Positive Ricci curvature on highly connected manifolds (Q1700287)

The authors construct metrics of positive Ricci curvature on highly connected odd-dimensional manifolds. Let \(M\) be a manifold of dimension \(4k-1\geq7\). Assume that \(M\) is \((2k-2)\)-connected (i.e., \(\pi_jM=0\) for \(j\leq 2k-2\)) and \((2k-1)\) parallelisable (i.e., the tangent bundle restricted to some \(2k-1\) skeleton is trivial). The authors show: Theorem 1. Let \(M\) be a manifold of dimension \(4k-1\geq7\) which is \((2k-2)\)-connected and \((2k-1)\) parallelisable. Then there exists a homotopy sphere \(\Sigma^{4k-1}\) such that the connected sum of \(M\) with \(\Sigma\) admits a metric of positive Ricci curvature. From this they derive the following consequence Theorem 2. All 2-connected 7-dimensional manifolds and all 4-connected 11-dimensional manifolds admit metrics of positive Ricci curvature. The authors also provide the following result in dimension \((4k+1)\). Theorem 3. Let \(N\) be a \((2k-1)\)-connected manifold of dimension \(4k+1\) which is \(2k\) parallelisable and so that \(H^*(N;\mathbb{Z})\) is torsion-free. Then there is a homotopy sphere \(\Sigma\) of dimension \(4k+1\) such that the connected sum of \(N\) with \(\Sigma\) admits a metric of positive Ricci curvature. The authors use bordisms to establish these results; the main technique involved is a new topological description of highly connected manifolds using plumbing. The authors provide a good geometric introduction to the subject in the first section. In the second section, they discuss plumbing. The third section deals with extended quadratic forms and their boundaries; these are complete invariants of handle bodies which are treated in the fourth section. The fifth section deals with highly connected manifolds of dimension \(4k-1\) and uses bordism theory. The sixth section discusses realizing linking forms as the boundaries of tree-like forms. The seventh section treats tree-like plumbings and their boundaries in other dimensions. The final section discusses symmetric forms which are stably tree-like.