The Becker-Gottlieb Transfer: a Geometric Description

Abstract

In this note, we examine geometric aspects of the Becker-Gottlieb transfer in terms of the Umkehr and index maps, and rework some classic index theorems, using the cohomological formulae of the Becker-Gottlieb transfer. The results are natural from the homotopy-theoretic point of view; they reveal subtle geometric information in the Umkehr map, and demonstrate the beauty of the Atiyah-Singer index theorem for families and its generalizations.