On the Hopf conjecture with symmetry (Q1945758)

Positively-curved spaces have been of interest since the beginning of global Riemannian geometry. There are few known examples and few topological obstructions to any given manifold admitting a positively curved metric. In fact, all known closed, simply connected examples in dimensions larger than 24 are spheres and projective spaces, and all known obstructions to positive curvature for closed, simply connected manifolds are already obstructions to nonnegative curvature. One famous conjectured obstruction to positive curvature was made by Heinz Hopf. It states that closed, even-dimensional manifolds admitting positive sectional curvature have positive Euler characteristic. This conjecture holds in dimensions two and four. In this paper, the author proves that if \(M^n\), where \(n\equiv 0 \mod 4\), is a connected, closed Riemannian manifold with positive sectional curvature, and admits an effective, isometric torus \(T^r\)-action with \(r\geq 2\log_2n-2\), then its Euler characteristic satisfyies \(\chi(M^n)>0\). Using the action of the Steenrod algebra on cohomology, the author also proves that if \(M^n\) is a closed, one-connected, positively curved manifold that contains a pair of totally geodesic, transversely intersecting submanifolds of codimensions \(k_1\leq k_2\) such that \(k_1+3\,k_2\leq n\), then its rational cohomology ring \(H^*(M;\mathbb Q)\) is \(\text{gcd}(4, k_1, k_2)\)-periodic. As a conclusion, it follows that if \(\text{gcd}(4, k_1, k_2)=1\), then \(M^n\) is a rational homology sphere and if \(\text{gcd}(4, k_1, k_2)=2\), then \(M^n\) has the rational cohomology of \(\mathbb S^n\) or \(\mathbb{CP}^{n/2}\). Also, it is proven that if \(M^n\) is a closed, one-connected manifold such that its integral cohomology ring \(H^*(M;\mathbb Z)\) is \(k\)-periodic with \(3k\leq n\), then the rational cohomology ring \(H^*(M; \mathbb Q)\) is \(\text{gcd}(4, k)\)-periodic.