Becker-Gottlieb Transfer

I gave a talk during the June 2013 algebraic topology summer course at MSRI. Can you see how nervous I am? notes

I spoke about the Becker Gottlieb transfer map as described in

J. Becker, D. Gottlieb, “The transfer map and fiber bundles” Topology , 14 (1975) pp. 1–12 link

In Gottlieb’s 1975 paper, he proved the existence of a transfer homomorphism involving the total space and the base space of a fiber bundle. The transfer homomorphism is between the nth cohomology of the total space, and the nth cohomology of the base space.

The transfer map is defined s.t. the composition of the induced pullback of the projection and the transfer map is just multiplication by the Euler characteristic of the fiber.
$\hat{\tau} p*: H*(B) \rightarrow H*(B)$

A natural question to pose is whether or not this transfer homomorphism was induced by some map from $B$ to $E$. This would make the transfer homomorphism geometric.

The paper ends with a proof of the Adams conjecture. It is an important result dealing with real vector bundle isomorphism classes and cohomology operations in K-Theory. The original proof used algebraic geometry. Quillen then proved the Adams Conjecture using the Brauer Induction Theorem. Becker and Gottlieb’s proof is considered exceptional, because it only uses algebraic topology.

Their proof reduces the problem to elements of $KO(X)$ which are $2n$-dimensional vector bundles. It uses the transfer map by applying a splitting principal derived from the transfer map.