Towards a Browder theorem for spherical classes in $\Omega^lS^{n+l}$

According to Browder if $4n+2\neq 2^{t+1}-2$ then the Kervaire invariant of the cobordism class of a $(4n+2)$-dimensional manifold $M^{4n+2}$ vanishes and $M^{2^{t+1}-2}$ is of Kervaire invariant one if and only if $h_t^2\in\mathrm{Ext}(\mathbb{Z}/2,\mathbb{Z}/2)$ is a permanent cycle. On the other hand, according to Madsen if $4n+2\neq 2^t-2$ then $M^{4n+2}$ is cobordant to a sphere (hence of Kervaire invariant zero) and $M^{2^{t+1}-2}$ is not cobordant to a sphere (hence of Kervaire invariant one) if and only if certain element $p_{2^{t}-1}^2\in H_*QS^0$ is spherical. Moreover, it is known that $p_{2^t-1}^2$ is spherical if and only if $h_t^2$ is a permanent cycle in the Adams spectral sequence. Moreover, classes $p_{2n+1}^2\in H_*QS^0$ with $2n+1\neq 2^t-1$ are easily eliminated from being spherical. Hence, Browder's theorem admits a presentation and proof in terms of certain square classes being spherical in $H_*QS^0$ (see also work of Akhmetev and Eccles). In this note, we consider the problem of determining spherical classes $H_*\Omega^lS^{n+l}$ with $n>0$ and $4\leqslant l\leqslant +\infty$. We show (1) if $\xi^2\in H_*\Omega^lS^{n+l}$ is given with $\dim\xi+1\neq 2^t$ and $\dim \xi+1\equiv 2\textrm{ mod }4$ and $n>l$, then $\xi^2$ is not spherical. We refer to this as a generalised Browder theorem. We also present some partial results on the degenerate cases, corresponding to $\dim\xi\neq 2^t-1$, when $l>n$. (2) For $l\in\{4,5,6,7,8\}$ the only spherical classes in $H_*\Omega^lS^{n+l}$ arise from the inclusion of the bottom cell, or the Hopf invariant one elements. This verifies Eccles conjecture when restricted to finite loop spaces with $l<9$.

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