# Transitive Lie group actions and exotic smooth tori

Let $$M$$ be a smooth manifold homeomorphic to a torus. Suppose that $$M$$ admits a smooth transitive action by a finite dimensional Lie group $$G$$. Can we conclude that $$M$$ is diffeomorphic to the standard torus $$T^n \cong \mathbb{R}^n/\mathbb{Z}^n$$?

Note that in general not every manifold homeomorphic to a torus is diffeomorphic to the standard torus. For example exotic smooth tori are discussed here

https://mathoverflow.net/questions/133797/do-there-exist-exotic-4-tori

• I'm guessing this question arose from the comments on your MO question. The argument by Vitali Kapovitch there shows that if such an action exists on an exotic torus, then $G$ is necessarily non-compact. Feb 20, 2022 at 9:42
• @MichaelAlbanese yes everything you say is true. However it turns out no $G$ can ever act transitively on an exotic torus. See the answer I just posted. All results due to paper of Gorbatsevich. Feb 20, 2022 at 11:33

In fact something much stronger is true. A compact smooth homogeneous $$K(\pi,1)$$ with fundamental group $$\pi$$ nilpotent is determined up to diffeomorphism by $$\pi$$. See Corollary 2.7 and the discussion above Theorem 2.1 in [Gorbatsevich, ON LIE GROUPS, TRANSITIVE ON COMPACT SOLVMANIFOLDS].

So, in particular, a compact smooth homogeneous $$K(\mathbb{Z}^n,1)$$ must be diffeomorphic to the torus $$T^n$$ with the standard smooth structure.

The intuition here is that if $$G$$ acts transitively on a compact $$K(\pi,1)$$ and $$\pi$$ is solvable then we expect that the maximal solvable subgroup of $$G$$ will also act transitively (and it is a theorem of Auslander that compact solvmanifolds are determined up to diffeomorphism by $$\pi$$). Compare this to theorem of Montgomery that if $$G$$ acts transitively on a compact manifold $$M$$ with $$\pi_1(M)$$ compact (i.e. finite since the fundamental group is discrete) then the maximal compact subgroup of $$G$$ also acts transitively. Anyway the idea here is just that knowing what kind of Lie groups can act transitively on your manifold often tells you cool stuff about the topology of the manifold.

So that's the intuition but the details are a bit complicated.

Gorbatsevich conjectures that any compact smooth homogeneous $$K(\pi,1)$$ with $$\pi$$ solvable is determined up to diffeomorphism by $$\pi$$. The idea is to prove that such a space admits a transitive action by a solvable Lie group and then use the result of Auslander that a compact solvmanifold is determined up to diffeomorphism by its fundamental group. But he says that he can't figure out how to make the proof work :(

He does, however, prove that a compact smooth homogeneous $$K(\pi,1)$$ with $$\pi$$ solvable is determined up to homeomorphism by $$\pi$$ [Proposition 1.5].

For the case of diffeomorphism, which is what concerns us here, he does the proof for the weaker case of $$\pi$$ virtually nilpotent (which is good enough for us because a torus even has $$\pi$$ abelian!). He notes that every hypersolvable group is virtually nilpotent so this covers some important solvable non-nilpotent cases anyway.