Hypersurface representation and the image of
the double $S^{3}$–transfer

Let $\nu^{*}\in\pi^{s}_{18}(S^{0})$ be an element of order $8$ in the stable homotopy groups of spheres. Throughout the paper, $\pi_{i}(X)$ (resp. $\pi^{s}_{i}(X)$) denotes the homotopy group (resp. the stable homotopy group) of a space $X$, and we use the notations of Toda [10] for elements of $\pi^{s}_{*}(S^{0})$. Then, using a generators $\nu\in\pi^{s}_{3}(S^{0})\cong\Z/24$ and $\sigma\in\pi^{s}_{7}(S^{0})\cong\Z/240$, $\nu^{*}$ is represented by the Toda bracket $\langle\sigma,2\sigma,\nu\rangle=-\langle\sigma,\nu,\sigma\rangle$ with no indeterminacy.

As is known, $\nu^{*}$ is not in the image of the homomorphism $J_{0}\co\pi_{18}(SO)\to\pi^{s}_{18}(S^{0})$ induced by the stable $J$–map $J\co SO\to\Omega^{\infty}\Sigma^{\infty}S^{0}$. We shall show that $\nu^{*}$ is in the image of the homomorphism $J_{1}\co\pi_{19}(\Sigma SO)\to\pi^{s}_{18}(S^{0})$ induced by the adjoint map $\Sigma SO\to\Omega^{\infty}\Sigma^{\infty}S^{1}$ to $J$. Actually, we prove that $\nu^{*}$ is in a double $S^{3}$–transfer image which is a subgroup of $\mbox{Im}(J_{1})$.

Eccles [3] has made clear that $\mbox{Im}(J_{1})$ consists of elements represented by framed hypersurfaces. Such study has also done by Rees [9]. Our result gives the following.

Here, a framed hypersurface of dimension $18$ is a closed manifold of dimension 18 embedded in $S^{19}$ and framed in $S^{N}$ for sufficiently large $N$, and the theorem means that $\nu^{*}$ is the framed cobordism class of such a framed hypersurface when we regard $\pi^{s}_{18}(S^{0})$ as the framed cobordism group $\Omega^{fr}_{18}$.

The Bott map $B\co\Sigma^{3}BSp\to SO$ is the adjoint map to the homotopy equivalence $BSp\times\Z\to\Omega^{3}SO$, where $SO=\bigcup_{n}SO(n)$ is the rotation group and $BSp$ is the classifying space of the symplectic group $Sp=\bigcup_{n}Sp(n)$. Let $\mathbb{H}P^{\infty}=\bigcup_{n}\mathbb{H}P^{n}$ be the infinite dimensional quaternionic projective space. Then, $\mathbb{H}P^{\infty}=BSp(1)$, and there is an inclusion map $i\co\mathbb{H}P^{\infty}\to BSp$. We define a map $t_{H}$ as the composition

We denote by $t_{H}^{s}\co\mathbb{H}P^{\infty}\to S^{-3}$ a stable map adjoint to $t_{H}$. It is not certain whether $t_{H}^{s}$ is stably homotopic to the stable map constructed by the Boardman’s transfer construction [1] on the principal $S^{3}$–bundle over $\mathbb{H}P^{\infty}$. However, since $t_{H}$ has the following property in common with the Boardman’s $S^{3}$–transfer map, we call $t_{H}$ and $t_{H}^{s}$ the $S^{3}$–transfer maps.

Let $\mu\co\Omega^{\infty}\Sigma^{\infty}S^{0}\wedge\Omega^{\infty}\Sigma^{\infty}S^{0}\to\Omega^{\infty}\Sigma^{\infty}S^{0}$ be the multiplication on the infinite loop space $\Omega^{\infty}\Sigma^{\infty}S^{0}$ given by composition, and $\mu^{\prime}\co\Sigma\Omega^{\infty}\Sigma^{\infty}S^{0}\wedge\Omega^{\infty}\Sigma^{\infty}S^{0}\to\Omega^{\infty}\Sigma^{\infty}S^{1}$ the adjoint map to $\mu$. Then, we set

and call $t_{H}(2)$ a double $S^{3}$–transfer map. Then, our main result may be stated as follows:

Applying a result due to Eccles and Walker [4], we have $\text{\rm Im}(t_{H}(2)_{*})\subset\text{\rm Im}(J_{1})$ (see \fullrefProp2 and Corollary2.2). Hence, it turns out that $\nu^{*}$ is represented by a framed hypersurface of dimension $18$ as is stated in the first theorem.

In Carlisle et al [2], effective use of the $S^{1}$–transfer homomorphism $\tau\co\pi^{s}_{i-1}(\C P^{\infty})\to\pi^{s}_{i}(S^{0})$ shows that certain elements are represented by framed hypersurfaces. In this respect, $\nu^{*}$ is not in the image of the double $S^{1}$–transfer homomorphism $\tau_{2}\co\pi^{s}_{16}(\C P^{\infty}\wedge\C P^{\infty})\to\pi^{s}_{18}(S^{0})$. In fact, the image of $\tau_{2}$ in $\pi^{s}_{18}(S^{0})$ is equal to $\Z/4\{2\nu^{*}\}$ by Imaoka [5, Theorem 10].

By Toda [10], $\pi_{16}^{s}(S^{0})=\Z/2\{\eta^{*}\}\oplus\Z/2\{\eta\circ\rho\}$. Here, $\eta\in\pi^{s}_{1}(S^{0})\cong\Z/2$ is a generator, $\eta^{*}$ is an element in the Toda bracket $\langle\sigma,2\sigma,\eta\rangle$ and $\rho$ is a generator of the image of $J_{0}\co\pi_{15}(SO)\to\pi_{15}^{s}(S^{0})$. Mahowald [6] has constructed an important family $\eta_{i}\in\pi_{2^{i}}^{s}(S^{0})$ for $i\geq 3$, and $\eta^{*}\equiv\eta_{4}\pmod{\eta\circ\rho}$.

By \fullrefLem1, we see the following:

Mukai [8, Theorem 2] has shown that both $\nu^{*}$ and $\eta^{*}$ are in the image of the $S^{1}$–transfer homomorphism $\tau\co\pi_{i-1}^{s}(\C P^{\infty})\to\pi_{i}^{s}(S^{0})$. Morisugi [7, Corollary E] has shown that all Mahowald’s elements are in the image of the $S^{3}$–transfer homomorphism $\pi_{*}^{s}(Q_{\infty})\to\pi_{*}^{s}(S^{0})$ given for the quaternionic quasi-projective space $Q_{\infty}=\bigcup_{n}Q_{n}$.

In contrast with these, we have the following:

We also remark the following noted by Eccles [3], where $J_{2}\co\pi_{18}(\Sigma^{2}SO)\to\pi_{16}^{s}(S^{0})$ is the homomorphism induced by the map $\Sigma^{2}SO\to\Omega^{\infty}\Sigma^{\infty}S^{2}$ adjoint to the stable $J$–map.

In \fullrefsec2 we prove \fullrefLem1 and \fullrefThA, and in \fullrefsec3 we prove \fullreftH20, \fullrefPropB and \fullrefeta*2.

We first prove \fullrefLem1.

Let $H(m)\co\Sigma SO\wedge SO\to\Sigma SO$ be the map defined by the Hopf construction on the multiplication $m\co SO\times SO\to SO$, and $J^{*}\co\Sigma SO\to\Omega^{\infty}\Sigma^{\infty}S^{1}$ the adjoint map to the stable $J$–map $J\co SO\to\Omega^{\infty}\Sigma^{\infty}S^{0}$. Then, Eccles and Walker have shown the following [4, Proposition 2.2], in which $\mu^{\prime}\co\Sigma(\Omega^{\infty}\Sigma^{\infty}S^{0}\wedge\Omega^{\infty}\Sigma^{\infty}S^{0})\to\Omega^{\infty}\Sigma^{\infty}S^{1}$ is the adjoint map to the multiplication $\mu\co\Omega^{\infty}\Sigma^{\infty}S^{0}\wedge\Omega^{\infty}\Sigma^{\infty}S^{0}\to\Omega^{\infty}\Sigma^{\infty}S^{0}$.

Recall that the homomorphism $J_{1}\co\pi_{i}(\Sigma SO)\to\pi^{s}_{i-1}(S^{0})$ is induced by $J^{*}$. Since $(J\wedge J)\circ(B\circ i\wedge B\circ i)$ is equal to $t_{H}\wedge t_{H}$, by composing it with the maps of \fullrefProp2, $J^{*}\circ H(m)\circ\Sigma(B\circ i\wedge B\circ i)\co\Sigma(\Sigma^{3}\mathbb{H}P^{\infty}\wedge\Sigma^{3}\mathbb{H}P^{\infty})\to\Omega^{\infty}\Sigma^{\infty}S^{1}$ is homotopic to $t_{H}(2)=\mu^{\prime}\circ\Sigma(t_{H}\wedge t_{H})$. Hence, we have the following.

Let $\beta_{i}\in H_{4i}(\mathbb{H}P^{\infty};\thinspace\Z)\cong\Z$ be a generator. Then, we have $H_{12}(\mathbb{H}P^{\infty}\wedge\mathbb{H}P^{\infty};\thinspace\Z)=\Z\{\beta_{1}\otimes\beta_{2}\}\oplus\Z\{\beta_{2}\otimes\beta_{1}\}$.

We remark that

Then, using \fullrefLem1,

and thus we have \fullreftH20.

Lastly, we prove \fullrefPropB and \fullrefeta*2.