# The Reinterpretation of Davis-Lueck Equivariant Homology in Terms of $L$-theory

# The Reinterpretation of Davis-Lueck Equivariant Homology in Terms of $L$-theory

The $K$-theory $K_n(\mathbb{Z}G)$ and quadratic $L$-theory $L_n(\mathbb{Z}G)$ functors provide information about the algebraic and geometric topology of a smooth manifold $X$ with fundamental group $G=\pi_1(X, x_0)$. Both $K$- and $L$-theory are difficult to compute in general and assembly maps give important information about these functors. Ranicki developed a combinatorial version of assembly by describing $L$-theory over additive bordism categories indexed over simplicial complexes. The chain duality defined for such categories also has an interpretation as a Verdier duality.

In this talk, I will present current work with Jim Davis where we define an equivariant version of Ranicki’s local/global assembly map and identify this assembly map with the map on equivariant homology defined by Davis and Lueck. Furthermore, I will discuss some applications. In particular, it is known that the $L$-theoretic Farrell-Jones conjecture holds for $G = H \rtimes_{\alpha} \mathbb{Z}$ assuming that it holds for the group $H$. Nonetheless, a satisfactory proof of this often-used result has never been given. I will give insight into how we use our investigation of the equivariant assembly maps to prove this result.