34 problems

Let $\Gamma_{n,s}$ be the graph automorphism group with parameters $n$ and $s$, and let $2n+s-3$ be its virtual cohomological dimension. Below-vcd vanishing conjecture. All classes…

Let $k$ be the field appearing in the power-series congruence subgroup, let $n,m$ be positive integers, let $p$ be a prime, and write $C_{n,m}(k[[t_1,t_2,\dots,t_m]])$ for the corr…

For each integer $i\geq1$, let … be the homology class constructed from Kontsevich's graph-homology construction. Morita's conjecture. The classes $\mu_i$ are non-trivial for all…

The homological $n$-$(n+1)$-$(n+2)$ conjecture. If $N_{1}$ is of type $\operatorname{FP}_n$, $G_1$ and $G_2$ are of type $\operatorname{FP}_{n+1}$, and $Q$ is of type…

Let $2\leqslant k\leqslant m$ be integers. For groups $G_{1},\ldots,G_{m}$, a subgroup $P\leqslant G_{1}\times\cdots\times G_{m}$ virtually surjects on $k$ factors if, for every…

Let $\pi$ be a finite $2$-group and let $N\triangleleft T\trianglelefteq W\triangleleft\pi$ be normal subgroups such that $\pi/N$ is cyclic, $T/N\cong\mathbb Z/{2}$, and…

Let $n=p_1\cdots p_l$, with $l>1$, be the prime decomposition of $n$ such that $1<r_{p_1}\leq\cdots\leq r_{p_l}$; if $r_{p_j}=r_{p_{j+1}}$ for some $j$, assume that $p_j<p_{j+1}$.…

Let $F$ be an infinite field and let $M_\bullet$ be the exact complex of $\operatorname{GL}_n(F)$-modules used to construct the first-quadrant spectral sequence…

Let $p$ be a regular odd prime, let $\zeta_p$ be a primitive $p$th root of unity, set $R=\mathbb Z[\zeta_p,1/p]$, and let $k=\mathbb F_p$. Let $D_1(R)$ denote the diagonal subgroup…

Let $GL_j(R)$ be the general linear group and let $D_j(R)\subset GL_j(R)$ be its diagonal subgroup. For group homology with coefficients in $k$, the canonical inclusion induces … H…

Cyclic-group homology conjecture.

Let $F_n$ be a free group of rank $n$, and let $IA_n$ be the kernel of the canonical epimorphism ${\rm Aut}(F_n)\twoheadrightarrow GL_n(\mathbb{Z})$. For each $d\in\mathbb{N}$, let…

Let $F_n$ be a free group of rank $n$, and let $IA_n$ be the kernel of the canonical epimorphism ${\rm Aut}(F_n)\twoheadrightarrow GL_n(\mathbb{Z})$. For each $d\in\mathbb{N}$, let…

Let $G$ be a group. Write $\langle G \rangle^{\infty}$ for the conjugacy classes of infinite-order elements, and for $g\in G$ let $C_g$ be its centralizer and…

Let $(G_p)_{p\geqslant 0}$ be a family of groups with multiplication, and let $\mathit{SP}_n$ denote the splitting complex associated with this family. Suppose that…

Final unstable homology conjecture. For every $m\geqslant 1$, the group $H_m(\operatorname{GL}_{2m}(\mathbb{F}_2);\mathbb{F}_2)$ is a single copy of $\mathbb{F}_2$ generated by the…

Assume $(p_0)_*(\mu_k)=0$, and construct a cycle $u\in Z_{4k+1}(\mathrm{GL}(2k+3,\mathbb{Z});\mathbb{Q})$ by bounding the relevant abelian cycle in $\mathrm{Out}\,F_{2k+3}$ and its…

Let $\mu_k\in H_{4k}(\mathrm{Out}\,F_{2k+2};\mathbb{Q})$ be the Morita class, and let $\overset{\circ}{\beta}_{2k+1}$ be the secondary class defined at the critical rank $2k+2$, wi…

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

Let $\phi$ be a gluing and let $A_\phi$ be its assembly map for the homology groups of the graph automorphism groups $\Gamma_{n,s}$. A class is positive-dimensional when it has hom…

Odd-dimensional injectivity conjecture. This map is injective for all $k$. The target lies in codimension $2$ because the virtual cohomological dimension of…

Let $\mathcal{X}_{2m,0}$ be the space appearing in the modular-form construction, and let … be the map obtained by gluing two rank-$2$ graphs along all their leaves. Injectivity co…

Let $\Gamma_{n,s}$ denote the relevant graph automorphism group, and let $A_\phi$ be the assembly map associated to a gluing $\phi$ producing the graph $X_{n,s}$. Write $2n+s-3$ fo…

Let $M$ be a closed oriented $n$-dimensional manifold, let $\pi=\pi_1(M)$, and let $B\pi$ be a classifying space. If $f\colon M\to B\pi$ classifies the universal bundle, call $M$ r…

Let $\mathbf{Grp}$ be the category of groups, let $\mathbf{Zinb}^c$ denote coZinbiel coalgebras, and let $\mathbf{Com}^c$ denote cocommutative coalgebras. For a group $G$, write…