 The " restricted " version of Hilbert 's fifth problem ( solved by Gleason ) asks , more specifically , whether every locally Euclidean topological group is a Lie group . That is , if a group G has the structure of a topological manifold , can that structure be strengthened to a real analytic structure , so that within any neighborhood of an element of G , the group law is defined by a convergent power series , and so that overlapping neighborhoods have compatible power series definitions ? Prior to Gleason 's work , special cases of the problem had been solved by Luitzen Egbertus Jan Brouwer , John von Neumann , Lev Pontryagin , and Garrett Birkhoff , among others . 
