Hostname: page-component-7c8c6479df-8mjnm Total loading time: 0 Render date: 2024-03-17T12:05:09.654Z Has data issue: false hasContentIssue false

The spectral sequence of an extraordinary cohomology theory

Published online by Cambridge University Press:  24 October 2008

C. R. F. Maunder
Affiliation:
University of Southampton

Extract

Given an ‘extraordinary cohomology theory’, that is, a cohomology theory satisfying all the axioms of Eilenberg and Steenrod (7) except the dimension axiom, it is well known that there exists a spectral sequence relating the ordinary cohomology of a space with the extraordinary theory (see, for example, (3) in the case of K*(X)). Obviously, it would be useful to know the differentials in this spectral sequence, and it is the purpose of this paper to identify them in terms of cohomology operations defined by certain k-invariants. We shall make use of E. H. Brown's recent theorem (5) on the representability of extraordinary cohomology theories, to construct a second spectral sequence, in which the differentials are readily identifiable, which we shall prove is isomorphic to the usual one.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1963

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., On Chern characters and the structure of the unitary group. Proc. Cambridge Philos. Soc. 57 (1961), 189199.Google Scholar
(2)Adem, J.The iteration of Steenrod squares in algebraic topology. Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 720726.Google Scholar
(3)Atiyah, M. F., and Hirzebruch, F., Vector bundles and homogeneous spaces. Proceedings of Symposia in Pure Mathematics, vol. III (American Mathematical Society, Providence, R.I., 1960), pp. 738.Google Scholar
(4)Atiyah, M. F., and Hirzebruch, F., Analytic cycles on complex manifolds. Topology, 1 (1962), 2545.Google Scholar
(5)Brown, E. H., Cohomology theories. Ann. of Math. 75 (1962), 467484.Google Scholar
(6)Dold, A., Relations between ordinary and extraordinary homology. Colloquium on Algebraic Topology, Aarhus, 1962 (mimeographed notes), pp. 2–9.Google Scholar
(7)Eilenberg, S., and Steenrod, N.Foundations of algebraic topology (Princeton, 1952).Google Scholar
(8)Kahn, D. W. Induced maps for Postnikov systems. Colloquium on Algebraic Topology, Aarhus, 1962 (mimeographed notes), pp. 47–51.Google Scholar
(9)Massey, W. S.Exact couples in algebraic topology. Ann. of Math. 56 (1952), 363396; 57 (1953), 248–286.CrossRefGoogle Scholar
(10)Peterson, F. P., Some results on cohomotopy groups. American J. Math. 78 (1956), 243258.CrossRefGoogle Scholar
(11)Peterson, F. P., Functional cohomology operations. Trans. American Math. Soc. 86 (1957), 197211.CrossRefGoogle Scholar
(12)Peterson, F. P. and Stein, N., Secondary cohomology operations: two formulas. American J. Math. 81 (1959), 281305.CrossRefGoogle Scholar
(13)Puppe, D., Homotopiemengen und ihre induzierten Abbildungen. I. Math. Z. 69 (1958), 199344.Google Scholar
(14)Whitehead, J. H. C.Combinatorial homotopy. I. Bull. American Math. Soc. 55 (1949), 213245.Google Scholar