Gleason’s Contribution to the Solution of Hilbert’s Fifth Problem Richard Palais * May 15, 2009 1 What IS Hilbert’s Fifth Problem? Andy Gleason is probably best known for his work contributing to the solu- tion of Hilbert’s Fifth Problem. We shall discuss this work below, but first we need to know just what the “Fifth Problem” is. In its original form it asked, roughly speaking, whether a continuous group action is analytic in suitable coordinates. But as we shall see, the meaning has changed over time. As Hilbert stated it in his lecture delivered before the International Congress of Mathematicians in Paris in 1900[Hi], the Fifth Problem is linked to Sophus Lie’s theory of transformation groups[L], i.e., Lie groups acting as groups of transformations on manifolds. The “groups” that Lie dealt with were really just neighborhoods of the identity in what we now call a Lie group, and his group actions were only defined locally, but we will ignore such local versus global considerations in what follows. However it was cru- cial to the techniques that Lie used that his manifolds should be analytic and that both the group law and the functions defining the action of the group on the manifold should be analytic – that is, given by convergent power series. For Lie, who applied his theory to such things as studying the symmetries of differential equations, the analyticity assumptions were natural enough. But Hilbert wanted to use Lie’s theory as part of his logical foundations of geometry, and for this purpose Hilbert felt that analyticity was unnatural, and perhaps superfluous. So Hilbert asked if analyticity could be dropped in favor of mere continuity. More precisely, if one only assumed a priori * Richard Palais is Professor Emeritus at Brandeis University and Adjunct Professor at the University of California at Irvine. His email address is palais@uci.edu 1