Kosniowski Conjecture

The Kosniowski conjecture asks for a universal linear lower bound, in terms of the dimension, on the number of isolated fixed points of a circle group action on a unitary manifold whose class in ordinary complex cobordism is nonzero. A proposed sharp form asserts , where is the Euler characteristic, equal in this setting to the number of fixed points (Lü et al. 2026).

Lü et al. (2026) report an infinite family of real 18-dimensional unitary manifolds with circle group actions, four isolated fixed points, and nonzero classes in complex cobordism. These would violate the proposed sharp inequality because . The claimed construction has not been independently verified here. It concerns stable complex structures and does not by itself refute the corresponding assertion for almost complex structures on the actual tangent bundle. Nor would this family rule out every possible universal linear bound.