5 problems

Let $\phi:\Gamma\to\pi$ be a homomorphism of discrete groups. The LS-category $\operatorname{cat}\phi$ is the least integer $k$ such that a classifying-space map $B\Gamma\to B\pi$…

Let $d52f$ be a class of finite groups, let $A$ be an $d52f$-separable normal subgroup of a finite group $G$, and let $G\rightarrow G/A$ be the canonical epimorphism. The homomorph…

Let $S=S_{g,p}$ be a surface with $p\ge 0$ and $g>1$. For positive integers $m$ and $n$ with $m\neq n$, consider homomorphisms between the braid groups $B_n(S)$ and $B_m(S)$. Nonex…

Flipping criterion. The morphism $\psi$ is invertible if and only if it is flippable and sends the integral of ${\mathcal D}(G)$ to the integral of ${\mathcal D}(H)$.

Rank-two isometry conjecture. For every $g\in F_2$, one has