Cancellative semigroups on manifolds. (English) Zbl 0626.22001

Brown, D. R.; Houston, R. S.

Semigroup Forum 35, 279-302 (1987).

This paper gives a positive solution to Hilbert’s fifth problem for cancellative semigroups on a manifold. Indeed a connected, paracompact locally Euclidean space supporting a cancellative topological semigroup structure carries a unique analytic structure such that the semigroup multiplication is analytic. Moreover, there is a Lie group attached to it into which local patches of the semigroup can be embedded in a compatible fashion. Considering how little is actually known on open subsemigroups of Lie groups, one recognizes how difficult the subject matter is. Even after further work in the same line, many unsettled questions remain.
The paper has been in the making since the mid-seventies, and some people in the field knew of its existence. A natural tool is invariance of domain. The solution of Hilbert’s fifth problem is needed in its strongest form, namely the local one due to R. Jacoby [Ann. Math., II. Ser. 66, 36-69 (1957; Zbl 0084.032)]. A considerable amount of semigroup theory needs to be organized in order to define local quotient groups attached to certain pairs of open sets in the manifold S. It are these local quotient groups which turn out to be local Lie groups, all being locally isomorphic to one and the same simply connected Lie group G. This is the source of the analytic structure on S.
The authors show that a homomorphism \(S\to G\) exists which is a local diffeomorphism provided that S is simply connected. (It has been shown in the meantime by Weiss and the reviewer that even without this hypothesis there will be always a certain covering semigroup of S allowing such a homomorphism into G [see the reviewer and W. Weiss, Semigroup Forum 37, 93-111 (1988)].) Semigroup morphisms between cancellative semigroups on manifolds are always analytic. The assignment of the Lie group if functorial.
A systematic treatment of the theory will also become available in “Lie groups, convex cones, and semigroups”, Chapter VII, by J. Hilgert, the reviewer, and J. D. Lawson (to appear).

Reviewer: K.H.Hofmann

MSC:

22A15	Structure of topological semigroups
20M99	Semigroups
22E05	Local Lie groups
22A20	Analysis on topological semigroups
58H99	Pseudogroups, differentiable groupoids and general structures on manifolds

Keywords:

Hilbert’s fifth problem; cancellative semigroups; manifold; topological semigroup; Lie group; local quotient groups; local Lie groups

Citations:

Zbl 0084.032

References:

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