On spherical classes in $H_{\ast }QX$

Abstract

We give an upper bound on the set of spherical classes in $H_{\ast }QX$ when $X=P,S^1$. This is related to the Curtis conjecture on spherical classes in $H_{\ast }{Q}_0{S}^0$. The results also provide some control over the bordism classes on of immersions when $X$ is a Thom complex.