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Working at the prime $2$, Curtis conjecture predicts that, in positivedimensions, spherical classes in $H_*QS^0$ only arise from Hopf invariant oneand Kervaire invariant one elements. Eccles conjecture states that, in positivedimensional, for a path connected space $X$, a class in $H_nQX$ is spherical ifits either stably spherical or it arises from a stable map $S^n\to X$ which isdetected by a primary operation in its mapping cone. (i) We use Hopf invariantone result to verify Eccles conjecture on some finite loop spaces of spheres,namely, for $X=S^k$ , we completely determine spherical classes in$H_*(\Omega^dS^{k+d};\mathbb{Z}/2)$ for specific values of $d$, $k$ showingthat a spherical classes in these cases only do arise from Hopf invariant oneelements. (ii) We completely determine spherical classes in homology of single,double, and triple loop spaces of spheres, namely $\Omega S^{n+1}$, $\Omega^2S^{n+2}$, and $\Omega^3S^{n+3}$ with $n>0$. These computations, verify Ecclesconjecture on the finite loop spaces that we have considered. We also recordsome observations on the relation between two conjectures. The latterconjecture for $X=S^k$, with $k>0$, provides some evidence for the former to betrue.

Spherical classes in some finite loop spaces of spheres