COMMUTATORS OF EQUIVARIANT HOMEOMORPHISMS

ON G-MANIFOLDS WITH ONE ORBIT TYPE

Tomasz Rybicki

Let M be a manifold and let a compact and connected Lie group G act on M with one orbit type. The long-standing problem of the perfectness of the identity component of the compactly supported equivariant homeomorphism group of M is solved in the affirmative under the assumption that the orbit codimension is greater than one. The result makes use of a well-known theorem of J.N. Mather [4] on the simplicity of homeomorphism groups on manifolds.

In other categories analogous results are known. Namely, a theorem for equivariant C^{r}-diffeomorphisms, where r = 1, …, ∞ and r − 1 is different than the orbit space dimension, is due to A. Banyaga [3] for G = T^{n} , and is due to K. Abe and K. Fukui [1] for an arbitrary compact Lie group G. Recently the latter authors showed in [2] such a theorem for Lipschitz homeomorphisms. However, the proofs of these analogs were much easier than the present one as they made use of a stability property for diffeomorphisms or Lipschitz homeomorphisms. This property is no longer true in the topological category. On the other hand, a clue part of the proof of our result is no longer true even in the C^{1}-case. The theorem seems to be a starting point for computing the first homology of the equivariant homeomorphism group for more complicated G-manifolds.

References

1. K. Abe, K. Fukui, On commutators of equivariant diffeomorphisms, Proc. Japan Acad. 54, Ser.A (1978), 53-54.

2. K. Abe, K. Fukui, On the structure of the group of Lipschitz homeomorphisms and its subgroups, J. Math. Soc. Japan 53 (2001), 501-511.

3. A. Banyaga, On the structure of the group of equivariant diffeomorphisms, Topology 16 (1977), 279-283.

4. J. N. Mather, The vanishing of the homology of certain groups of homeomorphisms, Topology 10 (1971), 297-298.

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