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According to Browder if $4n+2\neq 2^{t+1}-2$ then the Kervaire invariant ofthe 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 theother hand, according to Madsen if $4n+2\neq 2^t-2$ then $M^{4n+2}$ iscobordant to a sphere (hence of Kervaire invariant zero) and $M^{2^{t+1}-2}$ isnot cobordant to a sphere (hence of Kervaire invariant one) if and only ifcertain element $p_{2^{t}-1}^2\in H_*QS^0$ is spherical. Moreover, it is knownthat $p_{2^t-1}^2$ is spherical if and only if $h_t^2$ is a permanent cycle inthe 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'stheorem admits a presentation and proof in terms of certain square classesbeing spherical in $H_*QS^0$ (see also work of Akhmetev and Eccles). In thisnote, 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 partialresults 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 Hopfinvariant one elements. This verifies Eccles conjecture when restricted tofinite loop spaces with $l<9$.

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