group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The Becker-Gottlieb transfer is a variant of push-forward in generalized cohomology of cohomology theories along proper submersions of smooth manifolds.
The Becker-Gottlieb transfer operation has been refined to differential cohomology in (Bunke-Gepner 13).
Its compatibility in differential algebraic K-theory with the differential refinement of the Borel regulator is the content of the transfer index conjecture (Bunke-Tamme 12, conjecture 1.1, Bunke-Gepner 13, conjecture 5.3).
For the moment see at regulator – Becker-Gottlieb transfer for more.
See e.g. (Haugseng 13, def. 3.9).
The original articles
James Becker, Daniel Gottlieb, The transfer map and fiber bundles, Topology , 14 (1975) (pdf, doi:10.1016/0040-9383(75)90029-4)
(which also gives a proof of the Adams conjecture).
James Becker, Daniel Gottlieb, Vector fields and transfers Manuscr. Math. , 72 (1991) pp. 111–130 (pdf, doi:10.1007/BF02568269)
Interpretation in terms of dualizable objects:
Albrecht Dold, Dieter Puppe, Duality, Trace and Transfer, Proceedings of the Steklov Institute of Mathematics, 154 (1984) 85–103 [mathnet:tm2435, pdf]
James Becker, Daniel Gottlieb: A History of Duality in Algebraic Topology [pdf, pdf]
Review:
Dai Tamaki, Akira Kono, Section 4.5 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)
Discussion in the context of differential algebraic K-theory is in
Ulrich Bunke, Georg Tamme, section 2.1 of Regulators and cycle maps in higher-dimensional differential algebraic K-theory (arXiv:1209.6451)
Ulrich Bunke, David Gepner, Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory (arXiv:1306.0247)
In prop. 4.14 of
Becker-Gottlieb transfer was identified with the Umkehr map induced from a Wirthmüller context in which in addition $f_\ast$ satisfies its projection formula (a “transfer context”, def.4.9)
The article
claimed to establish the functoriality of the Becker-Gottlieb transfer for fibrations with finitely dominated fibers on the level of homotopy categories (without higher coherences), but contained an unfixable mistake (cf. Corrigendum to ‘The transfer is functorial’).
Last revised on November 1, 2022 at 08:11:58. See the history of this page for a list of all contributions to it.