Marc Hoyois: Multiplicative transfers in the cohomology of algebraic varieties

Given a finite covering space E → B, there is in ordinary cohomology an additive transfer map H^n(E) → H^n(B) which specializes to addition when E is a sum of copies of B. Such transfers exist more generally for any cohomology theory represented by an E_∞-space. More subtle is a multiplicative transfer map H^*(E) → H^*(B), generalizing the cup product, which comes from the fact that cohomology is represented by an E_∞-ring spectrum. There are similar phenomena in equivariant and motivic homotopy theory, but in these cases the analog of an E_∞-ring spectrum is a more complicated structure. It turns out that many motivic E_∞-ring spectra (eg those representing motivic cohomology, algebraic K-theory, and algebraic cobordism) possess this extra structure, and this leads to various multiplicative transfer maps in these cohomology theories. This is joint work with Tom Bachmann.