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.

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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.

In the first part we give necessary and sufficient conditions for the existence of a stable almost complex structure on a 10-manifold M with H 1 ( M ; Z) = 0 and no 2-torsion in H i ( M ; Z) for i = 2,3. Using the Classification Theorem of Donaldson we give a reformulation of the conditions for a 4-manifold to be almost complex in terms of Betti numbers and the dimension of the f-eigenspaces of the intersection form. In the second part we give general conditions for an almost complex manifold to admit infinitely many almost complex structures and apply these to symplectic manifolds, to homogeneous spaces and to complete intersections.In Theorem 1.2 we give necessary and sufficient conditions for a 10-manifold M to be stably almost complex if H l ( M ; Z ) =: 0 and H i ( M ; Z ) , i = 2,3, has no 2torsion. No assumption on w4(M) is made. The theorem simplifies if M has "nice" cohomology, such as the one of complete intersections (cf. Corollary 1.3).In Theorem 1.4 we give, using the Classification Theorem of Donaldson, a reformulation of the classical conditions for a 4 -manifold M to be almost complex in terms of the Betti numbers and the dimension of the feigenspaces of the intersection form.In the second part we study the question whether an almost complex 2n -dimensional manifold admits infinitely many almost complex structures. In Theorem 2.2 and Theorem 2.3 we give a simple answer for a large class of manifolds. We apply these theorems to symplectic manifolds (cf. Corollary 2.4), homogeneous spaces (cf. Corollary 2.5) and complete intersections (cf. Example 2.6). Germany e -mail: dessai4mathpool. Uni -Augsburg.DE D -86135 Augsburg

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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