mathrm{Ext}_A^{2,*}(\mathbb Z /p, \mathbb Z /p)$ has $\mathbb Z /p$-basis consisting of $\alpha _2$, $a_0^2$, $a_0h_i$ $(i>0)$, $g_i$ $(i\ge 0)$, $k_i$ $(i\ge 0)$, $b_i$ $(i\ge 0)$, and $h_ih_j$ $(j\ge i+2,i\ge 0)$ whose internal degrees are $2q+1$, $2$, $p^iq+1, p^{i+1}q+2p^iq$, $2p^{i+1}q+p^iq$, $p^{i+1}q$ and $p^iq+p^jq$, respectively. Aikawa [2] determined $\mathrm{Ext}_A^{3,*}(\mathbb Z /p,\mathbb Z /p)$ by virtue of the lambda algebra.

Let $M$ be the mod $p$ Moore spectrum given by the following cofibration

$$\begin{aligned} S\stackrel{p}{\longrightarrow }S\stackrel{i}{\longrightarrow }M\stackrel{j}{\longrightarrow }\Sigma S. \end{aligned}$$

(1.1)

Let $\alpha :\Sigma ^qM\longrightarrow M$ be the known Adams map and $K$ be its cofibre given by the cofibration

$$\begin{aligned} \Sigma ^qM\stackrel{\alpha }{\longrightarrow }M\stackrel{i^{\prime }}{\longrightarrow }K \stackrel{j^{\prime }}{\longrightarrow }\Sigma ^{q+1}M. \end{aligned}$$

(1.2)

This spectrum which we briefly write as $K$ is known to be the Smith-Toda spectrum $V(1)$. For $p>3$, Smith [9] showed that there exists a periodic map

$$\begin{aligned} \beta :\Sigma ^{(p+1)q}K\longrightarrow K \end{aligned}$$

which induces multiplication by $v_2$ in $K(2)$-theory. Let $V(2)$ be the cofibre of $\beta :\Sigma ^{(p+1)q}K\longrightarrow K$ given by the cofibration

$$\begin{aligned} \Sigma ^{(p+1)q}K\stackrel{\beta }{\longrightarrow }K\stackrel{\bar{i}}{\longrightarrow }V(2)\stackrel{\bar{j}}{\longrightarrow }\Sigma ^{(p+1)q+1}K. \end{aligned}$$

(1.3)

For $p\ge 7$, there exists the Smith-Toda map $\gamma :\Sigma ^{q(p^2+p+1)}V(2)\longrightarrow V(2)$ (see [10]).

Definition 1.1

For $s\ge 1$, the $\beta$-element $\beta _{s}$ is defined to be the composite map

$$\begin{aligned} jj^{\prime }\beta ^{s}i^{\prime }i\in \pi _{q[sp+(s-1)]-2}(S) \end{aligned}$$

and the $\gamma$-element $\gamma _{s}$ is defined to be the composite map

$$\begin{aligned} jj^{\prime }\bar{j}\gamma ^{s}\bar{i}i^{\prime }i\in \pi _{q[sp^{2}+(s-1)p+(s-2)]-3}(S). \end{aligned}$$

Theorem 1.2

• (1) ([9]) For $p\ge 5$ and $s\ge 1, \beta _{s}\ne 0$ in $\pi _{*}(S)$.

• (2) ([7]) For $p\ge 7$ and $s\ge 1, \gamma _{s}\ne 0$ in $\pi _{*}(S)$.

So far, not so many families of homotopy elements in $\pi _{*}S$ have been detected. For example, Cohen [3] detected a new element $\zeta _n\in \pi _{q(p^{n+1}+1)-3}(S)$ for $n\ge 1$, which is represented by $h_0b_n\in \mathrm{Ext}_A^{3,q(p^{n+1}+1)}(\mathbb Z /p , \mathbb Z /p)$ in the Adams spectral sequence.

Liu [5] also detected an infinite family of homotopy elements in the stable homotopy groups of spheres and obtained the following

Theorem 1.3

([5, Theorem 1.1]) Let $p\ge 7$ and $3\le s<p$. Then the product element

$$\begin{aligned} b_1g_0\widetilde{\gamma }_{s}\in \mathrm{Ext}_A^{s+4,*}(\mathbb Z /p, \mathbb Z /p) \end{aligned}$$

is a permanent cycle in the Adams spectral sequence and converges nontrivially to a homotopy element $\varphi _s\in \pi _{*}(S)$.

In this paper, we consider the composite element $\beta _1\varphi _s$ and show its nontriviality under some conditions. The main result can be stated as follows:

Theorem 1.4

Let $p\ge 7$ and $4\le s<p$. Then the composite map $\beta _1\varphi _s$ is nontrivial in $\pi _{*}(S)$.

The paper is arranged as follows: after recalling some knowledge on the May spectral sequence in Sect. 2, we make use of the May spectral sequence and the Adams spectral sequence to prove Theorem 1.4 in Sect. 3.

2 Preliminaries on the May spectral sequence

The May spectral sequence is one of our main tools in this paper. For the sake of completeness, we briefly recall some knowledge on the May spectral sequence.

From [8], there is the May spectral sequence $\{E_{r}^{s,t,*},d_{r}\}$ which converges to $\mathrm{Ext}_{A}^{s,t}(\mathbb Z /{p},\mathbb Z /{p})$ with $E_{1}$-term

$$\begin{aligned} E_1^{*,*,*}=E(h_{m,i}|m>0,i\ge 0)\otimes P(b_{m,i}|m>0, i\ge 0)\otimes P(a_n|n\ge 0), \end{aligned}$$

(2.1)

where $E$ is the exterior algebra, $P$ is the polynomial algebra, and

$$\begin{aligned} h_{m,i}\in E_1^{1,2(p^m-1)p^i,2m-1}, b_{m,i}\in E_1^{2,2(p^m-1)p^{i+1},p(2m-1)}, a_n\in E_1^{1,2p^n-1,2n+1}.\nonumber \\ \end{aligned}$$

(2.2)

One has

$$\begin{aligned} d_r:E_r^{s,t,u}\longrightarrow E_r^{s+1,t,u-r} \end{aligned}$$

(2.3)

for $r\ge 1$ and if $x\in E_r^{s,t,*}$ and $y\in E_r^{s^{\prime },t^{\prime },*}$, then

$$\begin{aligned} d_r(x\cdot y)=d_r(x)\cdot y+(-1)^sx\cdot d_r(y). \end{aligned}$$

(2.4)

There also exists a graded commutativity in the May spectral sequence as follows:

$$\begin{aligned} x\cdot y = (-1)^{ss^{\prime }+tt^{\prime }} y\cdot x \end{aligned}$$

for $x,y =h_{i,j},b_{k,l},$ or $a_n$. The first May differential $d_1$ is given by

$$\begin{aligned} \left\{ \begin{array}{l} d_1(h_{i,j})=\displaystyle \sum \limits _{0<k<i}^{}h_{i-k,k+j}h_{k,j},\\ d_1(a_i)=\displaystyle \sum \limits _{0\le k<i}h_{i-k,k}a_k,\\ d_1(b_{i,j})=0. \end{array} \right. \end{aligned}$$

(2.5)

For convenience, we define $\text{ dim}~x=s$, $\text{ deg}~x=t$ and $\mathrm{M}(x)=u$ for any element $x\in E_{1}^{s,t,u}$.

For each integer $t\ge 0$, it can be always expressed uniquely as

$$\begin{aligned} t=q(c_np^n+c_{n-1}p^{n-1}+\cdots +c_1p+c_0)+e, \end{aligned}$$

where $0\le c_i<p$ ($0\le i\le n$), $c_{n}>0$, $0\le e<q$. Suppose $g=x_1x_2\ldots x_m\in E_1^{\bar{s},t,*}$, where $m\le \bar{s}$, $x_{i}$ is one of $a_{k}$,$h_{l,j}$ or $b_{u,z}$, $0\le k\le n+1$, $0\le l+j\le n+1$, $0\le u+z\le n$, $l>0$, $j\ge 0$, $u>0$, $z\ge 0$. We also suppose $\text{ deg}~x_i=q(c_{i,n}p^n+c_{i,n-1}p^{n-1}+\cdots +c_{i,0})+e_i$, where $c_{i,j}=0$ or $1$, $e_i=1$ if $x_i=a_{k_i}$, or $e_i=0$. Then we have $\text{ dim}~g=\sum \nolimits _{i=1}^{m}\text{ dim}~x_i=\bar{s}$ and

$$\begin{aligned} \text{ deg}~g=t&= \left(\left(\sum \limits _{i=1}^{m}c_{i,n}\right)p^{n}+\cdots +\left(\sum \limits _{i=1}^{m}c_{i,2}\right)p^2+ \left(\sum \limits _{i=1}^{m}c_{i,1}\right)p+\left(\sum \limits _{i=1}^{m}c_{i,0}\right)\right) \\&+\left(\sum \limits _{i=1}^{m}e_i\right). \end{aligned}$$

Lemma 2.1

If $\left(\sum \nolimits _{i=1}^{m}c_{i,0}\right)-\left(\sum \nolimits _{i=1}^{m}e_i\right)=v$, then there exist integers $i_{v}>i_{v-1}>\cdots >i_1\ge 1$ and an element $\hat{g}$ such that $g=h_{i_v, 0}h_{i_{v-1},0}\ldots h_{i_1,0}\hat{g}$ up to sign.

Proof

By use of (2.2), there must exist $v$ elements of the form $h_{\mu ,0}$ at least among $g$. Note that $h_{i,0}^{2}=0$ for $i\ge 1$. Thus, there exist integers $i_k$ ($0<k<v$) and an element $\hat{g}$ such that $i_{v}>i_{v-1}>\cdots >i_1\ge 1$ and $g=h_{i_v,0}h_{i_{v-1},0}\ldots h_{i_1,0}\hat{g}$ up to sign. The lemma follows. $\square$

3 Proof of Theorem 1.4

Before showing our main theorem, we first give some important results on the May $E_r$-term ($r\ge 1$) and the Adams $E_2$-term.

The following Representation Theorem is due to X. Liu.

Lemma 3.1

([4, Theorem 1.1]) Let $p\ge 7$, $3\le s<p$. Then the element

$$\begin{aligned} a_3^{s-3}h_{3,0}h_{2,1}h_{1,2}\in E_1^{s,t(s),*} \end{aligned}$$

detects the third Greek letter element ${\tilde{\gamma }_{s}}\in \mathrm{Ext}_A^{s,t(s)}(\mathbb Z /p,\mathbb Z /p)$ in the May spectral sequence, where $t(s)=sp^2q+(s-1)p q+(s-2)q+s-3$ and ${\tilde{\gamma }_{s}}$ detects the $\gamma$-element $\gamma _{s}$ in the Adams spectral sequence.

The following lemma plays an important role in showing Theorem 1.4 and can be stated as follows:

Lemma 3.2

Let $p\ge 7$, $4\le s< p$ and $r\ge 1$. Then the May $E_1$-term

$$\begin{aligned} E_1^{s+6-r,t(s,r),*}=\left\{ \begin{array}{ll} G,\quad r=1,\\ 0,\quad r>1, \end{array} \right. \end{aligned}$$

where $t(s,r)=q[(s+1)p^2+(s+1)p+s]+s-r-2$ and $G$ is the $\mathbb Z /{p}$-module generated by $a_3^{s-3}h_{3,0}h_{2,0}h_{1,0}h_{2,1}b_{2,0}b_{1,1}, a_3^{s-3}h_{3,0}h_{2,0}h_{1,0}h_{1,2}b_{2,0}^2, a_3^{s-3}h_{3,0}h_{2,0}h_{1,0}h_{2,1}$ $h_{1,1}h_{1,2}b_{1,1} .$

Proof

When $r\ge s-1$, it is easy to check that that the May $E_1$-term

$$\begin{aligned} E_1^{s+6-r,t(s,r),*}=0. \end{aligned}$$

(3.1)

Therefore, we assume $1\le r\le s-2$ in the rest of the proof.

If $s=4$, then $r$ may equal $1$ or $2$ by $1\le r\le s-2$. We can show that

$$\begin{aligned} E_1^{9,t(4,1),*}&= \mathbb Z /p \{a_3h_{3,0}h_{2,0}h_{1,0}h_{2,1}b_{1,1}b_{2,0},a_3h_{3,0}h_{2,0}h_{1,0}h_{1,2}b_{2,0}^2, \\&\qquad \quad a_3h_{3,0}h_{2,0}h_{1,0}h_{2,1}h_{1,1}h_{1,2}b_{1,1}\} \end{aligned}$$

and

$$\begin{aligned} E_1^{8,t(4,2),*}=0 \end{aligned}$$

through easy computations. Thus, we assume $s\ge 5$ in the rest of the proof.

Case 1 $\;5\le s<p-1$. By replacing $\bar{s}$ and $t$ by $s+6-r$ and $t(s,r)$ in the argument given above Lemma 2.1 respectively, we have $\text{ dim}~g=\sum \nolimits _{i=1}^{m}\text{ dim}~x_i=s+6-r$ which implies that $m\le s+6-r\le s+5< p+4$. Thus, we have

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \sum \limits _{i=1}^{m}e_i=s-r-2,\quad&\displaystyle \sum \limits _{i=1}^{m}c_{i,0}=s,\\ \displaystyle \sum \limits _{i=1}^{m}c_{i,1}=s+1,\quad&\displaystyle \sum \limits _{i=1}^{m}c_{i,2}=s+1.\end{array}\right. \end{aligned}$$

(3.2)

Subcase 1.1 $\;2\le r\le s-2$. Note that

$$\begin{aligned} \sum \limits _{i=1}^{m}c_{i,0}-\sum \limits _{i=1}^{m}e_i=s-(s-r-2)=r+2>2+1. \end{aligned}$$

By Lemma 2.1, we have

$$\begin{aligned} E_1^{s+6-r,t(s,r),*}=0. \end{aligned}$$

Subcase 1.2 $\;r=1$. Note that in this case $\sum \nolimits _{i=1}^{m}c_{i,0}-\sum \nolimits _{i=1}^{m}e_{i}=s-(s-3)=3$. From Lemma 2.1, $g=\hat{g}h_{1,0}h_{2,0}h_{3,0}$ up to sign if $g$ exists, where $\hat{g}=x_1\ldots x_{m-3}$. For the above $\hat{g}$, we have:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \sum \limits _{i=1}^{m-3}e_i=s-3,\\ \displaystyle \sum \limits _{i=1}^{m-3}c_{i,0}=s-3,\\ \displaystyle \sum \limits _{i=1}^{m-3}c_{i,1}=s-1,\\ \displaystyle \sum \limits _{i=1}^{m-3}c_{i,2}=s.\\ \end{array} \right. \end{aligned}$$

(3.3)

Note that$\;m-3\le s+2$ and $\sum \nolimits _{i=1}^{m-3}c_{i,2}+\sum \nolimits _{i=1}^{m-3}e_{i}-(m-3)\ge s+(s-3)-(s+2)=s-5$. By (2.2) and degree reason, $\hat{g}$ must be of the form $\hat{g}=a_3^{s-5}\hat{\hat{g}}$ up to sign, where $\hat{\hat{g}}=x_{s-4}\ldots x_{m-3}\in E_1^{7,q(5p^2+4p+2)+2,*}$. By (2.1) and (2.2), we easily get that

$$\begin{aligned} E_1^{7,q(5p^2+4p+2)+2,*}&= \mathbb Z /p \{a_3^2 h_{2,1}b_{2,0}b_{1,1},a_3^2h_{1,2}b_{2,0}^2, a_3^2h_{2,1}h_{1,1}h_{1,2}b_{1,1},\\&\qquad \quad a_3a_0h_{3,0}h_{2,1}h_{1,2}b_{2,0}\}. \end{aligned}$$

Thus, we have that up to sign $g=a_3^{s-3}h_{3,0}h_{2,0}h_{1,0}h_{2,1}b_{2,0}b_{1,1}$, $a_3^{s-3}h_{3,0}h_{2,0}h_{1,0}h_{1,2}$ $b_{2,0}^2, a_3^{s-3}h_{3,0}h_{2,0}h_{1,0}h_{2,1}h_{1,1}h_{1,2}b_{1,1}$, $a_3^{s-4}a_0h_{3,0}^2h_{2,0}h_{1,0}h_{2,1}h_{1,2}b_{2,0}$ in which the last one is trivial by $h_{3,0}^2=0$.

Case 2 $\;s=p-1.$ By replacing $\bar{s}$ and $t$ by $p+5-r$ and $t(p-1,r)$ in the argument given above Lemma 2.1 respectively, we have $\text{ dim}~g=\sum \nolimits _{i=1}^{m}\text{ dim}~x_i=p+5-r$ which implies that $m\le p+5-r\le p+4$. Thus, we have:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \sum \limits _{i=1}^{m}e_i=p-3-r,\\ \displaystyle \sum \limits _{i=1}^{m}c_{i,0}=p-1,\\ \displaystyle \sum \limits _{i=1}^{m}c_{i,1}=\lambda _1 p,~~\lambda _1\ge 0,\\ \displaystyle \sum \limits _{i=1}^{m}c_{i,2}+\lambda _1=1+\lambda _2 p,~~\lambda _2\ge 0,\\ \displaystyle \sum \limits _{i=1}^{m}c_{i,3}+\lambda _2=1. \end{array} \right. \end{aligned}$$

(3.4)

From $\sum \nolimits _{i=1}^{m}c_{i,1}=\lambda _1 p$ and $m\le p+4$, we have $\lambda _1=0$ or $1$.

Subcase 2.1 $\;\lambda _1=0$. From $\sum \nolimits _{i=1}^{m}c_{i,0}-\sum \nolimits _{i=1}^{m}e_{i}=p-1-(p-3-r)=r+2$ and Lemma 2.1, we have that there would be $r+2$ factors of the form $h_{i,0}$ at least among $g$ if $g$ exists. Note that $\sum \nolimits _{i=1}^{m}c_{i,1}=0$ and $h_{i,0}^2$=0 for any $i\ge 1$. From (2.2), it follows that $g$ is impossible to exist in this case.

Subcase 2.2 $\;\lambda _1=1$. From $\sum \nolimits _{i=1}^{m}c_{i,2}=\lambda _2 p$, we have that $\lambda _2=0$ or $1$ by $m\le p+4$.

If $\lambda _2=0$, from (3.4) and (2.2) we can deduce that there must exist a factor $h_{1,3}$ or $b_{1,2}$ among $g$ if $g$ exists. Thus we can write $g$ as $g=x_1x_2\ldots x_{m-1}\bar{g}$, where $\bar{g}$=$h_{1,3}$ or $b_{1,2}$. Let $g_1=x_1x_2\ldots x_{m-1}$. By Lemma 2.1, $g_1$ is impossible to exist. Thus, $g$ is impossible to exist. If $\lambda _2=1$, by an argument similar to that used in Case 1, we have that, $g$ exists only when $r=1$, and up to sign $g=a_3^{p-4}h_{3,0}h_{2,0}h_{1,0}h_{2,1}b_{2,0}b_{1,1}$, $a_3^{p-4}h_{3,0}h_{2,0}h_{1,0}h_{1,2}b_{2,0}^2$, $a_3^{p-4}h_{3,0}h_{2,0}h_{1,0}h_{2,1}h_{1,1}h_{1,2}b_{1,1}$.

The Proof of Lemma 3.2 is completed. $\square$

From the above lemma, we easily have the following:

Theorem 3.3

Let $p\ge 7$, $4\le s< p$, $r\ge 2$, then we have

$$\begin{aligned} \mathrm{Ext}_A^{s+6-r,t(s,r)}(\mathbb Z /{p},\mathbb Z /{p})=0, \end{aligned}$$

where $t(s,r)=q[(s+1)p^2+(s+1)p+s]+s-r-2$.

Theorem 3.4

Let $p\ge 7$, $4\le s< p$. Then the product element $b_{0}b_1g_{0}\widetilde{\gamma }_{s}$ is nontrivial in the Adams $E_2$-term.

Proof

It is known that $b_{1,i}$ and $h_{2,0}h_{1,0}$ are permanent cocycles in the May spectral sequence and converge to $b_i$ and $g_0$ for $i\ge 1$, respectively. From Lemma 3.1, $\widetilde{\gamma }_{s}$ is represented by $a_3^{s-3}h_{3,0}h_{2,1}h_{1,2}$ in the May spectral sequence. Thus,

$$\begin{aligned} b_{0}b_1g_{0}\widetilde{\gamma }_{s}\in \mathrm{Ext}_A^{s+6,*}(\mathbb Z /{p},\mathbb Z /{p}) \end{aligned}$$

is represented by

$$\begin{aligned} b_{1,0}b_{1,1}h_{2,0}h_{1,0}a_3^{s-3}h_{3,0}h_{2,1}h_{1,2} \end{aligned}$$

for $s\ge 4$ in the May spectral sequence. Now we show that nothing can hit the permanent cocycle $b_{1,0}b_{1,1}h_{2,0}h_{1,0}a_3^{s-3}h_{3,0}h_{2,1}h_{1,2}$ under the May differential $d_r$ for $r\ge 1$.

Denote the generators $b_{1,0}b_{1,1}h_{2,0}h_{1,0}a_3^{s-3}h_{3,0}h_{2,1}h_{1,2}$, $a_3^{s-3}h_{3,0}h_{2,0}h_{1,0}h_{2,1}b_{2,0}$ $b_{1,1}, a_3^{s-3}h_{3,0}h_{2,0}h_{1,0}h_{1,2}b_{2,0}^2$ and $a_3^{s-3}h_{3,0}h_{2,0}h_{1,0}h_{2,1}h_{1,1}h_{1,2}b_{1,1}$ by $\mathcal G$, $\mathcal G _1$, $\mathcal G _2$ and $\mathcal G _3$, respectively. Since $\mathrm{M}(\mathcal G )=2p+7s-8>p+7s-7=\mathrm{M} (\mathcal G _3)$, we have that in the May spectral sequence $d_r(\mathcal G _3)\not =\mathcal G$ for any $r\ge 1$ up to a nonzero scalar. For the others, the possibilities are $d_{2p-1}(\mathcal G _1)=\mathcal G$ and $d_{4p-3}(\mathcal G _2)=\mathcal G$ up to a nonzero scalar. Note that up to sign $d_1(\mathcal G _1)=a_3^{s-3}h_{3,0}h_{2,0}h_{1,0}h_{1,1}h_{1,2}b_{2,0}b_{1,1}+\cdots \not =0$ and $d_1(\mathcal G _2)=a_3^{s-4}a_1h_{3,0}h_{2,0}h_{1,0}h_{2,1}h_{1,2}b_{2,0}^2+\cdots \not =0$. Thus, $\mathcal G _1$ and $\mathcal G _2$ both die at the May $E_2$-term. From the above discussion, we have that nothing can hit $\mathcal G$ under the May differential $d_r$ for $r\ge 1$. This completes the Proof of Theorem 3.4. $\square$

Now we are in a position to complete the Proof of Theorem 1.4.

Proof of Theorem 1.4

It is known that $b_0\in \mathrm{Ext}_A^{2,pq}(\mathbb Z /p,\mathbb Z /p)$ is a permanent cycle in the spectral sequence and converges nontrivially to the $\beta$-family $\beta _1\in \pi _{pq-2}(S)$. Moreover, from Theorem 1.3 we have that $b_1g_{0}\widetilde{\gamma }_{s}\in \mathrm{Ext}_A^{s+4,*}(\mathbb Z /p,\mathbb Z /p)$ detects the homotopy element $\varphi _s\in \pi _{*}(S)$ in the Adams spectral sequence. Thus, the composite map

$$\begin{aligned} \beta _1\circ \varphi _s \end{aligned}$$

is represented up to a nonzero scalar by

$$\begin{aligned} b_0b_1g_{0}\widetilde{\gamma }_{s} \end{aligned}$$

in the Adams spectral sequence. From Theorem 3.4, the product element $b_0b_1g_{0}\widetilde{\gamma }_{s}$ is nontrivial in the Adams $E_2$-term. Meanwhile, from Theorem 3.3, $b_0b_1g_{0}\widetilde{\gamma }_{s}$ does not bound in the Adams spectral sequence. Consequently, $b_0b_1g_{0}\widetilde{\gamma }_{s}$ is a permanent cycle in the Adams spectral sequence and converges nontrivially to the homotopy element $\beta _1\circ \varphi _s$. This finishes the Proof of Theorem 1.4. $\square$