Euler characteristic positivity under a seven-torus action

Establish that every even-dimensional closed manifold admitting a Riemannian metric with positive second intermediate Ricci curvature invariant under a torus action of rank seven has positive Euler characteristic.

Background

The paper explains that proving a suitable analogue of the missing b3 Lemma would remove the congruence restriction used in the representation-theoretic argument. This would allow the authors to apply their rank-five matroid result and reduce the torus-rank requirement in the geometric application by two, motivating the proposed rank-seven statement.

The conjecture concerns closed even-dimensional manifolds with positive second intermediate Ricci curvature and a T^7-invariant metric. It asserts positivity of the Euler characteristic, extending the paper’s proved T^9-invariant result to a lower symmetry rank.