9 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 $G$ be a group, let $E_{\mathcal{VCY}}(G)$ be the classifying $G$-space for the family of virtually cyclic subgroups, and let $\mathcal{A}$ be an additive $G$-category with inv…

Let $G$ be a group, let $R$ be a ring with involution, and let $\mathbf{L}^{\langle-\infty\rangle}(R)$ be the associated $L$-theory spectrum, whose $n$th homotopy group is…

Let $(B_*,\beta)$ be a chain bundle with algebraic Wu class $\widehat v_{2k}(\beta)$, and let $\operatorname{ev}$ denote evaluation on homology. Let $\Sigma$, $\Sigma_8$, $\Lambda$…

Let $(B_*,\beta)$ be a chain bundle, let $q\geq0$, and let $\eta_\beta\colon W_0(Q_\beta(q))\to L^{2q}(B_*,\beta)$ be the natural homomorphism. Conjecture on $\eta_\beta$. If…

Let $\mathrm{KR}(\mathcal{C},D,\eta)$ be the genuine $C_2$-spectrum of real algebraic K-theory associated to an exact category with duality, and let…

Passage conjecture. The maps $j_2$ and $j_3$ appearing in the diagram are bijective. The preceding discussion establishes several related comparison maps and split-injectivity resu…

For a torsion-free group $G$, let $EG$ be a classifying space for free $G$-actions, let $H_n^G(EG;\mathbb{L}^{\langle-\infty\rangle}_{\mathbb{Z}})$ denote equivariant homology with…

The surgery-theoretic Fibered Isomorphism Conjecture. The assembly map in $L^{\langle -\infty\rangle}$-theory should be an isomorphism.