32 problems

Let $M$ be a closed manifold and let $S^n$ be the $n$-sphere. The Lusternik–Schnirelmann category $\operatorname{cat}(X)$ is the least integer $k$ such that $X$ admits an open cove…

Let $M$ and $N$ be closed orientable manifolds, and let $f:M\to N$ be a map of degree one. The Lusternik–Schnirelmann category $\operatorname{cat}(X)$ is the least integer $n$ such…

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

Rudyak's conjecture. There should be a Grossmann–Whitehead type inequality

The systolic-category lower-bound conjecture. Such a manifold should satisfy

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…

For a space $X$, write $\operatorname{cat}{X}$ for its Lusternik–Schnirelmann category, and let $S^n$ denote the $n$-sphere. Iwase's product-with-spheres conjecture. For any space…

Let $X$ be a CW complex. The monoidal topological complexity conjecture. The monoidal topological complexity satisfies … By definition, $\operatorname{TC}^M(X)$ is either…

Let $Y$ be a finite CW complex, let $m\ge 2$ be an integer, and let $k\ge 1$. Denote by $\operatorname{\mathsf{TC}}_m$ the $m$-th sequential topological complexity, and by $S^k$ th…

Let $Y$ be a closed manifold and let $m$ be a positive integer. Write $Y^m$ for the $m$-fold Cartesian product of $Y$, and let $\operatorname{cat}$ denote Lusternik–Schnirelmann ca…

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 $M$ and $N$ be closed manifolds, and let $M\sharp N$ denote their connected sum. Write $\operatorname{cat} X$ for the Lusternik–Schnirelmann category of a topological space $X$…

Let $X$ be a topological space, and let $\mathrm{cat}(X)$ and $\mathrm{TC}(X)$ denote its Lusternik–Schnirelmann category and topological complexity, respectively. For continuous m…

A simplicial complex has a geometric simplicial Lusternik–Schnirelmann category, and a strong collapse is a simplicial reduction under which this category is known to be non-decrea…

Scheerer–Stanley–Tanré conjecture. A finite CW-complex $X$ satisfies the Ganea conjecture if and only if

Let $X$ be a topological space, let $\Delta:X\rightarrow X\times X$ be the diagonal map, and let $X\times X/\Delta(X)$ denote its quotient. Let $\operatorname{TC}^M(X)$ be the mono…

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…

Let $Sp(n)$ be the symplectic group, and let ${\cal O}_k$ be the Grassmannian conjugacy classes appearing in the Hunziker–Sepanski bound, for $0\leq k\leq n$. Write…

Weighted-category conjecture. The weighted category of an orbifold groupoid coincides with the unweighted category of its inertia groupoid:

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

McGibbon's Mislin-genus conjecture. For every $Y\in\mathcal{G}(X)$,