Semigroups and Hilbert’s fifth problem. (English) Zbl 0823.22003

Hofmann, Karl H.

Math. Slovaca 44, No. 3, 365-377 (1994).

It was known and properly credited by Hilbert, but is has been widely forgotten or ignored by subsequent generations of historians and mathematicians, that aspects of Hilbert’s famous Fifth problem (appropriately stated and interpreted) are nearly identical in form to a problem raised implicitly in a broader context many decades earlier by N. H. Abel [J. Reine Angew. Math. 1, 11-15 (1826)]. By placing in parallel his own versions, reworked into modern language consistent with both men’s intentions, the present author evidently aims to cure some part of this collective amnesia.
Abel’s problem (1826). Determine all cancellable topological semigroup structures on a connected topological manifold.
Hilbert’s Fifth Problem (1900). Determine all topological group structures on a connected topological manifold.
After placing in their correct historical context the ideas which made possible Hilbert’s formulation of his Fifth Problem, the author moves to a quick but careful review of its solution, first for compact groups (H. Weyl, J. von Neumann, and L. S. Pontryagin), then for locally compact, connected Abelian groups (via Pontryagin-van Kampen duality), then for locally Euclidean groups (A. Gleason, D. Montgomery, and L. Zippin), finally for locally compact (almost) connected groups (K. Iwasawa, H. Yamabe). Much effort since the sixties has attached to the “structure theory of locally compact groups, which plays a fundamental role in the contemporary theory of the foundation of geometry, notably in the theory of locally compact connected projective planes...” The author suggests that this would hardly have surprised Hilbert, since already in 1900 he “spoke in the connection with Lie groups about the foundations of geometry.”
The final portion of the paper returns to the broader (semigroup) context suggested by Abel’s work. An important contribution which has received due attention is a theorem of R. Jacoby [Ann. Math., II. Ser. 66, 36-59 (1957; Zbl 0084.032)] to the effect that a locally Euclidean local group is a local Lie group. On the basis of this and work of D. R. Brown and R. S. Houston, the author and W. Weiss [Semigroup Forum 37, 93- 111 (1988; Zbl 0635.22003)] were able to achieve a “systematic clarification of the semigroup situation”, showing (here the author writes informally, taking some liberties in the interest of brevity) that “cancellative topological connected locally euclidean semigroups are analytic and closely tied to a Lie group.” Referring finally to recent extended publications of J. Hilbert, K. H. Hofmann, J. D. Lawson, and K.- H. Neeb, he concludes that “the Abel-Hilbert problem has led us to analytic semigroups which illustrate the phenomenon that a consistent continuation of the program initiated by Hilbert’s Fifth problem breaks the boundaries of group theory and has significant consequences in modern representation theory”.

Reviewer: W.W.Comfort (Middletown)

MSC:

22A15	Structure of topological semigroups
22-03	History of topological groups
01A55	History of mathematics in the 19th century
22D05	General properties and structure of locally compact groups
01A60	History of mathematics in the 20th century
22E05	Local Lie groups

Keywords:

Hilbert’s fifth problem; Abel’s problem; locally compact groups; locally Euclidean local group; local Lie group; analytic semigroups

Citations:

Zbl 0084.032; Zbl 0635.22003

References:

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