16 problems

Let $G$ be an algebraic group over $\mathbb R$, let … and let $p$ be a prime number. Regard $G(\mathbb C)$ as a $\Gamma$-space and let $G(\mathbb C)^{\operatorname{top}}$ denote it…

Let $G=C_2$. A $\operatorname{Rep}(C_2)$-complex is a $C_2$-space built by attaching cells of the form $D(\mathbf R^{p,q})$, where $p\geq q\geq0$; it is of finite type if, for each…

Let $V$ be a representation and let $H_{*,*}^{C_2}(SP_\infty S^V;\underline{\mathbf F_2})$ be the free bigraded module with generators $\mu_i$ of bidegrees $(p_i,q_i)$. Finiteness…

Family Eilenberg–Ganea conjecture. For every family $\mathcal{F}$ of subgroups of $G$,

Family-stabiliser tree conjecture. The inequality $\operatorname{cd}_{\mathcal{F}}G\leqslant 1$ holds if and only if $G$ acts on a tree whose stabilisers generate $\mathcal{F}$.

For a group $G$, consider Bredon cohomology with respect to the family of finite subgroups. A group is locally free if every finitely generated subgroup is free. Bredon analogue of…

A group is non-virtually cyclic if it is not virtually cyclic, and it is of type $\underline{\underline{\operatorname{FP}}}_\infty$ when the trivial Bredon module admits a projecti…

Let $G$ be a group. Write $\mathcal{F}\mathrm{cd}(G)$ for the cohomological dimension relative to the family of finite subgroups, and let $\underline{\mathrm{cd}}(G)$ denote the Br…

Let $G$ be a finite group and work over the complex numbers. Let $\mathbb{Z}_{G}(n)$ denote the equivariant motivic complex obtained from the presheaf with equivariant transfers fr…

Let $\Gamma$ be a group and let $\mathcal{F}$ be the family of finite subgroups of $\Gamma$. Write $\mathcal{F}$-$\underline{\operatorname{cd}}\Gamma$ for the relative cohomologica…

Let $\mathfrak{F}$ be the class of finite groups. A group $G$ belongs to ${\scriptstyle \mathbf H}_{1}\mathfrak{F}$ if it admits a finite-dimensional contractible $G$-…

Finiteness equivalence conjecture.

Let $G$ be a ${{\scriptstyle\mathrm{L}\bf H}}\mathfrak{F}$-group, where $\mathfrak{F}$ is either the class of finite groups or the class of virtually cyclic groups. Here, having ju…

Let $\mathfrak{F}$ be the class of finite or virtually cyclic groups. Kropholler's class ${{\scriptstyle\bf H}}\mathfrak{F}$ and the generalized class…

Let $G$ be an elementary amenable group, and let $\uu EG$ denote a classifying space for the family of virtually cyclic subgroups. Bredon geometric-dimension conjecture. In light o…

Let $G$ be a group and let $\frakF$ be a semi-full family of subgroups of $G$. Write $\cd_{\frakF}G$ for the Bredon cohomological dimension and $\gd_{\frakF}G$ for the Bredon geome…