On the Eta Invariant, Stable Positive Scalar Curvature, and Higher $\hat{A}$-Genera

We show that the eta invariant provides obstructions to the existence, and stable existence, of positive scalar curvature metrics on odd dimensional closed connected spin smooth manifolds. We also prove, using the eta invariant, that the "stable existence" of a positive scalar curvature metric implies the vanishing of some higher $\hat{A}$-genera.

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