We prove that the Â-genus vanishes on certain non-spin manifolds. Namely, Â(M) vanishes on any oriented, compact, connected, smooth manifold M with finite second homotopy group and endowed with non-trivial (isometric) smooth S¹ actions. This result extends that of Atiyah and Hirzebruch on spin manifolds endowed with smooth S¹ actions [1] to manifolds which are not necessarily spin.

We prove such vanishing by means of the elliptic genus defined by Ochanine [23, 24], showing that it also has the special property of being "rigid under S¹ actions" on these (not necessarily spin) manifolds.

We conclude with a non-trivial application of this new vanishing theorem by classifying the positive quaternion-Kähler 12-manifolds. Namely, we prove that every quaternion-Kähler 12-manifold with a complete metric of positive scalar curvature must be a symmetric space.

By applying the symplectic cutting operation to cotangent bundles, one can construct a large number of interesting symplectic cones. In this paper we show how to attach algebras of pseudodifferential operators to such cones and describe the symbolic properties of the algebras.

We classify the biquotient manifolds which are either rational homology spheres or Cheeger manifolds (connected sums of two rank-one symmetric spaces). Also: there are only finitely many 2-connected biquotient manifolds in each dimension.

We classify compact surfaces with torsion-free affine connections for which every geodesic is a simple closed curve. In the process, we obtain completely new proofs of all the major results [4] concerning the Riemannian case. In contrast to previous work, our approach is twistor-theoretic, and depends fundamentally on the fact that, up to biholomorphism, there is only one complex structure on ℂℙ₂.