finite dimensional group is locally a manifold or product with a cantor set

Question

Is it the case that every compact topological group with finite topological dimension is locally homeomoprhic to a cantor set or $ \mathbb{R}^n $ or a product of a cantor set with $ \mathbb{R}^n $?

This is a follow up to:

Is every locally compact group/ homogeneous space locally homeomorphic to the Cantor set or an n-torus?

Answer 1

All topological spaces considered below are supposed to be Hausdorff.

Theorem. ([Pon, Th. 69]) Each compact group $G$ of finite dimension $r$ locally splits in a direct product of an $r$-dimensional Lie group $L$ and a zero-dimensional closed normal subgroup $N$ of $G$. More precisely, the group $G$ contains a subset $L$, homeomorphic to an open $r$-dimensional cube and zero-dimensional normal subgroup $N$ such that
a) Each element of $l\in L$ commutes with each element $n\in N$;
b) A set $V=LN$ is a neighborhood of the identity of $G$;
c) Each element $v\in V$ has a unique representation $v=ln$, where $l\in L$, $n\in N$, and both $l$ and $n$ are continuous functions of $v$;
d) If $l_1$ and $l_2$ are elements of $L$ such that $l_1l_2\in V$ then $l_1l_2\in L$, so $L$ is a local group with respect to the multiplication of $G$;
e) The local group $L$ is a Lie group.
The above directly implies that $G/N$ is a local Lie group. It turns out that we can put as $N$ any zero-dimensional closed normal subgroup of $G$ such that $G/N$ is a Lie group.

Concerning locally compact groups, I can tell on Abelian case. According to [Pon], a topological group $G$ is of compact origin if it has a compact neighborhood of the identity $V$ such that $G$ is its smallest subgroup containing $V$. It is easy to check that for each compact neighborhood $V$ of a locally compact group $G$, $\bigcup_{n\in\Bbb Z} V^n$ is an open subgroup of compact origin of $G$. I guess (I didn’t find a definition in [DPS]) that locally compact groups of compact origin are exactly locally compact compactly generated groups in sense of [DPS]. By Theorem 3.3.1 from [DPS], every compactly generated locally compact Abelian group is isomorphic to a product $\Bbb R^n\times \Bbb Z^m\times C$, where $m$ and $n$ are non-negative integers and $C$ is a compact Abelian group. Also by Theorem 51 and Example 69 from [Pon], an Abelian group $G$ of compact origin splits into a direct sum of a compact group and some number of copies of groups $\Bbb R$ and $\Bbb Z$, and in this decomposition the compact summand is defined uniquely as the largest compact subgroup of $G$. Also, by [Theorem 3.3.10, DPS], each locally compact Abelian group has a closed subgroup $H$ containing an open compact subgroup of $G$ such that $G\cong \Bbb R^n\times H$ for some non-negative integer $n$ (I guess “$\cong$” means “is isomorphic to”). The chapter on locally compact Abelian groups of [Pon] is finished by a claim that a locally compact locally connected finitely-dimensional second countable Abelian topological group is a Lie group. I guess [Pon] book does not considers a general case of non-Abelian locally compact groups, because the following chapters of [Pon] are devoted to the notion of a Lie group, structure of compact topological groups, locally isomorphic groups, Lie groups and algebras, and structure of compact Lie groups.

References

[DPS] Dikran N. Dikranjan, Ivan R. Prodanov, Luchezar N. Stoyanov. Topological Groups, Marcel Dekker, New-York, 1990.

[Pon] Lev Pontrjagin, Continuous groups, 2nd ed., M., (1954) (in Russian).

Answer 2

To complete the story in the case of general (Hausdorff) locally compact finite dimensional groups $G$. In section 4.9.3 of

Montgomery, Deane; Zippin, Leo, Topological transformation groups, Mineola, NY: Dover Publications (ISBN 978-0-486-82449-9). xi, 289 p. (2018). ZBL1418.57024.

it is proven that there exists a local (continuous) isomorphism of $G$ to the product of a compact totally disconnected group and a local Lie group.