nLab Becker-Gottlieb transfer

Idea

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.

Definition

See e.g. (Haugseng 13, def. 3.9).

Related concepts

• sheaf with transfer

References

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)

• eom, Becker-Gottlieb transfer

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

• Rune Haugseng, The Becker-Gottlieb Transfer Is Functorial (arXiv:1310.6321) (withdrawn now)

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

• John Klein, Cary Malkiewich, The transfer is functorial, arXiv:1603.01872,

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’).