71 problems

Let $\mathcal{I}_g$ be the Torelli subgroup of the mapping class group of the genus-$g$ oriented surface, consisting of mapping classes acting trivially on…

Let $G$ be a residually nilpotent group. Write $H_1(G)$ and $H_2(G)$ for its integral first and second homology groups, and let $\operatorname{Hom}(H_2(G),\mathbb Z)$ denote the gr…

Let $F$ be an infinite field and let $n\geq3$. Consider the sequence … … Here $\alpha_1$ and $\alpha_2$ are the two inclusion maps specified in the source. Exactness conjecture. Th…

Let $\operatorname{PSL}(2,\mathbb C)^\delta$ denote $\operatorname{PSL}(2,\mathbb C)$ with the discrete topology, and let $K$ range over all concrete number fields. The groups…

Let $F$ be a finite field, let $n\ge 3$, and let … be the subgroup of matrices congruent to the identity modulo $t$. Write $H_1(K)$ for its first homology group and…

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…

Let $F_n$ be the free group of rank $n$, and consider the rational homology $H_*(\operatorname{Out}(F_n);\mathbb Q)$. Let Morita's cycles and cycles obtained from a natural general…

Let $\Gamma_1,\dots,\Gamma_n$ be non-abelian limit groups and let $S\subseteq \Gamma_1\times\dots\times\Gamma_n$ be a subdirect product that intersects each $\Gamma_i$ non-triviall…

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…

Let $\mathsf G$ be an algebraic group defined over the integers, let $k$ be an algebraically closed field, and let $l$ be a prime. Write $H_n(\mathsf G(k),\mathbb{Z}/l)$ for the gr…

Let $F$ be an infinite field and let $n$ be a positive integer. The natural homomorphism … is considered, where $H_n({\rm inc})$ denotes the map induced by the standard inclusion…

Rational injectivity conjecture. The map above is an isomorphism if $\Gamma$ is torsion-free. The displayed map is already known to be injective under the hypotheses discussed imme…

Johnson's conjecture. These homomorphisms are isomorphisms for all $k$ and sufficiently large $g$.

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 $\kappa^h_j$ denote the surgery-theoretic obstruction maps discussed in the paper. Oozing conjecture. The original oozing conjecture asserts that $\kappa^h_j=0$ for every…

Let $\mathcal{I}_3$ denote the Torelli group of a closed genus $3$ surface, and let $\mathbb{Z}$ denote the integers. Non-finite generation conjecture. The group … is not finitely…

Let $G_1\to\dots\to G_n\to G_{n+1}\to\dots$ be a sequence of groups satisfying homological stability, and let $W_n$ denote the associated destabilization complexes. Converse conjec…

Let $G$ be a finite-dimensional Lie group, and write $G^\delta$ for its underlying discrete group. Milnor's conjecture. The map induced by the identity $G^\delta\to G$, … is an iso…

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 $\mathcal{I}_3$ be the Torelli group of a closed oriented surface of genus $3$, consisting of mapping classes acting trivially on first homology. Its second integral homology g…

Let $F$ be a non-cyclic free group, and let $\widehat F$ denote its free pronilpotent completion. For a group $G$, write $H_2(G)$ for its integral second homology and…

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 $S_g$ be a closed oriented surface of genus $g$, let $\mathop{\mathrm{Mod}}\nolimits(S_g)$ be its mapping class group, and let $\mathcal{I}_g$ be the Torelli group, consisting…