Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the bordism ring of complex projective space

Author: Claude Schochet
Journal: Proc. Amer. Math. Soc. 37 (1973), 267-270
MSC: Primary 55B20; Secondary 57A20
MathSciNet review: 0307222
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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}\vert r \geqq 0\} $. The ring structure, however, is not known. It is shown that the elements $ \{ {\beta _{{p^r}}}\vert 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))$.

References [Enhancements On Off] (What's this?)

  • [1] J. F. Adams, Quillen's work on formal groups and complex cobordism, Lectures, University of Chicago, Chicago, Ill., 1970. MR 1170568 (93c:01046)
  • [2] M. Kamata, On the ring structure of $ {U_\ast }(BU(1))$, Osaka J. Math. 7 (1970), 417-422. MR 43 #1204. MR 0275448 (43:1204)
  • [3] D. Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 1293-1298. MR 40 #6565. MR 0253350 (40:6565)
  • [4] C. Schochet, On the structure of graded formal groups of finite characteristic, Proc. Cambridge Philos. Soc. (to appear). MR 0340321 (49:5076)
  • [5] G. W. Whitehead, Generalized homology theories, Trans. Amer. Math. Soc. 102 (1962), 227-283. MR 25 #573. MR 0137117 (25:573)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55B20, 57A20

Retrieve articles in all journals with MSC: 55B20, 57A20

Additional Information

Keywords: Complex bordism, complex cobordism, oriented spectrum, graded formal group, Hopf algebra over a ring
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society