We prove that the Becker-Gottlieb transfer is functorial up to homotopy, for
all fibrations with finitely dominated fibers. This resolves a lingering
foundational question about the transfer, which was originally defined in the
late 1970s in order to simplify the proof of the Adams conjecture. Our approach
differs from previous attempts in that we closely emulate the geometric
argument in the case of a smooth fiber bundle. This leads to a
"multiplicative'" description of the transfer, different from the standard
presentation as the trace of a diagonal map.Comment: This is the final preprint version. The article is to appear in the
Advances in Mathematic