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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
DOI: https://doi.org/10.1090/S0002-9939-1973-0307222-2
MathSciNet review: 0307222
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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))$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0307222-2
Keywords: Complex bordism, complex cobordism, oriented spectrum, graded formal group, Hopf algebra over a ring
Article copyright: © Copyright 1973 American Mathematical Society