Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T07:37:22.174Z Has data issue: false hasContentIssue false

On the relation of connective K-theory to homology

Published online by Cambridge University Press:  24 October 2008

Larry Smith
Affiliation:
The University of Virginia

Extract

Let us denote by k*( ) the homology theory determined by the connective BU spectrum, bu, that is, in the notations of (1) and (9), bu2n = BU(2n,…,∞), bu2n+1 = U(2n + 1,…, ∞) with the spectral maps induced via Bott periodicity. The resulting spectrum, bu, is a ring spectrum. Recall that k*(point) ≅ Z[t], degree t = 2. There is a natural transformation of ring spectra

inducing a morphism

of homology functors. It is the objective of this note to establish: Theorem. Let X be a finite complex. Then there is a natural exact sequence

where Z is viewed as a Z[t] module via the augmentation

and, is induced by η*in the natural way.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1970

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.CrossRefGoogle Scholar
(2)Adams, J. F.Lectures on generalized homology theories. Springer.Verlag Lecture Notes in Mathematics No. 99 (1969).CrossRefGoogle Scholar
(3)Boardman, J. M.Notes on stable homotopy theory (Warwick University, 1966).Google Scholar
(4)Burdick, R. O., Conner, P. E. and Floyd, E. E.Chain theories and their derived homology. Proc. Amer. Math. Soc. 19 (1968), 11151118.CrossRefGoogle Scholar
(5)Conner, P. E. and Smith, L.On the complex bordism of finite complexes. I.H.E.S. Journal de Mathematique, No. 37 (1970).Google Scholar
(6)Landweber, P. S.On the complex bordism and cobordism of infinite complexes. Bull. A.M.S. 76 (1970), 650654.CrossRefGoogle Scholar
(7)Novikov, S. P.The method of algebraic topology from the point of view of cobordism theory. Izv. Akad. Nauk Armjan SSR., Ser. Mat. 31 (1967), 855951.Google Scholar
(8)Vogt, R.Lectures on Boardman's stable category (Aarhus University, 1969).Google Scholar
(9)Whiteread, G. W.Generalized homology theories. Trans. Amer. Math. Soc. 102 (1962), 227283.CrossRefGoogle Scholar
(10)Stong, R. E.Lectures on cobordism theory (Princeton University Press, 1969).CrossRefGoogle Scholar