11 problems

For the compact symplectic group $Sp(n)$, let $\operatorname{cat}(Sp(n))$ denote its Lusternik–Schnirelmann category. Symplectic-group category conjecture. … This is described in t…

Let $G$ be a simply-connected compact Lie group with … where each $H_i$ is a simple Lie group. Product formula conjecture. The weak category, ordinary category, and strong category…

Farber–Oprea conjecture. The formal power series $f_X(t)$ represents a rational function of the form

Let $M$ and $N$ be closed orientable $n$-manifolds, and let $f:M\to N$ be a map of degree $\pm1$. Write $\operatorname{\mathsf{dcat}}$ for the distributional category of a manifold…

Coincidence conjecture. The analog and distributional notions of category and topological complexity coincide on the class of metrizable spaces.

Let $X$ be a finite topological space, let $PX$ be its path space, and let $\operatorname{tc}(X)$ denote its topological complexity. Let $\operatorname{cat}(X)$ denote its Lusterni…

For the special orthogonal group $\mathrm{SO}(n)$, let $\operatorname{cat}(\mathrm{SO}(n))$ denote its Lusternik–Schnirelmann category and let…

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 $X$ be a space and let $n>2$. Define $LS_2(X)\leq n-2$ when $X$ has Lusternik–Schnirelmann category at most $n$ and admits an open cover $U_1,\ldots,U_n$ such that each map…

Category conjecture. For all $n$ and $k$,

Let $\operatorname{Sp}(n)$ be the compact symplectic group, let $\mathcal O_k$ be the distinguished conjugacy class associated with the $k$-th vertex of the fundamental alcove, and…