20 problems

Let ${\mathcal H}^?_*$ be an equivariant homology theory with values in $R$-modules, let $G$ be a group, and let ${\mathcal C}$ be a family of subgroups of $G$. Let…

Let $\pi$ be a group, and let $A^L_\pi$ and $A^K_\pi$ denote the assembly maps in $L$-theory and $K$-theory, respectively. Their rationalizations are obtained by tensoring with the…

Fibered homotopy K-theory isomorphism conjecture. The group $G$ satisfies the (fibered) $\mathit{KH}$-isomorphism conjecture for $R$ if the pair $(G,\mathcal{FIN})$ satisfies the (…

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 $G$ be a group, let $R$ be a regular ring, and let $BG$ be a classifying space for $G$. The spectrum $\mathbf{K}(R)$ represents algebraic $K$-theory, and the assembly map is th…

Let $\Gamma$ be a countable group acting properly and isometrically on the Euclidean space $\mathbb{E}_{\Gamma}$, and let $\mu_{\mathrm{IP}}$ denote the assembly map for the equiva…

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

Meta-Isomorphism Conjecture. The induced map

Let ${\mathbf S}\colon\operatorname{SPACES}\to\operatorname{SPECTRA}$ respect weak equivalences and disjoint unions. Let $G$ be a group and ${\mathcal F}$ a family of subgroups of…

Let $\Gamma$ be a group, let $(G,\psi)$ be a group over $\Gamma$ with homomorphism $\psi\colon G\to\Gamma$, let ${\mathcal H}^?_*$ be an equivariant homology theory over $\Gamma$,…

Let ${\mathbf S}\colon\operatorname{SPACES}\to\operatorname{SPECTRA}$ be a covariant functor respecting weak equivalences and disjoint unions. Let ${\mathcal C}$ be a class of grou…

Let ${\mathbf S}\colon\operatorname{SPACES}\to\operatorname{SPECTRA}$ be a covariant functor respecting weak equivalences and disjoint unions. Let $G$ be a group, ${\mathcal F}$ a…

Let ${\mathbf S} \colon \operatorname{SPACES}\to \operatorname{SPECTRA}$ be a covariant functor respecting weak equivalences and disjoint unions. For a group $G$, let…

Let $\cF$ be a family of groups, let $G$ be a countable discrete group, and let $W$ be a free $G$-CW-complex. Let $E_\cF G$ denote the classifying space of $G$ for the family $\cF$…

Novikov conjecture for K-theory. The assembly map

Rationalized Farrell-Jones conjecture. For every group $G$ and every $n\in\mathbb{Z}$, this rationalized Farrell-Jones assembly map is an isomorphism.

Hsiang's conjecture. For every torsion-free group $G$ and every $n\in\mathbb{Z}$, this rationalized classical assembly map is an isomorphism.

Weinberger–Yu's conjecture. The group $\mathcal M_{\gamma_1,\ldots,\gamma_n}$ has rank $n$, and every nonzero element of it lies outside the image of the assembly map

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

Let $G$ be a discrete group and $R$ a ring. Let $\mathcal{\mathrm{fbc}}$ denote the family of finite-by-cyclic subgroups, $\mathcal{\mathrm{all}}$ the family of all subgroups, and…