25 problems

Let $F$ be an elliptic space with minimal model $\Lambda V$, and let $F\to X\to B$ be a fibration. The condition that $\dim V^{\mathrm{odd}}=\dim V^{\mathrm{even}}$ is imposed on t…

Equality conjecture. An $s$-motion planning algorithm for $\operatorname{\mathbb{R}P}^m$ with $\operatorname{secat}(\pi_s)+1$ $s$-local rules can be constructed out of such a map.…

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…

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…

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

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…

Proposed cohomological lower bound. If

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

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–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…

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…

Let $f:G\to H$ be a homomorphism between discrete groups. Let $Bf:BG\to BH$ be a map between Eilenberg–MacLane spaces inducing $f$ on fundamental groups, and let…

Let $b\geq 1$, $d\geq 2$ be integers, and let $\alpha\in(0,1)$. For a family $\mathcal{A}$ of sets in $\mathbb{R}^d$, write $TC_{\lceil d/2\rceil}(\mathcal{A})$ for its topological…

Let $G$ be a connected graph, and let $V_{\ge3}$ denote the set of vertices of $G$ of degree at least $3$. Assume that $|V_{\ge3}|\ge2$ and let $n\ge2|V_{\ge3}|$. Farber's conjectu…

Let $D$ be the disc, let $C(D,n)$ be its unordered configuration space, and let $B_n$ be the braid group on $n$ strands. The disc braid-group conjecture asserts that … This would f…

Let $P_n$ be the pure braid group of the disc, and let $[P_n,P_n]$ denote its commutator subgroup. The cohomological-dimension conjecture asserts that … This value would make the l…

Assume $m\equiv2^e-1\pmod{2^{e+1}}$ with $e\geq0$, so $e$ is the length of the block of consecutive ones ending the binary expansion of $m$. Let…

Let $X$ be a locally finite simplicial complex, and let $\operatorname{TC}(X)$ denote its topological complexity and $\operatorname{TC}^M(X)$ its monoidal topological complexity. T…

Let $f:Y\rightarrow X$ be a map. A homotopy retraction of $f$ is a map $r:X\rightarrow Y$ such that $rf\simeq \operatorname{id}_Y$. The sectional category of $f$, denoted by…

Let $\alpha(m)$ denote the number of ones in the binary expansion of $m$, and let $b(n,e)$ be the smallest positive integer $r$ for which there is a $\mathbb{Z}_{2^e}$-biequivarian…

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