Transformation groups in the theory of topological loops

Abstract

In a recent paper [5] the author has applied the techniques of topological transformation group theory to the study of certain topological loops. The purpose of this note is to show more explicitly the close connection between topological loops and topological transformation groups. It is shown that for every transformation group G which acts reasonably on a space X so that (a) there is a global cross-section a from the coset space GIG, into G for pEX, (b) 5(G/G,) is a strongly transitive collection of homeomorphisms of X, and (c) G, is compact, such a space X may be given a binary operation so that it becomes a topological loop with a left invariant uniformity. Conversely, it is shown that in certain cases a topological loop L allows a transformation group G to act reasonably on L with a cross-section from G/G1 for the identity 1 of L satisfying (a) and (b) above. In certain cases (c) will also be satisfied. The reader may consult K. H. Hofmann [3] for the appropriate terminology in topological loops, with the following exceptions. A topological loop L has a left invariant uniformity (see [5 ]) when there is a uniform structure 'IU on L compatible with its topology so that 'I has a base a of entourages satisfying (x, y) EB if and only if (ax, ay) EB for all BE 6 and a EL. An invariant uniformity is a left invariant uniformity which also satisfies a right invariant condition. If a and b are elements of a loop, the unique solutions x and y to the equations ax =b and ya = b will be designated in this note by a(-I)b and ba(-1), respectively. The parentheses embracing -1 are used as a reminder that a(1)6 is not in general the product of 'a- and b as in a group. (This is at least the third notation appearing in print for these solutions. At the present this seems to be the most satisfactory.) The reference for transformation group topics is Montgomery and Zippin [7]. If G is a transformation group acting on a space X, and if G, is the isotropy or stability group at pEX, then there is a canonical mapping 7r: G/Gp--X defined by r(gG.) =g(p). Whenever ir is a homeomorphism, G is said to act reasonably on X (see [8]). The canonical mapping Xis continuous, onto, and one-one when G acts effectively and transitively [7, p. 43], but not in general open. A cross-section from G/G, into G is a continuous mapping a of the coset