# On the Bott periodicity, J-homomorphisms, and $H_*Q_0S^{-k}$

Zare, Hadi (2011) On the Bott periodicity, J-homomorphisms, and $H_*Q_0S^{-k}$. [MIMS Preprint]

The Curtis conjecture predicts that the only spherical classes in $H(Q_0S^0; Z/2)$ are the Hopf invariant one and the Kervaire invariant one elements. We consider Sullivan's decomposition $Q_0S^0 = J \times \cokerJ$ where $J$ is the fibre of $\psi^q - 1$ ($q = 3$ at the prime 2) and observe that the Curtis conjecture holds when we restrict to $J$. We then use the Bott periodicity and the $J$-homomorphism $SO \rightarrow Q_0S^0 to define some generators in$H(Q_0S^0; Z/p)$, when$p$is any prime, and determine the type of subalgebras that they generate. For$p = 2$we determine spherical classes in$H_*( \Omega^k_0J; Z/2)$. We determine truncated subalgebras inside$H_*(Q_0-k}; Z/2)$. Applying the machinery of the Eilenberg-Moore spectral sequence we dene classes that are not in the image of by the$J$-homomorphism. We shall make some partial observations on the algebraic structure of$H_*(\Omega^k_0 \coker J; Z/2)$. Finally, we shall make some comments on the problem in the case equivariant$J\$-homomorphisms.