For every integer n≥ 1, there exists a constant C(n) such that, for every closed n-dimensional manifold M admitting Riemannian metrics g_i with sec_g_i≥ -1/i and operatornamediam(M,g_i)≤ 1, the fundamental group pi_1(M) contains an abelian subgroup of index at most C(n).

Fukaya–Yamaguchi and Kapovitch–Petrunin–Tuschmann conjectures

These conjectures predict strong, dimension-dependent control of fundamental groups for manifolds with almost nonnegative curvature. The latest claim treats only manifolds equipped with torus actions, not the conjectures in full generality.

Known results

Bruè (2025) proved the Fukaya–Yamaguchi conclusion for nonnegative curvature in dimensions $n \le 4$ , obtaining an abelian subgroup of index at most $C(n)$ .
Kapovitch–Petrunin–Tuschmann proved the relevant uniform-index nilpotence conclusion under a specified tower-of-fiber-bundles condition.

August 27, 2026 claimed symmetry result

The preprint Torus actions, almost non-negative curvature and fundamental groups claims proofs of both conjectures in the presence of torus actions. This is a genuine restricted-setting advance, but the proofs are not yet peer reviewed or independently verified.

Current status (as of August 2026): The Fukaya–Yamaguchi conjecture is known in dimensions $n \le 4$ , and the Kapovitch–Petrunin–Tuschmann conclusion is known in a conditional bundle setting; both conjectures remain open in general, while the torus-action claims are unverified.

Fukaya–Yamaguchi and Kapovitch–Petrunin–Tuschmann conjectures

Primary source

Additional references

Progress summary

Solutions 0