Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T00:46:08.605Z Has data issue: false hasContentIssue false
  • English
  • Français

On Adams' splitting of K-theory and complex cobordism

Published online by Cambridge University Press:  24 October 2008

Idar Hansen
Idar Hansen
Affiliation:
University of Kentucky
Get access Rights & Permissions [Opens in a new window]

Extract

In (2) Adams gave a splitting of complex K-theory with coefficients in the ring R(d) of rationals a/b such that b contains no prime p with p ≡ 1 (mod d). The splitting comes from a complete set of projection operators on K(X; R(d)). One of the operators is then used to obtain a stable, multiplicative idempotent ε on complex cobordism with coefficients in the same ring R(d) and hence a splitting of the representing spectrum MUR(d). However, the idempotent is initially defined over the rational numbers and work is needed to show that it actually gives an operation on MUR(d). Since Novikov (6) has shown that multiplicative cobordism operations are distinguished by their values on the generator ω ∈ MU2CP∞, it is natural to seek an explicit formula for ε(ω) which wi11 show that ε gives an operation over the subring R(d).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 1 , January 1978 , pp. 113 - 116
Copyright
Copyright © Cambridge Philosophical Society 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1) Adams, J. F. Vector fields on spheres. Ann. of Math. 75 (1962), 603–632.CrossRefGoogle Scholar
(2) Adams, J. F. Lectures on generalised cohomology. In: Category theory, homology theory and their applications III (Seattle 1968), pp. 1–38 (Lecture Notes in Mathematics, No. 99 Berlin–Heidelberg–NewYork: Springer, 1969).Google Scholar
(3) Adams, J. F. Stable homotopy and generalised homology (Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 1974).Google Scholar
(4) Atiyah, M. F. and Tall, D. O. Group representations, λ-rings andthe J-homomorphism. Topology 8 (1969), 253–297.CrossRefGoogle Scholar
(5) Bott, R. Lectures on K(X) (New York, Amsterdam: W. A. Benjamin, 1969).Google Scholar
(6) Novikov, S. P. The methods of algebraic topology from the viewpoint of cobordism theory (Russian). Izv. Akad. Nauk SSSR., Ser. Mat. 31 (1967), 855–951. (English translation: Math. USSR, Izvestija 1 (1967), 827–913.)Google Scholar
(7) Quillen, D. On the formal group laws of unoriented and complex cobordism theory. Bull. Amer. Math. Soc. 75 (1968), 1293–1298.CrossRefGoogle Scholar