We show that in each dimension 4n + 3, n ≥ 1, there exist infinite sequences of closed smooth simply connected manifolds M of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension seven, and our result also holds for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, in conjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such M the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold M × R also has infinitely many path components.
We show that the indices of certain twisted Dirac operators vanish on a Spin-manifold M of positive sectional curvature if the symmetry rank of M is ≥ 2 or if the symmetry rank is one and M is two connected. We also give examples of simply connected manifolds of positive Ricci curvature which do not admit a metric of positive sectional curvature and positive symmetry rank. 1Note that the indexÂ(M, T M ) does not vanish for the quaternionic plane. Since the symmetry rank of HP 2 (with its standard metric) is three the lower bound on the dimension of M is necessary.We believe that the vanishing ofÂ(M, T M ) also holds under weaker symmetry assumptions. For 2-connected manifolds we show Theorem 1.2. Let M be a closed 2-connected manifold of dimension > 8. If M admits a metric of positive curvature with effective isometric S 1 -action then A(M ) andÂ(M, T M ) vanish.The proofs of Theorem 1.1 and Theorem 1.2 are rather indirect. For both statements we study the fixed point manifold of isometric cyclic subactions. The Bott-Taubes-Witten rigidity theorem [57,53,8] for elliptic genera implies that
Abstract. We give a survey on geometric properties of the Witten genus. The survey focuses on relations between the Witten genus, group actions and positive curvature.
We prove that an effective action of a torus T on a homotopy CP m is linear if m < 4 · rk(T ) − 1. Examples show that the bound is optimal. Combining this with a theorem of Hattori we conclude that the total Pontrjagin class of such a manifold is given by the usual formula (1 + x 2 ) m+1 .
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