23 problems

Homological stability conjecture. The cohomology group

Homological stability conjecture. If $A$ has homological stability, then there is a range of dimensions tending to infinity below which $t_b$ induces an isomorphism on homology. Mo…

Let $X$ be an irreducible smooth complex variety. For a partition $\lambda$ of $k$, let $W_\lambda(X)$ denote the symmetric complement associated with the stratum indexed by…

Let $F$ be an infinite field, let $k$ be either $\mathbb{Q}$ or $\mathbb{Z}/l$ for a prime $l$, and assume $K_2^M(F)\otimes k=0$. Let $\mathfrak{t}_r^{(k)}(F)$ denote the homologic…

Let $F$ be an algebraically closed field. Here $\mathfrak{p}_n(F)$ denotes the primitive part of the homology group associated with $\operatorname{GL}_n(F)$, and $l$ is a prime. Th…

Exactness conjecture. This sequence is exact.

Alpert–Kahle–MacPherson's conjecture. The inclusion of $SF_n(R_{w,h})$ into $F_n(\mathbb{R}^2)$ induces an isomorphism

Let $g\geq2$ be an integer and let $G$ be a finite group. Let $\operatorname{CHur}_g^G$ be the algebraic stack over $\mathbb C$ whose objects are geometrically connected finite éta…

Let $M$ be an $n$-manifold, and let $\mathrm{BDiff}_c^{r}(M)^{\delta}$ denote the classifying space of the discrete group of compactly supported $C^r$ diffeomorphisms, while…

Let $\pi:\mathcal{X}\to B$ be a Fano fibration and let $M$ range over the Manin components of $\operatorname{Sec}(\mathcal{X}/B)$, with degree tending to infinity. Stability conjec…

Folklore conjecture. For each $k$, there exists some $G_k$ such that for $g \geq G_k$, the homology group $\operatorname{H}_k(\operatorname{\mathcal{I}}_g;\mathbb{Q})$ is finite-di…

For fixed $n$, let $\operatorname{Sel}_n^d(\mathbb{C})$ be the Selmer space associated with the universal family of Weierstrass models of degree parameter $d$. A Landesman homologi…

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…

Integral homology conjecture. The map $\mathcal{G}^1_n(X)\to\mathcal{G}_n(X)$ induces an isomorphism on homology in degrees $2*\leq n-3$.

Let $H$ and $G$ be finitely generated groups, and let $F:U\! f\mathcal{G}\to \mathbb{Z}\operatorname{-Mod}$ be a coefficient system of finite degree. For each $n$, consider the hom…

Let $(\mathcal{C},\oplus,0)$ be a homogeneous category and let $A$ and $X$ be objects of $\mathcal{C}$. Set $G_n=\operatorname{Aut}(A\oplus X^{\oplus n})$, and let…

Let $X$ be an irreducible smooth complex variety, let $^j\lambda$ denote the partition obtained by adjoining $j$ parts equal to $1$ to a partition $\lambda$, and let…

Let $i$ be a nonnegative integer, let $bb$ be a partition, and let $X$ be an irreducible smooth complex variety. For each $j$, let $\bar{w}^c_{1^j\lambda}(X)$ denote the complement…

Let $\Gamma_n(p)$ and $\Gamma^{\operatorname{Sp}}_{2n}(p)$ denote the congruence subgroups considered in the source, with homology over $\mathbb{F}_p$ carrying the natural actions…

Let $G$ be a symmetric group on more than two letters, and let $c$ be the conjugacy class of transpositions. For each integer $i\geq 2$, consider the rational homology of the conne…

Stability conjecture. For fixed $L \geq 2$ and $k \geq 1$, there exists some $N$ such that if $g \geq N$ and $\gamma$ is a simple closed nonseparating curve on $\Sigma_g$, then thi…

Isomorphism conjecture. For fixed $L \geq 2$ and $k \geq 1$, there exists some $N$ such that if $g \geq N$, then this map is an isomorphism.