Let X be a smooth closed manifold. Write Diff(X) for its diffeomorphism group, BDiff(X) for its classifying space, and pi_k and H_k(-;Z) for homotopy and integral homology groups. Finite generation conjecture. If pi_1(X) is finite and dim X≠ 4, then pi_k(Diff(X)) and H_k(BDiff(X);Z) should be finitely generated for all k≥ 0. No counterexample is currently known; the conjecture concerns finiteness phenomena for diffeomorphism groups and their classifying spaces outside dimension four, under a finite fundamental-group hypothesis.

Finite generation conjecture for diffeomorphism groups of closed manifolds

The conjecture predicts finite generation of all homotopy groups of $\mathrm{Diff}(X)$ and homology groups of $B\mathrm{Diff}(X)$ when $\pi_1(X)$ is finite and $\dim X\ne 4$ . No source identifies its proposer or a complete resolution.

Known results

Kupers (2021) proved the conjecture for closed manifolds of even dimension at least $6$ with finite fundamental group, including finite generation of homotopy, homology, and cohomology groups.
Earlier work handled the all-degree finite-generation statement for $2$ -connected manifolds.
Baraglia, Auckly--Ruberman, and Konno produced infinitely generated diffeomorphism-group homotopy or homology in dimension $4$ ; these examples are outside the conjecture.

April 2026 survey update

Konno’s survey reiterates that no counterexample is known in the stated range and that even-dimensional cases are established. The supplied sources contain no claimed proof or counterexample covering all remaining dimensions.

Current status (as of August 2026): The conjecture is known in even dimensions at least $6$ , while dimension $4$ has counterexamples outside its scope and the remaining cases, including odd dimensions, are open.

Finite generation conjecture for diffeomorphism groups of closed manifolds

Primary source

Progress summary

Solutions 0