Assembly maps in topology

Speaker: Guglielmo Nocera – Institut des Hautes Études Scientifiques (IHÉS)

Abstract: The Borel conjecture states that, for a discrete group $G$, any two closed topological manifolds homotopy equivalent to $BG$ are indeed homeomorphic to each other. The truth of the conjecture can be checked by inspecting whether the so-called assembly maps are equivalences. Informally, these maps relate certain $G$-equivariant homology theories to $K$-theory and $L$-theory of the group ring $\mathbb{Z}[G]$. In the special case of $G=\mathbb{Z}$, they are also related to classical splitting results in $K$- and $L$-theory of Laurent polynomials. We will try to give a self-contained account of these problems and their state-of-the-art, briefly explaining how higher categorical tools are nowadays used in tackling them.