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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Highly connected manifolds of positive $p$-curvature
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by Boris Botvinnik and Mohammed Labbi PDF
Trans. Amer. Math. Soc. 366 (2014), 3405-3424 Request permission

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 $\Omega ^{\operatorname {spin}}_*$, $\Omega ^{\mathrm {string}}_*$, 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 $\Omega ^{\mathrm {string}}_*\otimes \mathbf {Q}$ 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
  • MR Author ID: 235944
  • Email: botvinn@math.uoregon.edu
  • Mohammed Labbi
  • Affiliation: Department of Mathematics, College of Science, University of Bahrain, 32038, Bahrain
  • Email: labbi@sci.uob.bh
  • Received by editor(s): January 30, 2012
  • Published electronically: March 14, 2014
  • © Copyright 2014 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 3405-3424
  • MSC (2010): Primary 53C20, 57R90; Secondary 81T30
  • DOI: https://doi.org/10.1090/S0002-9947-2014-05939-4
  • MathSciNet review: 3192601