Proper actions of Lie groups and Hilbert's fifth problem
by
Sören Illman
University of Helsinki, Department of Mathematics, University of Helsinki, Finland

Abstract for the International Colloquium on Topology, Gyula, Hungary, August 9-15, 1998.

Hilbert asks in his fifth problem the following. Suppose G is a locally euclidean topological group and M is a locally euclidean space, and let

F\colon G ×M ® M     (1)

be a continuous action of G on M. Is it then possible to choose the coordinates in G and M so that the action F becomes real analytic? In other words is it possible to give the topological manifolds G and M real analytic structures so that F is real analytic?

In the special case when G = M and

F\colon G ×G ® G     (2)

is the multiplication in the group G the answer to Hilbert's question is yes, and this was proved by Gleason, and Montgomery and Zippin in 1952. This result is the famous result which says that every locally euclidean topological group is a Lie group. In the case when G is compact this was proved by von Neumann in 1933, and in the case when G is commutative by L.S. Pontryagin in 1939.

The answer to Hilbert's original question, whether the action F in (1) can be made real analytic, is in general no. R.H. Bing constructed in 1952 an example of a continuous action of Z₂ on R³ which could not be differentiable and hence not either real analytic. In 1954 Montgomery and Zippin extended the example of Bing to give an example of a continuous action of S¹ on R⁴ which could not be differentiable. In fact the answer to Hilbert's question is no even in the case when G is the trivial group {e}, since there exist topological manifolds which cannot have any differentiable structure, and hence also no real analytic structure. The first example of such a topological manifold with no smooth structure is due to Kervaire, and from 1960.

Recall that an action of a Lie group G on a locally compact space X is said to be proper if G_[A] = {g Î G | gA ÇA ¹ Æ} is a compact subset of G for every compact subset A of X. We say that an action of G on X is Cartan, if each point in X has a compact neighborhood A such that G_[A] is compact. Thus every proper action of G is Cartan, but the converse does not hold. In the case when G is a discrete group the notion of a proper action coincides with the classical notion of a properly discontinuous action. In 1995 we proved the following result, see S. Illman, Every proper smooth action of a Lie group is equivalent to a real analytic action: a contribution to Hilbert's fifth problem, Ann. of Math. Stud. 138 (1995), 189-220.

Theorem A. Let G be a Lie group which acts on a C¹-smooth manifold M by a C¹-smooth and Cartan action. Then there exists a real analytic structure on M, compatible with the given smooth structure, such that the action of G becomes real analytic.

This result gives a positive answer to the general question in Hilbert's fifth problem, and the answer is best possible since it is known that there exist smooth, in fact C^¥-smooth, non-Cartan actions of Lie groups which cannot be made real analytic. Note that Theorem A in particular applies in the case of proper actions, and thus when G is a discrete group in the case of properly discontinuous actions.

When G is compact every action of G is proper, and the result in Theorem A was proved by T. Matumoto and M. Shiota in 1987. However their method of proof cannot be applied when G is non-compact.

The result in Theorem A is important for it shows that every Cartan (and hence also every proper) smooth action of a Lie group can be made real analytic, and thus that such actions are more rigid than at first hand is apparent. As one application of Theorem A we mention that we use it in proving that every smooth proper G-manifold M, where G is an arbitrary Lie group, has a G-equivariant triangulation.

Date received: June 15, 1998