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The spectral sequence of an extraordinary cohomology theory

Published online by Cambridge University Press:  24 October 2008

C. R. F. Maunder
C. R. F. Maunder
Affiliation:
University of Southampton
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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
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 3 , July 1963 , pp. 567 - 574
Copyright
Copyright © Cambridge Philosophical Society 1963

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References

REFERENCES

(1)Adams, J. F., On Chern characters and the structure of the unitary group. Proc. Cambridge Philos. Soc. 57 (1961), 189–199.Google Scholar
(2)Adem, J.The iteration of Steenrod squares in algebraic topology. Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 720–726.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. 7–38.Google Scholar
(4)Atiyah, M. F., and Hirzebruch, F., Analytic cycles on complex manifolds. Topology, 1 (1962), 25–45.Google Scholar
(5)Brown, E. H., Cohomology theories. Ann. of Math. 75 (1962), 467–484.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), 363–396; 57 (1953), 248–286.CrossRefGoogle Scholar
(10)Peterson, F. P., Some results on cohomotopy groups. American J. Math. 78 (1956), 243–258.CrossRefGoogle Scholar
(11)Peterson, F. P., Functional cohomology operations. Trans. American Math. Soc. 86 (1957), 197–211.CrossRefGoogle Scholar
(12)Peterson, F. P. and Stein, N., Secondary cohomology operations: two formulas. American J. Math. 81 (1959), 281–305.CrossRefGoogle Scholar
(13)Puppe, D., Homotopiemengen und ihre induzierten Abbildungen. I. Math. Z. 69 (1958), 199–344.Google Scholar
(14)Whitehead, J. H. C.Combinatorial homotopy. I. Bull. American Math. Soc. 55 (1949), 213–245.Google Scholar