On the bordism ring of complex projective space

Abstract:

The bordism ring $M{U_\ast }(C{P^\infty })$ is central to the theory of formal groups as applied by D. Quillen, J. F. Adams, and others recently to complex cobordism. In the present paper, rings ${E_\ast }(C{P^\infty })$ are considered, where E is an oriented ring spectrum, $R = {\pi _\ast }(E)$, and $pR = 0$ for a prime p. It is known that ${E_\ast }(C{P^\infty })$ is freely generated as an R-module by elements $\{ {\beta _r}|r \geqq 0\}$. The ring structure, however, is not known. It is shown that the elements $\{ {\beta _{{p^r}}}|r \geqq 0\}$ form a simple system of generators for ${E_\ast }(C{P^\infty })$ and that $\beta _{{p^r}}^p \equiv {s^{{p^r}}}{\beta _{{p^r}}}\bmod ({\beta _1}, \cdots ,{\beta _{{p^{r - 1}}}})$ for an element $s \in R$ (which corresponds to $[C{P^{p - 1}}]$ when $E = MU{Z_p})$. This may lead to information concerning ${E_\ast }(K(Z,n))$.