# Homogeneous spaces that are homotopy tori

Let $G$ be a compact Lie group, and let $H$ be a closed subgroup such that $G/H$ is homotopy equivalent to a torus. Is it true that $H$ is normal and $G/H$ is isomorphic to a torus as a Lie group?

One line of attack would be to reduce to the case where $G$ is connected, then to the case where $G$ is the product of a torus with a simply connected group, then try to use the classification of simply connected groups. A proof like that would be OK, but I would actually prefer a more uniform proof that just works directly with $G$.

• I would have guessed this question was answered by Wu-Chung Hsiang, or someone from that era. I believe Wu-Chung answered the analogous question of homotopy spheres which are homogenous (or quotients of homogenous spaces by isometries). – Ryan Budney Dec 3 '14 at 18:54

Reduce to the case when $G$ is connected.
Because $H$ is compact, $\pi_0(H)$ is finite, so by the homotopy long exact sequence $\pi_1(G)\to \pi_1(G/H) = \mathbb Z^n$ maps onto a finite index subgroup, hence $H^1(G/H, \mathbb Z) \to H^1(G, \mathbb Z)$ is surjective. Fix an invariant metric on $G$ and, applying Hodge theory, represent a basis of $H^1(G/H, \mathbb R)$ by harmonic $1$-forms.
The harmonic $1$-forms are unique, so since the cohomology classes are $G$-invariant, the $1$-forms are $G$-invariant. Hence they give a homomorphism from the Lie algebra of $G$ to the abelian Lie algebra of rank $n$. This gives a homomorphism from the universal cover of $G$ to $\mathbb R^n$ and hence a, after modding out by the lattice corresponding to the homology of $G/H$, a homomorphism from $G$ to a torus.
The image of $H$ in this torus is a closed subgroup, hence a finite union of tori. It cannot contain a nontrivial torus or else there would be a nontrivial map from $\pi_1$ of $H$ to $\pi_1$ of that nontrivial torus to $\pi_1$ of $G/H$. So it is a finite union of points. But the map from $\pi_0$ of $H$ to the torus factors through $H_1(G/H)$, hence factors through the zero map. So the image of $H$ is trivial. If $H'$ is the kernel of the map to this torus, then $G/H \to G/H'$ is a homotopy equivalence. Its also a fibration, so the fiber is contractible. The fiber is a compact manifold, so it's a point. Hence $H=H'$ and $G/H=G/H'$ is a torus.