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 give an elementary proof of the rational Hurewicz theorem and compute the rational cohomology groups of Eilenberg–MacLane spaces and the rational homotopy groups of spheres. Instead of using the Serre spectral sequence, we only assume the classical Hurewicz theorem, and give a short proof of the rational Gysin and Wang long exact sequences, which are applied inductively to the path fibration of Eilenberg–MacLane spaces.

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A ®nitely iterated pullback of path ®brations is called a tower. A pyramid is de®ned as a ®nite sequence of maps with zero-homotopies of their successive compositions, and also with a system of higher zero-homotopies for all successive compositions of lower zerohomotopies. For the category of towers and the category of pyramids , we de®ne maps P X 3 and T X 3 which connect both concepts. The map P is an enhancement of the associated chain complex of a tower, which forms the bottom sequence of the associated pyramid. The map T is de®ned using a glueing construction for higher zero-homotopies. We show for a pyramid Y of length r that the associated chain complex of TY is homotopy equivalent to the r À 1-fold looping of the chain complex of Y . This gives in particular a characterization of the admissible chain complexes over the Steenrod algebra in the sense of Maunder: a chain complex comes from a higher cohomology operation if and only if its Toda bracket vanishes.1991 Mathematics Subject Classi®cation: 55P35, 55P42, 55S20, 55S45.Brought to you by |