Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-30T20:21:50.509Z Has data issue: false hasContentIssue false
  • English
  • Français

On the K-theory of the loop space of a Lie group

Published online by Cambridge University Press:  24 October 2008

Francis Clarke
Francis Clarke
Affiliation:
Department of Pure Mathematics, University College of Swansea, Wales
Get access Rights & Permissions [Opens in a new window]

Extract

Let G be a simply connected, semi-simple, compact Lie group, let K* denote Z/2-graded, representable K-theory, and K* the corresponding homology theory. The K-theory of G and of its classifying space BG are well known, (8),(1). In contrast with ordinary cohomology, K*(G) and K*(BG) are torsion-free and have simple multiplicative structures. If ΩG denotes the space of loops on G, it seems natural to conjecture that K*(ΩG) should have, in some sense, a more simple structure than H*(ΩG).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 1 , July 1974 , pp. 1 - 20
Copyright
Copyright © Cambridge Philosophical Society 1974

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)Atiyah, M. F. and Hirzebruch, F.Vector bundles and homogeneous spaces. Proc. Sympos. Pure Math. A.M.S. 3 (1961), 7–38.CrossRefGoogle Scholar
(2)Borel, A. and Siebenthal, J. DE.Les sous-groupes fermés de rangmaximum des groupes de Lie clos. Comment. Math. Helv. 23 (1949), 200–221.CrossRefGoogle Scholar
(3)Bott, R.An application of the Morse theory to the topology of Lie groups. Bull. Soc. Math. France 84 (1956), 251–281.Google Scholar
(4)Bott, R.The space of Loops on a Lie group. Michigan Math. J. 5 (1958), 35–61.CrossRefGoogle Scholar
(5)Clarke, F. W. Ph.D. thesis, University of Warwick (1971).Google Scholar
(6)Gitler, S. and Lam, K. Y.The K-theory of Stiefel manifolds. Lecture Notes in Mathematics, vol. 168 (1970) (Springer-Verlag; Berlin-Heidelberg-New York).Google Scholar
(7)Hammond, J.On the use of certain differential operators in the theory of equations. Proc. London Math. Soc. 14 (1883), 119–129.Google Scholar
(8)Hodgkin, L.On the K-theory of Lie groups. Topology 6 (1967), 1–36.CrossRefGoogle Scholar
(9)Hodgkin, L. Künneth formula spectral sequences (to appear).Google Scholar
(10)Husemoller, D.Fibre bundles (McGraw-Hill; New York, 1966).CrossRefGoogle Scholar
(11)Liu, C. L.Introduction to combinatorial mathematics (McGraw-Hill; New York, 1968).Google Scholar
(12)MacMahon, P. A.Combinatory analysis (Cambridge, 1915).Google Scholar
(13)Milnor, J.On axiomatic homology theory. Pacific J. Math. 12 (1962), 337–341.CrossRefGoogle Scholar
(14)Milnor, J. and Moore, J. C.On the structure of Hopf algebras. Ann. of Math. 81 (1965), 211–264.CrossRefGoogle Scholar
(15)Petrie, T.The weakly complex bordism of Lie groups. Ann. of Math. 88 (1968), 371–402.CrossRefGoogle Scholar
(16)Pittie, H. V.Homogeneous vector bundles on homogeneous spaces. Topology 11 (1972), 199–203.CrossRefGoogle Scholar
(17)Quillen, D.Rational homotopy theory. Ann. of Math. 90 (1969), 205–295.CrossRefGoogle Scholar
(18)Snaith, V. P.On the K-theory of homogeneous spaces and conjugate bundles of Lie groups. Proc. London Math. Soc. 22 (1971), 562–584.CrossRefGoogle Scholar
(19)Smith, L.On the Künneth theorem. I. Math. Z. 116 (1970), 94–140.CrossRefGoogle Scholar