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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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Kreck-Stolz invariants for quaternionic line bundles
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by Diarmuid Crowley and Sebastian Goette PDF
Trans. Amer. Math. Soc. 365 (2013), 3193-3225 Request permission


We generalise the Kreck-Stolz invariants $s_2$ and $s_3$ by defining a new invariant, the $t$-invariant, for quaternionic line bundles $E$ over closed spin-manifolds $M$ of dimension $4k-1$ with $H^3(M; \mathbb Q) = 0$ such that $c_2(E)\in H^4(M)$ is torsion. The $t$-invariant classifies closed smooth oriented $2$-connected rational homology $7$-spheres up to almost-diffeomorphism, that is, diffeomorphism up to a connected sum with an exotic sphere. It also detects exotic homeomorphisms between such manifolds.

The $t$-invariant also gives information about quaternionic line bundles over a fixed manifold, and we use it to give a new proof of a theorem of Feder and Gitler about the values of the second Chern classes of quaternionic line bundles over $\mathbb H P^k$. The $t$-invariant for $S^{4k-1}$ is closely related to the Adams $e$-invariant on the $(4k-5)$-stem.

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Additional Information
  • Diarmuid Crowley
  • Affiliation: Hausdorff Research Institute for Mathematics, Universität Bonn, Poppelsdorfer Allee 82, D-53115 Bonn, Germany
  • Email:
  • Sebastian Goette
  • Affiliation: Mathematisches Institut, Universität Freiburg, Eckerstr. 1, 79104 Freiburg, Germany
  • Email:
  • Received by editor(s): January 5, 2011
  • Received by editor(s) in revised form: October 22, 2011, and October 25, 2011
  • Published electronically: November 20, 2012
  • Additional Notes: The second author was supported in part by SFB-TR 71 “Geometric Partial Differential Equations”
  • © Copyright 2012 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 3193-3225
  • MSC (2010): Primary 58J28, 57R55; Secondary 57R20
  • DOI:
  • MathSciNet review: 3034463