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Highly connected manifolds of positive $p$-curvature

Boris Botvinnik, Mohammed Labbi
2014 Transactions of the American Mathematical Society  
We study and in some cases classify highly connected manifolds which admit a Riemannian metric with positive p-curvature. The p-curvature was defined and studied by the second author in earlier papers. It turns out that the positivity of p-curvature is preserved under surgeries of codimension at least p + 3. This gives a key to reducing a geometrical classification problem to a topological one, in terms of relevant bordism groups and index theory. In particular, we classify 3-connected
more » ... -connected manifolds with positive 2-curvature in terms of the bordism groups Ω spin * , Ω string * , and by means of an α-invariant and a Witten genus φ W . Here we use results from Anand Dessai (2009), which provide appropriate generators of the ring Ω string * ⊗ Q in terms of "geometric CaP 2 -bundles", where the Cayley projective plane CaP 2 is a fiber and the structure group is F 4 which is the isometry group of the standard metric on CaP 2 .
doi:10.1090/s0002-9947-2014-05939-4 fatcat:5qu2az5wjnafpgun7g4r4aywg4