Topological Zimmer conjecture for low-dimensional compact manifolds

Let $n\geq 3$, and let $M^r$ be a connected compact $r$-dimensional manifold with $r<n-1$. A group action of $\mathrm{SL}_{n}(\mathbb{Z})$ on $M$ by homeomorphisms is equivalently a homomorphism

$$\mathrm{SL}_{n}(\mathbb{Z})\longrightarrow \operatorname{Homeo}(M).$$

Topological Zimmer conjecture. Every such action factors through a finite group; equivalently, every homomorphism

$$\mathrm{SL}_{n}(\mathbb{Z})\longrightarrow \operatorname{Homeo}(M)$$

has finite image.

This is the topological form of Zimmer's program for special linear groups. The survey records proofs in several cases, including one-dimensional manifolds, tori, spheres, products of two spheres, flat manifolds, nilpotent manifolds, and certain orientable manifolds, but the general statement remains open.