A note on the double quaternionic transfer and its 𝑓–invariant

von Bodecker, Hanno

doi:10.1090/proc/12940

A note on the double quaternionic transfer and its $f$–invariant
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by Hanno von Bodecker

Proc. Amer. Math. Soc. 144 (2016), 2731-2740

DOI: https://doi.org/10.1090/proc/12940

Published electronically: November 4, 2015

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Abstract:

It is well known that for a line bundle over a closed framed manifold, its sphere bundle can also be given the structure of a framed manifold, usually referred to as a transfer. Given a pair of lines, the procedure can be generalized to obtain a double transfer. We study the quaternionic case, and derive a simple formula for the $f$–invariant of the underlying bordism class, enabling us to investigate its status in the Adams–Novikov spectral sequence. As an application, we treat the situation of quaternionic flag manifolds.

References

J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21–71. MR 198470, DOI 10.1016/0040-9383(66)90004-8
A. Baker, D. Carlisle, B. Gray, S. Hilditch, N. Ray, and R. Wood, On the iterated complex transfer, Math. Z. 199 (1988), no. 2, 191–207. MR 958648, DOI 10.1007/BF01159652
Mark Behrens and Gerd Laures, $\beta$-family congruences and the $f$-invariant, New topological contexts for Galois theory and algebraic geometry (BIRS 2008), Geom. Topol. Monogr., vol. 16, Geom. Topol. Publ., Coventry, 2009, pp. 9–29. MR 2544384, DOI 10.2140/gtm.2009.16.9
Ulrich Bunke and Niko Naumann, The $f$-invariant and index theory, Manuscripta Math. 132 (2010), no. 3-4, 365–397. MR 2652438, DOI 10.1007/s00229-010-0351-7
D. Carlisle, P. Eccles, S. Hilditch, N. Ray, L. Schwartz, G. Walker, and R. Wood, Modular representations of $\textrm {GL}(n,p)$, splitting $\Sigma (\textbf {C}\textrm {P}^\infty \times \cdots \times \textbf {C}\textrm {P}^\infty )$, and the $\beta$-family as framed hypersurfaces, Math. Z. 189 (1985), no. 2, 239–261. MR 779220, DOI 10.1007/BF01175047
P. E. Conner and E. E. Floyd, The relation of cobordism to $K$-theories, Lecture Notes in Mathematics, No. 28, Springer-Verlag, Berlin-New York, 1966. MR 0216511
Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and modular forms, Aspects of Mathematics, E20, Friedr. Vieweg & Sohn, Braunschweig, 1992. With appendices by Nils-Peter Skoruppa and by Paul Baum. MR 1189136, DOI 10.1007/978-3-663-14045-0
Mitsunori Imaoka, Hypersurface representation and the image of the double $S^3$-transfer, Proceedings of the Nishida Fest (Kinosaki 2003), Geom. Topol. Monogr., vol. 10, Geom. Topol. Publ., Coventry, 2007, pp. 187–193. MR 2402784, DOI 10.2140/gtm.2007.10.187
K. Knapp, Rank and Adams filtration of a Lie group, Topology 17 (1978), no. 1, 41–52. MR 470960, DOI 10.1016/0040-9383(78)90011-3
K. Knapp, Some applications of $K$-theory to framed bordism: E-invariant and transfer., Ph.D. thesis, Bonn: Univ. Bonn, Mathematisch-Naturwissenschaftliche Fakultät (Habil.), 115 p. , 1979.
Gerd Laures, The topological $q$-expansion principle, Topology 38 (1999), no. 2, 387–425. MR 1660325, DOI 10.1016/S0040-9383(98)00019-6
Gerd Laures, On cobordism of manifolds with corners, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5667–5688. MR 1781277, DOI 10.1090/S0002-9947-00-02676-3
Peter Löffler and Larry Smith, Line bundles over framed manifolds, Math. Z. 138 (1974), 35–52. MR 346826, DOI 10.1007/BF01221883
Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres. 2nd ed., Providence, RI: AMS Chelsea Publishing., 2004.
Wilhelm Singhof and Dieter Wemmer, Parallelizability of homogeneous spaces. II, Math. Ann. 274 (1986), no. 1, 157–176. MR 834111, DOI 10.1007/BF01458022
Hanno von Bodecker, On the geometry of the $f$-invariant, Ph.D. thesis, Ruhr-Universität Bochum, 2008.
—, The beta family at the prime two and modular forms of level three, arXiv:0912.3082 [math.AT], 2009.