Monday

A natural way to generalize the Lichnerowicz obstruction to positive scalar curvature on spin manifolds is to find curvature conditions which imply that some twisted Dirac operators have vanishing index. Such vanishing has topological consequences for the manifold, in terms of its rational cobordism type. To make these generalizations most interesting, the curvature conditions should be as weak as possible, easily computable, and, ideally, invariant under appropriate surgeries. Following this scheme and inspired by recent works of Petersen and Wink, we determine linear inequalities on the eigenvalues of curvature operators that imply vanishing of the twisted hat A genus on a closed Riemannian spin manifold, where the twisting bundle is any prescribed parallel bundle of tensors. For instance, this yields a new obstruction to existence of Einstein metrics with 5-positive curvature operator on certain 8-manifolds, e.g., on HP^2. (This is based on joint work with Jackson Goodman.)