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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Highly connected manifolds of positive $ p$-curvature


Authors: Boris Botvinnik and Mohammed Labbi
Journal: Trans. Amer. Math. Soc. 366 (2014), 3405-3424
MSC (2010): Primary 53C20, 57R90; Secondary 81T30
Published electronically: March 14, 2014
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Abstract: 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 manifolds with positive $ 2$-curvature in terms of the bordism groups $ \Omega ^{\operatorname {spin}}_*$, $ \Omega ^{\textup {string}}_*$, and by means of an $ \alpha $-invariant and a Witten genus $ \phi _W$. Here we use results from Anand Dessai (2009), which provide appropriate generators of the ring $ \Omega ^{\textup {string}}_*\otimes \mathbf {Q}$ in terms of ``geometric $ {\mathbb{C}}{\mathbf a}\mathbf {P}^2$-bundles'', where the Cayley projective plane $ {\mathbb{C}}{\mathbf a} \mathbf {P}^2$ is a fiber and the structure group is $ F_4$ which is the isometry group of the standard metric on $ {\mathbb{C}}{\mathbf a}\mathbf {P}^2$.


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Additional Information

Boris Botvinnik
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
Email: botvinn@math.uoregon.edu

Mohammed Labbi
Affiliation: Department of Mathematics, College of Science, University of Bahrain, 32038, Bahrain
Email: labbi@sci.uob.bh

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-05939-4
PII: S 0002-9947(2014)05939-4
Received by editor(s): January 30, 2012
Published electronically: March 14, 2014
Article copyright: © Copyright 2014 American Mathematical Society