53 problems

Iwase–Sakai conjecture.

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

Let an arrangement of finitely many complex central hyperplanes have complement $M$. Let the zero-divisors-cup-length of $H^*(M,\mathbf C)$ mean the largest number of zero-divisors…

Topological complexity conjecture. If $\Gamma$ has at least two essential vertices and $n\geq 2m(\Gamma)$, then

Let $C_n(\mathbf R^m)$ denote the configuration space of $n$ ordered distinct points in $\mathbf R^m$, and let ${\rm\bf {TC}}$ denote topological complexity. Conjecture. For even…

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 $Q_{8m}$ denote the generalized quaternion group of order $8m$ acting freely on $S^3$, and let $r$-th sequential topological complexity be denoted by $\mathsf{TC}_r$. Sequentia…

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 $F(\mathbb{R}^2,n)$ denote the ordered configuration space of $n$ distinct points in the plane, and let $\mathrm{TC}^{\Sigma_s}[F(\mathbb{R}^2,s+1)\to F(\mathbb{R}^2,s)]$ denot…

Let $X$ be a compact Polish space with uncountably many points. An $A'$-computable presentation is a presentation of $X$ as a separable metric space computable relative to the Turi…

Let $\widetilde G_{n,3}$ denote the oriented Grassmann manifold of oriented $3$-planes in $\mathbb{R}^n$, and let $W_n$ be the relevant associated space whose zero-divisor cup-leng…

Proposed cohomological lower bound. If

Knudsen–Weinberger conjecture. Their invariants coincide with the corresponding distributional invariants on metrizable spaces.

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

Let $f=(f_1,\ldots,f_p):(\mathbb{R}^n,0)\to(\mathbb{R}^p,0)$ be an analytic map germ with $n>p\geq 2$, where $p$ is odd, and let … be its nonempty link. Let … be the associated Mil…

Let $f:(\mathbb{R}^n,0)\to(\mathbb{R}^p,0)$ be an analytic map germ admitting a Milnor fibration … Here $0<\eta\ll\epsilon\leq\epsilon_0$, where $\epsilon_0$ is a Milnor radius for…

Let … be the space of nonsingular plane cubic curves, and let … be the space of a cubic together with a flex point. The projection $(F,p)\mapsto F$ is a 9-fold covering, whose Schw…

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…

Let $\mathbb{S}^1_{2^k}$ be the Khalimsky circle with $2^k$ points, where $k\geq 1$, and let $\operatorname{tc}(X)$ denote the topological complexity of a space $X$. Tanaka's conje…

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

Farber's conjecture. The tree restriction should be superfluous: equality should hold in the upper bound

Farber–Kaufman–Schütz rationality conjecture. The function $\mathcal{F}_X(x)$ is rational with a single pole of order $2$ at $x=1$; equivalently, there exists $P\in\mathbb{Z}[X]$ s…

PL-regularity conjecture. For any reasonable class of finite PL functions, a $0$-cell is far more likely to be PL regular than PL critical.

Generic total H-complexity conjecture. With probability $1$, the total $H$-complexity of $F$ is less than or equal to the number of $0$-cells of $\mathcal{C}(F)$.

Coarse bounded H-complexity conjecture. The coarse bounded sublevel $H$-complexity (respectively, the coarse bounded superlevel $H$-complexity) of $F$ is a lower bound for the glob…