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On the sphere of origin of infinite families in the homotopy groups of spheres. (English) Zbl 0176.52502
The object of this paper is to prove some general theorems on the existence of families of elements in the homotopy groups of spheres, and deduce from these a best possible result on the sphere of origin of the elements with nontrivial \(e\)-invariant in the \(p\)-primary component for \(p>2\). Recall that \(e\colon G_r \to \mathbb Q/\mathbb Z\), where \(G_r\) is the stable \(r\)-stem and that \(G_r \approx (\text{Im}\, J)\oplus(\ker e)\), for \(p>2\). We conjecture that the elements constructed are in the image of the stable \(J\)-homomorphism. This is true mod \(p\).
Let \(\{A,B\}\) denote the set of stable homotopy classes of maps from \(A\) to \(B\). If \(\alpha\in\{A,B_q\}\) where \(B_q\) is a Moore space of homological dimension \(q\), we write \(\alpha_t\) for any desuspension of \(\alpha\) to a map \(\alpha_t\colon S^{t-q}A\to B_t\). If such an \(\alpha_t\) exists we will say that \(\alpha\) exists on \(B_t\) If \(\alpha\in\{B_p,B_q\}\) write \(\vert\alpha\vert = p-q\). Throughout this paper, \(\alpha\colon S^p\to S^q\) will be a fixed map and we will pick a fixed homotopy \(H\colon m\alpha\simeq 0\) where the order of \(\alpha\) divides \(m\). We assume \(m\) and \(\vert\alpha\vert = p-q\) are both odd.
Theorem 5.4. With the above assumptions there exist maps \(\alpha(r)\colon S^{r(\vert\alpha\vert +1)+q-1} \to S^q\) such that \[ (1)\quad m\alpha\simeq 0,\quad (2)\quad \alpha(1) = \alpha,\quad (3)\quad (r+s)\alpha(r)_{2q}\alpha(s)\simeq 0, \] \[ (4)\quad \alpha(r+s)_q\in\{\alpha(r)_q, m\iota, \alpha(s)\},\quad (5) \quad r \alpha(r+s)_{2q}\in \{(r+s)\alpha(r)_{2q},\alpha(s),m\iota\}.\]
Theorem 5.8. With the above assumptions, suppose \(\alpha_{q+1} =n\beta\) where \(n\) is odd. Choose a homotopy \(\bar E: mn\beta\simeq 0\) and define \(\beta(s)\) as above. Then there exists \(\gamma\colon S^t\to S^{2q+k}\) such that \(m \gamma=\beta(m)_{2q+k}\), where \(t=m(\vert\alpha\vert+1)+2q+k-1\). Furthermore, \(\pm\gamma\) belongs to the Toda bracket:
\[\left\{ \alpha_{2q+k}, \begin{pmatrix} 2\beta \\ -m\iota\end{pmatrix}, (mn\iota, \alpha), \beta(m-2)\right\}. \]
Theorem 6.1. In addition to the above assumptions assume \(q\) is odd. If \(m\equiv 3\pmod 9\) assume \((S^2\alpha)\alpha_1 =0\) where \(\alpha_1\in \pi_{p+5}(S^{p+2})\) is any element of order 3. Then there exist elements \(\alpha^{(t)}\) in the \(m^t(\vert\alpha\vert+ 1) -1\) stem on \(S^{q+t (q-1)}\) and extensions \(\dot\alpha^{(t)}\colon Y_{m^{t+1}}\to S^{q+1(q-1)}\) such that (a) \(\alpha^{(0)} =\alpha\), (b) \(m\alpha^{(t)}(r) =S^{q-1} \alpha^{(t-1)} (r m)\), where \(Y_{(m^{t+1})}\) is a Moore space with attaching map \(m^{t+1}\iota\). In the \(p\)-component, \(p>2\) the only stems in which the \(e\)-invariant could possibly be nonzero are those of the form \(2k(p-1) -1\), and there it takes on values \(-p^{-s}\) if and only if \(p^{s-1}\mid k\).
Main Theorem 6.2. In the \(2k(p-1)-1\) stem of the component, \(p>2\), there exist elements \(\alpha\) such that \(e(\alpha)\equiv -p^{-t}\pmod {\mathbb Z}\) on \(S^{t+1}\) but not on \(S^{2t}\) if \(p^{t-1}\mid k\). The nonexistence of these elements on \(S^{2t}\) has been proven by P. Hoffman using K-theory. It is not hard to show that these elements are stable in the image of \(J \bmod p\).
Reviewer: Brayton Gray

57-XX Manifolds and cell complexes
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