Abstract. The Hopf conjecture states that an even-dimensional, positively curved Riemannian manifold has positive Euler characteristic. We prove this conjecture under the additional assumption that a torus acts by isometries and has dimension bounded from below by a logarithmic function of the manifold dimension. The main new tool is the action of the Steenrod algebra on cohomology.Positively curved spaces have been of interest since the beginning of global Riemannian geometry. Unfortunately, there are few known examples (see [15] for a survey and [8,5,7] for recent examples) and few topological obstructions to any given manifold admitting a positively curved metric. In fact, all known simply connected examples in dimensions larger than 24 are spheres and projective spaces, and all known obstructions to positive curvature for simply connected manifolds are already obstructions to nonnegative curvature.One famous conjectured obstruction to positive curvature was made by H. Hopf in the 1930s. It states that even-dimensional manifolds admitting positive sectional curvature have positive Euler characteristic. This conjecture holds in dimensions two and four by the theorems of Gauss-Bonnet or Bonnet-Myers (see [3,4]), but it remains open in higher dimensions.In the 1990s, Karsten Grove proposed a research program to address our lack of knowledge in this subject. The idea is to study positively curved metrics with large isometry groups. This approach has proven to be quite fruitful (see [14,6] for surveys). Our main result falls into this category: Theorem 1. Let M n be a closed Riemannian manifold with positive sectional curvature and n ≡ 0 mod 4. If M admits an effective, isometric T r -action with r ≥ 2 log 2 n, then χ(M ) > 0.Previous results showed that χ(M n ) > 0 under the assumption of a linear bound on r. For example, a positively curved n-manifold with an isometric T r -action has positive Euler characterisic if n is even and r ≥ n/8 or if n ≡ 0 mod 4 and r ≥ n/10 (see [10,11]).Theorem 1 easily implies similar results where the assumption on the symmetry rank, i.e. rank(Isom(M )), is replaced by one where the symmetry degree, i.e. dim(Isom(M )), is large or the cohomogeneity, i.e. dim(M/ Isom(M )), is small (see Section 6).A key tool is Wilking's connectedness theorem (see [12]), which has proven to be fundamental in the study of positively curved manifolds with symmetry. The theorem relates the topology of a closed, positively curved manifold with that of its totally geodesic submanifolds of small codimension. Since fixed-point sets of isometries are totally geodesic, this becomes a powerful tool in the presence of symmetry.Part of the utility of the connectedness theorem is to allow proofs by induction over the dimension of the manifold. Another important implication is a certain periodicity in cohomology. By using the action of the Steenrod algebra on cohomology, we refine this periodicity in some cases. For example, we prove:

Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics.In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. We apply our results to prove obstructions to symmetric spaces, products of manifolds, and connected sums admitting positively curved metrics with symmetry.

Simply-connected manifolds of positive sectional curvature $M$ are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipticity to obtain several topological properties of the underlying manifold. These results include a small upper bound on the Euler characteristic and confirmations of famous conjectures by Hopf and Halperin under additional torus symmetry. We prove several cases (including all known even-dimensional examples of positively curved manifolds) of a conjecture by Wilhelm

A famous conjecture of Hopf is that the product of the two-dimensional sphere with itself does not admit a Riemannian metric with positive sectional curvature. More generally, one may conjecture that this holds for any nontrivial product. We provide evidence for this generalized conjecture in the presence of symmetry.Comment: 10 page

This is the first part of a series of papers where we compute Euler characteristics, signatures, elliptic genera, and a number of other invariants of smooth manifolds that admit Riemannian metrics with positive sectional curvature and large torus symmetry. In the first part, the focus is on even-dimensional manifolds in dimensions up to 16. Many of the calculations are sharp and they require less symmetry than previous classifications. When restricted to certain classes of manifolds that admit non-negative curvature, these results imply diffeomorphism classifications. Also studied is a closely related family of manifolds called positively elliptic manifolds, and we prove the Halperin conjecture in this context for dimensions up to 16 or Euler characteristics up to 16.2010 Mathematics Subject Classification. 53C20 (Primary), 57N65 (Secondary).