Journal of Lie Theory Volume 8 (1998) 183–185 C ◯1998 Heldermann Verlag

Compact Lie Groups with isomorphic Homotopy Groups

Sven Boekholt^{∗}

Communicated by K. H. Hofmann

Abstract. Compact connected Lie groups with isomorphic higher homotopy groups are locally isomorphic.

In [6] H. Scheerer has shown that compact, connected Lie groups are locally isomorphic if they are homotopy equivalent (see [4] for a conceptually different proof). Obviously, the converse is not true. If G is a compact, connected Lie group which is not simply connected and G̃ is its universal covering group, then these two groups do not even have isomorphic fundamental groups: π_{1}(G) is isomorphic to the kernel of the covering G̃ → G (see [2], (5.8)–(5.10)), while π_{1}(G̃) is trivial. But the covering yields for every n > 1 an isomorphism π_{n}(G̃) → π_{n}(G) (see [2], (7.12)). So the question arises wether it suffices to replace homotopy equivalence in Scheerer’s result by isomorphy of the homotopy groups. As will be shown, the answer to this question is “yes”. The main step to this result was done by H. Toda in [9]:

Theorem 1. Two simply connected, compact (and hence semi-simple) Lie groups are isomorphic to each other if and only if they have isomorphic homotopy groups for each dimension.

The key to the generalization of H. Toda’s theorem to all compact, connected Lie groups is a result due to A. Borel which seems to be not as well known as it would deserve to. In [1] he has shown that a compact, connected Lie group is homeomorphic to the topological direct product of its commutator subgroup and the connected component of its center. This result was generalized to arbitrary compact, connected groups by H. Scheerer (see [7]), and K. H. Hofmann remarked that every compact, connected group is even a semi-direct product of its commutator subgroup and an abelian subgroup (see [8] and [3]). However, we will need only A. Borel’s weaker version of the result.

Theorem 2. Let G_{1} and G_{2} be two compact, connected Lie groups with isomorphic homotopy groups in each dimension. Then G_{1} and G_{2} are locally isomorphic.

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag Proof. Let i ∈ {1, 2}. As G_{i} is compact, its Lie algebra 𝔤_{i} is reductive (see [10], Theorem 4.11.7). Thus 𝔤_{i} is the direct sum of its center 𝔠_{i} and its derived algebra 𝔇𝔤_{i} = [𝔤_{i}, 𝔤_{i}], which is semi-simple in this situation (see [10], Theorem 3.16.3). The analytic subgroup of G_{i} corresponding to 𝔇𝔤_{i} is its commutator subgroup DG_{i} (see [10], Theorem 3.18.8), while the connected component C_{i} of the center of G_{i} corresponds to 𝔠_{i}. Now by [1], 3.2 the group G_{i} is homeomorphic to C_{i}×DG_{i}. For the homotopy groups this means π_{n}(G_{i}) ≅ π_{n}(C_{i}) × π_{n}(DG_{i}) (see [2], (7.11)). The compact, semi-simple group DG_{i} has a compact universal covering group D̃G_{i} (see [10], Theorem 4.11.6). Therefore, the fundamental group π_{1}(DG_{i}) is finite (see [2], (5.8)–(5.10)), whereas the torus group C_{i} has a fundamental group isomorphic to ℤ^{d}, where d = dim C_{i} = dim 𝔠_{i} (see [2], (4.4) and (4.8)). So the dimension of the abelian Lie algebra 𝔠_{i} is determined by the torsion-free rank of the fundamental group of G_{i}, and we get 𝔠_{1} ≅ 𝔠_{2} . It remains to show that 𝔇𝔤_{1} and 𝔇𝔤_{2} are isomorphic too. Since every torus has trivial higher homotopy groups (see [2], (7.14) and (7.11)), we get π_{n}(G_{i}) ≅ π_{n}(DG_{i}) for n > 1. The covering of DG_{i} by its universal covering group D̃G_{i} yields an isomorphism π_{n}(D̃G_{i}) → π_{n}(DG_{i}) in every dimension n > 1 (see [2], (7.12)). Hence the compact, simply connected Lie groups D̃G_{1} and D̃G_{2} have isomorphic homotopy groups in each dimension, and Theorem 1 says that D̃G_{1} and D̃G_{2} are isomorphic. Consequently, their Lie algebras 𝔇𝔤_{1} and 𝔇𝔤_{2} are isomorphic as well.

∗ The author is grateful to Dr. Markus Stroppel for helpful discussions.

The groups U_{2} ℂ and SU_{2} ℂ × SO_{2} ℝ are homeomorphic but not isomorphic. Therefore, one cannot omit the word “locally” in Theorem 2. A semi-simple example for this situation is given by the groups SO_{3} ℝ × SU_{2} ℂ and SO_{4} ℝ ≅ SO_{3} ℝ ⋉ SU_{2} ℂ. In [9] H. Toda presents an example showing that homology and cohomology do not suffice to determine compact Lie groups up to local isomorphism. A survey on homotopy theory of Lie groups can be found in [5].

References

[1] Borel, A., Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, Tohoku Math. J. (2) 13 (1961) 216–240.

[2] Greenberg, M. J., and J. R. Harper, “Algebraic topology, a first course,” Addison–Wesley, Reading, Mass., 1981.

[3] Hofmann, K. H., Sur la décomposition semidirecte des groupes compacts connexes, Symposia Mathematica 24 (1975), 471–476.

[4] Hubbuck, J. R., and R. M. Kane, The homotopy types of compact Lie groups, Israel J. Math. 51 (1985), 20–26.

[5] Mimura, M., Homotopy theory of Lie groups, in: James, I. M. (Ed.): “Handbook of algebraic topology,” North–Holland, Amsterdam, 1995, 951–991.

[6] Scheerer, H., Homotopieäquivalente kompakte Liesche Gruppen, Topology 7 (1968), 227–232.

[7]

, Sur la topologie des groupes compacts connexes, C.R. Acad. Sc. Paris, Série A 274 (1972), 1441–1443.

[8]

, Restklassenräume kompakter zusammenhängender Gruppen mit Schnitt, Math. Ann. 206 (1973), 149–155.

[9] Toda, H., A note on compact semi-simple Lie groups, Japan J. Math. 2 (1976), 355–358.

[10] Varadarajan, V. S., “Lie groups, Lie algebras and their representations,” Springer-Verlag, Berlin etc., 1984.

Mathematisches Institut B Universität Stuttgart 70550 Stuttgart Germany boekholt@mathematik.uni-stuttgart.de

Received August 18, 1997 and in final form September 1, 1997