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Transactions of the American Mathematical Society

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Kreck-Stolz invariants for quaternionic line bundles

Authors: Diarmuid Crowley and Sebastian Goette
Journal: Trans. Amer. Math. Soc. 365 (2013), 3193-3225
MSC (2010): Primary 58J28, 57R55; Secondary 57R20
Published electronically: November 20, 2012
MathSciNet review: 3034463
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Abstract: 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

Sebastian Goette
Affiliation: Mathematisches Institut, Universität Freiburg, Eckerstr. 1, 79104 Freiburg, Germany

Keywords: $𝜂$-invariant, quaternionic line bundle, $7$-manifold, smooth structure, Kirby-Siebenmann invariant
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”
Article copyright: © Copyright 2012 American Mathematical Society