Contributed talk: Norm, Assembly and Coassembly

A major open problem in topology are (rational) injectivity results about assembly in $K$- and $L$-theory, e.g. the Borel and Novikov conjectures.

Malkiewich showed that given a ring R and finite group $G$ there exists a so called coassembly map going from the K-theory of the group ring $R[G]$ to the homotopy fixed points of the K-theory spectrum of $R$ equipped with the trivial action. Moreover the composition of the assembly map with this coassembly map agrees with the norm map. This implies that for a finite group G, the assembly map is rationally and $K(n)$-locally injective. In recent work with Alex M\"uller and Holger Reich, we generalize this fact to arbitrary additive invariants of either stable or Poincaré categories, while also allowing for twisted coefficients. In particular, Malkiewich's theorem is true more generally for $THH$, $TC$, Grothendieck-Witt and $L$-theory. The fundamental insight is that the content of the theorem is fundamentally about the interaction of the algebraic theory of G-actions with the algebraic theory of $E_\infty$-monoids. The general nature of this approach allows one to ask the question if there exists a generalization to more general groups, using the framework of dualizable categories.