Euler Characteristic of Closed Manifolds with Almost Nonnegative Curvature Operator

We study closed manifolds with almost nonnegative curvature operator and address a question of Herrmann--Sebastian--Tuschmann concerning the sign of their Euler characteristic. Our main result shows that if a closed $2n$-dimensional manifold admits an almost nonnegative curvature operator together with a uniform upper bound on the curvature operator, then its Euler characteristic is nonnegative. In addition, under an ANCO-type condition and assuming that the fundamental group is infinite, we prove vanishing results for the Euler characteristic, the signature, and, in the spin case, the $\widehat{A}$-genus, extending recent work of Chen--Ge--Han from almost nonnegative Ricci curvature to the curvature-operator setting.