On a Type of Subgroups of a Compact Lie Group

Yozô Matsushima

doi:10.1017/S0027763000009995

On a Type of Subgroups of a Compact Lie Group

Published online by Cambridge University Press: 22 January 2016

Yozô Matsushima

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Yozô Matsushima*: Affiliation:
Mathematical Institute, Nagoya University

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Let G be a connected compact Lie group and H a connected closed subgroup. Then H is an orientable submanifold of G and we may consider H as a cycle in G. In his interesting paper on the topology of group manifolds H. Samelson has proved that, if H is not homologous to 0, then the homology ring of the coset space G/H is isomorphic to the homology ring of a product space of odd dimensional spheres and the homology ring of G is isomorphic to that of the product of the spaces H and G/H. On the other hand, in a recent investigation of fibre bundles’ T. Kudo has shown that, if the homology ring of the coset space G/H is isomorphic to that of an odd dimensional sphere, then H is not homologous to 0.

Type: Research Article
Information: Nagoya Mathematical Journal , Volume 2 , February 1951 , pp. 1 - 15

DOI: https://doi.org/10.1017/S0027763000009995 [Opens in a new window]
Copyright: Copyright © Editorial Board of Nagoya Mathematical Journal 1951

References

¹⁾ Samelson, H., Beiträge zur Topologie der Gruppen-Mannigfaltigkeiten, Ann. of Math Vol. 42 (1941); Satz VI. We refer to this paper as [S]Google Scholar.

²⁾ The coefficients of the homology ring are rational numbers.

³⁾ Kudo, T., On the homological properties of fibre bundles, forthcoming in Journ. of the Institute of Polytechnics, Osaka City University Google Scholar.

⁴⁾ Montgomery, D. and Samelson, H.,. Transformation groups of spheres, Ann. of Math. Vol 44 (1943)Google Scholar. We refer to this paper as [M-S].

⁵⁾ Hopf, H., Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen, Ann. of Math. Vol. 42 (1941)Google Scholar and Hopf, H., Über den Rang geschlossener Lieschen Gruppen, Commet. Math. Helvet. Vol. 13 (1941)Google Scholar.

⁶⁾ See, [S], Satz VI. Note that by the results of Samelson, H. and Kudo, T. H is an S-subgroup if and only if H is not homologous to 0 and r(H) = r(G)-1Google Scholar.

⁷⁾ In this case we may consider G₁ o G₂ as the usual product of two normal subgroups Gi and G₂.

⁸⁾ For the definition and the properties of the minimal element, see H. Hopf, loc, oit. and [S].

⁹⁾ See, [S], Satz III. Korollar 1.

¹⁰⁾ The following proof is similar to the proof of Theorem I b) in [M-S], But we avoid to use a theorem of Gysin which played an essential role there.

¹¹⁾ See, [S], Satz III Korollar 3.

¹²⁾ See, [S], Satz VI.

¹³⁾ See, [S], Satz IV.

¹⁴⁾ The writer can not decide whether Theorem II is also valid for n<8 or not. Since every subgroups of rank 1 is not homologous to 0, Theorem II is not valid for n = 2 as we may show by an example. Cf. J. L. Koszul, C. R. Paris 225, p. 477 (1947), and H. Samelson, C. R, Paris 228, p. 630 (1949).

¹⁵⁾ See, Pontrjagin, L., Homologies in compact Lie groups, Ree. Math. N. S. Vol. 6 (1939) or [S]Google Scholar.

¹⁶⁾ For, by a theorem of E. Cartan, the 3-dimensional Betti number of any semi-simple group is not equal to 0.

¹⁷⁾ For, any toral subgroup of a compact connected Lie group G is conjugate to a subgroup of any maximal toral subgroup G. See, Weil, A., Démonstration topologique d’un théorème fondamental de Cartan. C. R. Paris 200 (1935)Google Scholar; Hopf, H. and Samelson, H., Ein Satz über die Wirkungsräume geschlossener Liescher Gruppen, Commet. Math. Helvet. Vol. 13 (1941)Google Scholar.

¹⁸⁾ Through in the following we denote by £ such a vector.

¹⁹⁾ Take an element h₁ of which is not contained in . Since the mapping u → [h₁, u] (u ∈ U) is a derivation of U. As the derivations of the simple Lie algebra U are inner, there exists an element u_o of U such that [h₁ , u] = [u_o, u] for every u ∈ It. But since [h₁, u] = 0 for every commutes with every element of . is a maximal abelian subalgebra of U and hence Then the element h₁ — m satisfies our condition.