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On the cohomology of loop spaces of compact Lie groups

Published online by Cambridge University Press:  18 May 2009

Howard Hiller
Howard Hiller
Affiliation:
Mathematisches Institut, Universität Göttingen, West Germany Department of Mathematics, Columbia University, New York, N.Y. 10027, U.S.A.
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Let G be a compact, simply-connected Lie group. The cohomology of the loop space ΏG has been described by Bott, both in terms of a cell decomposition [1] and certain homogeneous spaces called generating varieties [2]. It is possible to view ΏG as an infinite dimensional “Grassmannian” associated to an appropriate infinite dimensional group, cf. [3], [7]. From this point of view the above cell-decomposition of Bott arises from a Bruhat decomposition of the associated group. We choose a generator H ∈ H2(ΏG, ℤ) and call it the hyperplane class. For a finite-dimensional Grassmannian the highest power of H carries geometric information about the variety, namely, its degree. An analogous question for ΏG is: What is the largest integer Nk = Nk(G) which divides Hk ∈ H2k(ΏG, ℤ)?

Of course, if G = SU(2) = S3, one knows Nk = h!. In general, the deviation of Nk from k! measures the failure of H to generate a divided polynomial algebra in H*(ΏG, ℤ).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 26 , Issue 1 , January 1985 , pp. 91 - 99
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

REFERENCES

1.Bott, R. and Samelson, H., Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964–1029.CrossRefGoogle Scholar
2.Bott, R., The space of loops on a Lie group, Michigan Math. J. 5 (1958), 35–61.CrossRefGoogle Scholar
3.Garland, H. and Raghunathan, M., A Bruhat decomposition for the loop space of a compact Lie group: a new approach to results of Bott, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 4716–4717.CrossRefGoogle Scholar
4.Harris, B., On the homotopy groups of the classical groups, Ann. of Math. (2) 74 (1961), 407–413.CrossRefGoogle Scholar
5.Hiller, H., Geometry of Coxeter groups, Research Notes in Mathematics 54 (Pitman, 1982).Google Scholar
6.Hubbuck, J., Finitely generated cohomology Hopf algebras, Topology 9 (1970), 205–210.CrossRefGoogle Scholar
7.Kac, V. and Peterson, D., Infinite flag varieties and conjugacy theorems, preprint.Google Scholar
8.Proctor, R., Interactions between combinatorics, Lie theory and algebraic geometry via the Bruhat orders (Ph.D. thesis, MIT, 1980).Google Scholar
9.Thomas, E., Exceptional Lie groups and Steenrod squares, Michigan Math. J. 11 (1964), 151–156.CrossRefGoogle Scholar