Haydeé Herrera

The signature and the elliptic genus of π2-finite manifolds with circle actions

Abstract We prove the vanishing of various characteristic numbers, such as the signature and the A -genus, on manifolds with finite second homotopy group and which admit smooth S1-actions. More precisely, we prove the vanishing of various coefficientes of the elliptic genus on non-spin π2-finite manifolds when the S1-action either satisfies a “parity” condition or has isolated fixed points only.

Elliptic genera on non-spin Riemannian symmetric spaces with b2=0

We prove that the elliptic genera of the real Grassmannian Gr4(R2m+5), m≥1, and the space F4/Sp(3)Sp(1) are identically zero. These vanishings are consistent with the rigidity under S1 actions of the elliptic genera on these non-spin manifolds, and imply that their signatures are zero.

Generalized Elliptic Genus and Cobordism Class of Nonspin Real Grassmannians

We prove that Wittens generalized elliptic genusis rigid under certain group actions and constant(identically zero) on the real Grassmanniansℝr4ℝ2m+5, m ≥ 1, for appropriate vectorbundles.We also prove that all the Pontryagin numbers of theseGrassmannians are zero, i.e. they have cobordism class zero,which implies the vanishing of all genera.