Semigroups and Hilbert's fifth problem
Karl Heinrich Hofmann · TUbilio (Technical University of Darmstadt) · 1994
We revisit the full content of Hilbert's Fifth problem which asks whether topological groups on manifolds are automatically analytical.It is not too well know that this same problem has history in the case of topological semigroup, too.arrd this history can be traced back to Abel.We explain what is known in this regard and lead up to contemporary problems in the Lie theory of semigroups.Iii the year 1900, at the International Congress of Mathematicians in Paris, ) a v i (I II i 1 b ert formulated 23 Problems which, on the basis provided by he achievements of 19th century mathematics, would decisively influence the •ourse of the history of mathematics in the 20th century.Among these, one of he most familiar is the fifth; it deals with the transformation groups introduced luring the preceding decades by Sop h u s L i e who died in 1899.It is comnoitlv known that essential aspects of It i 1 b e r t 's Fifth Problem were solved >y papers published by G 1 e a son, and M o n t g o m e r y (f March 15, 1992) uid X i p p i n in the year 1952.What is less commonly known is the fact that I i 1 b e r t "s Fifth Problem has other aspects, among which there is a semigroup heoret ieal one.It is this aspect on which I shall focus in this essay ^ which I mi honored to dedicate to Stefan Schwarz, who among a generation >f pioneers of semigroup theory such as A .II .C 1 i f f o r d, P .D u b r e i 1 , \ .II i 1 1 e.V .V .V a gner.A .I) .W a. 1 I a c e promoted the algebraic and he topological theory of semigroups so significantly.