nLab Becker-Gottlieb transfer

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Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

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).

References

The original articles

Interpretation in terms of dualizable objects:

Review:

Discussion in the context of differential algebraic K-theory is in

In prop. 4.14 of

Becker-Gottlieb transfer was identified with the Umkehr map induced from a Wirthmüller context in which in addition f *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.