Leo Zippin

Topological transformation groups

1. Introduction This note will summarize some of the recent work on topological groups and discuss a few topics in transformation groups mainly in S 3 and S 4. In one aspect of this subject, namely the relation of general locally compact groups to Lie groups, information is now fairly complete. However in most other aspects the situation is far less satisfactory and in many areas known results are fragmentary. In particular the study of compact and locally compact groups acting on manifolds is filled with unexplored areas and this is true in both the differentiable and non-differentiable cases.

A theorem of Lie groups

Roughly, the theorem says that each subgroup near enough to G* can be transformed into G* by an appropriate element of G. This result can be regarded as a generalization of the known fact that Lie groups cannot have arbitrarily small subgroups (other than the identity), although it was not from this point of view that our interest arose. To make our meaning clear, assume that G* is an invariant subgroup so that the factor group G/G* is also a Lie group. If there were in G a subgroup H near G* it would go, by the homomorphism taking G into G/G*, into a subgroup near the identity of G/G*. The only subgroup of G/G* near the identity is the identity itself which means that if H is to be near G* it must actually be a subgroup of G*. We see that when G* is an invariant compact subgroup of G, the conclusion of the theorem is true in a trivial sense. Our proof of Theorem 1 in the more general situation is based on certain facts about the way in which G operates on the coset space G/G* which will be denoted by M. This is the space whose points are the cosets gG* of G* in G. The group G acts transitively on M which can be regarded as a Riemannian space and Cartan has shown that there exists in M a Riemannian metric for which G is a group of isometries. This fact will be of great importance in what follows. We begin, as we may, by supposing that M is endowed with a Riemannian metric invariant under G and, furthermore, we assume that M has been made into a metric space (Fréchet) in the usual way

A study of continuous curves and their relation to the Janiszewski-Mullikin theorem

1. In this paper we treat of continuous curvest in n-dimensional euclidean space; the arguments, excepting the use of inversion,t are established in more general space.? The principal theorems are devoted to the relation of such curves to the Janiszewski-Mullikin Theorem. This, stated generally, is to the effect that two bounded? subcontinua of a space, C, neither of which disconnects C, can disconnect C in their sum if and only if their product is not connected.** The theorem is shown to characterise, among bounded cyclicly connected t t continuous curves, the simple closed surface; t t among bounded continuous curves in general, those whose maximal cyclicly connected subsets are simple closed surfaces; among unbounded cyclicly connected continuous curves, the cylinder-trees;?? and, in general, unbounded

The Moore-Kline problem

It has been shown by Moore and Kline t that in order that a closed subset M of the euclidean plane be contained in an arc of the plane, it is necessary and sufficient that (1) M be compact, (2) the maximal connected subsets (components) of M be arcs or points, (3) no inner point of any arc of M be a limit point of the complement (in M) of that arc. A closed point set with these properties we shall call a Moore-Kline set (or M. K. set) and we shall say that a topologic space has the Moore-Kline (M. K.) property if every M. K. subset is contained in an arc of that space. Our problem is the characterisation of spaces which have this property, in the universe of generalised continuous curves: i.e., complete, metric, separable, connected, and locally connected spaces.§ The characterisation which we give is, in an equivalent form, also valid for certain non-metric spaces developed by R. L. Moore, and the space of Aronszajn.|| The paper contains an extension to generalised continuous curves of a recent theorem of G. T. Whyburn,^f with an independent proof. 1. We shall prove for generalised continuous curves C the equivalence of the two following properties :

A theorem on the rotation group of the two-sphere

Let R be the group of all rotations of euclidean three-dimensional space E about its origin. We shall take the domain of operation of R to be a euclidean two-sphere 5 with center at the origin. On the space Sj R is transitive. The group R is topological and, considered as a space, is homeomorphic to projective three-space. While studying the action of groups in certain spaces, the following theorem, which, as far as We know, is not in the literature, occurred to us.