18 problems

Farrell–Jones conjecture in $L$-theory. The assembly map is an isomorphism for all $n$. This conjecture predicts that the algebraic $L$-theory of the group ring can be computed fro…

Let $k$ be a regular ring, let $G$ be a torsionfree group, let $\mathrm{B}G$ denote its classifying space, and let $\mathbf{K}(k)$ denote the algebraic $K$-theory spectrum of $k$.…

A uniformly contractible Riemannian manifold is considered with its bounded L-theory assembly map. Bounded Farrell–Jones isomorphism conjecture. In dimensions higher than four, the…

Let $\mathcal F$ be one of the stable topological pseudo-isotopy, algebraic $K$-theory, or $L$-theory functors, with orientation data for the $L$-theory case. Let $\Gamma$ be a dis…

Let $p\colon\Gamma\to G$ be a surjective group homomorphism, and let $p^*\operatorname{vcyc}$ be the family of subgroups of $\Gamma$ whose images under $p$ are virtually cyclic. Le…

Farrell–Jones and Baum–Connes assembly conjectures. The maps $\operatorname{asmb}_i$ for $i=1,2,3$ are bijective. These comprise the rational and real assembly maps associated with…

Let $\pi$ be a torsion-free group, and let $\operatorname{Wh}_2(\pi)$ denote its second Whitehead group. Vanishing conjecture. It is conjectured that … for every torsion-free group…

Generalized opposite-category Farrell–Jones conjecture. The td-group $G$ satisfies this conjecture if, for every Hecke category with $G$-support, the displayed…

COP-Farrell-Jones conjecture. For every Hecke category with $G$-support $\mathcal{B}$ satisfying condition (Reg), the assembly map induced by the projection

COP-Farrell-Jones conjecture for Hecke algebras. For every uniformly regular ring $R$ with $\mathbb{Q} \subseteq R$, the projection

The linear K-theoretic Farrell–Jones conjecture. This assembly map is an equivalence.

Let $M$ and $N$ be smooth connected closed manifolds of dimension at least $5$. Write $N\approx_{\,{{\mathbb R}}\!\text{-}\rm diff} M$ for the relation of being smoothly…

Let $G$ be a group, let $\mathcal{FC}\!\mathit{yc}$ be the family of finite cyclic subgroups, let $(\mathcal{FC}\!\mathit{yc})$ denote their conjugacy classes, let $Z_GC$ be the ce…

Let $G$ be a group, let $\mathcal{F}$ be a family of subgroups of $G$, let $E$ be a functor from small $\mathbb{Z}$-linear categories to spectra that sends category equivalences to…

Let $G$ be a discrete group and let $\mathcal F$ be a family of subgroups of $G$. For a group homomorphism $\varphi:K{\longrightarrow} G$, define the induced family … Let…

A contracting self-similar group is a self-similar group whose action has a finite nucleus controlling all sufficiently deep sections. Nekrashevych's amenability conjecture. Every…

Meta-isomorphism conjecture. The assembly map $A_{\mathcal{F}}$ is an isomorphism for every $n\in\mathbb{Z}$.

Let $G$ be a group, $R$ a ring, and let $E_{\operatorname{fbc}}G$ denote the classifying $G$-space for the family of finite-by-cyclic subgroups, namely groups that surject onto a c…