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Hilbert’s fifth problem for local groups. (English) Zbl 1219.22004
The author has started two versions of Hilbert’s fifth problem written below after some definitions:
(a) A local group is a tuple $$(G,1,\iota,t, p)$$ (or simply, $$G$$ in short) where $$G$$ is a Hausdorff topological space with a distinguished element $$1\in G$$, an $$\iota:\Lambda\to G$$ (the inversion map) and $$p:\Omega\to G$$ (the product map) are continuous functions with $$\Lambda\subseteq G$$ and open $$\Omega\subseteq G\times G$$, such that $$1\in\Lambda$$, $$\{1\}\times G\subseteq\Omega$$, $$G\times \{1\}\subseteq\Omega$$, and for all $$x,y,z\in G$$:
(1) $$p(1, x)= p(x, 1)= x$$; (2) if $$x\in\Lambda$$, then $$(x, \iota(x))\in\Omega$$, $$(\iota(x), x)\in\Omega$$ and $$p(x,\iota(x))= p(\iota(x), x)= 1$$; (3) if $$(x, y), (y, z)\in\Omega$$ and $$(p(x, y),z)$$, $$(x, p(y,z))\in\Omega$$, then (A) $$p(p(x,y),z)= p(x,p(y,z))$$;
(b) $$G$$ is locally Euclidean if there is an open neighbourhood of 1 homeomorphic to an open subset of $$\mathbb{R}^n$$ for some $$n$$;
(c) $$G$$ is a local Lie group if $$G$$ admits a $$C^\infty$$ structure such that the maps $$\iota$$ and $$p$$ are $$C^w$$;
(d) with $$U$$ an open neighbourhood of 1 in $$G$$, the restriction of $$G$$ to $$U$$ is the local group $$G/U:= (U,\iota/\Lambda_U, p/\Omega_U)$$, where $$\Lambda_U:= \Lambda\cap U\cap \iota^{-1}(U)$$ and $$\Omega_U:= (U\times U)\cap p^{-1}(U)$$;
(e) $$G$$ is globalizable if there is a topological group $$H$$ and an open neighbourhood $$U$$ of $$1_H$$ in $$H$$ such that $$G= H/U$$.
Local $$H5$$ first form: If $$G$$ is a locally Euclidean local group, then some restriction of $$G$$ is a local Lie group. Local $$H5$$ second form: If $$G$$ is a locally Euclidean local group, then some restriction of $$G$$ is globalizable.
The equivalence of the two forms is established. The author follows the non-standard treatment of Hilbert’s fifth problem given by Hirschfeld; in dealing with local $$H5$$, the author has a simplified way in the non-standard method.

##### MSC:
 22A30 Other topological algebraic systems and their representations 22E99 Lie groups
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##### References:
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