On the Bott periodicity, J -homomorphisms, and H * Q 0 S −k

Abstract

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.