552 problems

Poincaré conjecture. Every closed simply connected $3$-manifold is homeomorphic to the $3$-sphere $S^3$. This conjecture was proved by Perelman as part of the resolution of the geo…

Let $X$ be a finite, simply-connected $CW$-complex. Call $X$ rationally elliptic if it has finitely many rational homotopy groups, and say that $X$ has a finite homotopy exponent a…

Let $G$ be a finite group and let $p$ be a prime. Let $\mathcal{S}_p(G)$ be the poset of all non-trivial $p$-subgroups of $G$, and let $\operatorname{O}_p(G)$ be the largest normal…

Configuration-space homotopy invariance conjecture. The homotopy type of $\mathrm{Conf}_n(X)$ depends only on the homotopy type of $X$.

Let $M$ be a manifold homotopy equivalent to $\mathbb{C}P^n$ and admitting a nontrivial circle action. Petrie's conjecture. Any homotopy equivalence between $M$ and $\mathbb{C}P^n$…

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 $\varphi\colon R\rightarrow S$ be a homomorphism of commutative rings of finite type, and let $\operatorname{D}_{n}(S|R;-)$ denote the $n$th André–Quillen homology functor on t…

Let $R$ be a Euclidean or local ring, and let $\operatorname{CB}(R^n)$ be the simplicial complex whose vertices are the proper nonzero direct summands of $R^n$, with simplices give…

Let $S$ be a surface equipped with a metric of constant curvature, and let $T$ be a triangulation of $S$. Write $\mathcal{GT}(S,T)$ for the space of geodesic triangulations associa…

Diaz–Park sharpness conjecture. For every $n\geq 1$,

Let $\mathcal{X}$ be a collection of topological spaces, and let $\prod\mathcal{X}$ denote the collection of spaces homotopy equivalent to a finite-type product of spaces in…

Webb's conjecture. The orbit space

Picard 2-group conjecture. It is conjectured that this algebraic structure captures part of the homotopy type of $\mathcal{M}(\mathcal{B})$.

Let $\cat{C}$ be a $tt$-category. The nerves of steel condition holds when the map … is a bijection. Nerves of Steel Conjecture. Every $tt$-category satisfies the nerves of steel c…

An $n$-type is a homotopy type whose homotopy groups vanish in degrees greater than $n$. An $n$-groupoid is an algebraic structure intended to encode homotopical information throug…

Let $\mathcal{B}$ be an anyon model, let $\mathcal{M}_{\mathcal{B}}$ be the moduli space of gapped systems with intrinsic topological order $\mathcal{B}$, and let…

Let $X_{cSigma}$ be a smooth toric variety satisfying conditions $({\rm homogenous}.1)$ and $({\rm homogenous}.2)$, and let $D=(d_1,\ldots,d_r)$ be a multidegree. Define … where…

Martino–Priddy conjecture. The groups $G$ and $H$ have isomorphic $p$-fusion systems if and only if

Let $d$ be an integer with $d\geq 3$, let $\alpha$ satisfy … and let $\Delta\sim\Delta(n,n^{-\alpha})$ be a random clique complex. Kahle's bouquet of spheres conjecture. With high…

Let $X$ be a simply connected finite CW complex whose cohomology with coefficients in ${\mathbb Z}/p{\mathbb Z}$ is non-trivial. Serre showed that $X$ has infinitely many non-zero…

Let $X\sim X(n,p)$ be a random simplicial complex, let $d\geq 3$, and fix … Set $p=n^{-\alpha}$. Bouquet-of-spheres conjecture. With high probability, $X$ is homotopy equivalent to…

Let $\tau$ be a collection of singularity types, let $X_\tau$ be the classifying space for $\tau$-cobordisms, let $\mathcal K_\tau$ be the Kazarian space, let $\mathcal K^+_\tau$ b…

Shareshian's conjecture. Every open interval in the lattice of subgroups of a finite group has the homotopy type of a wedge of spheres.

Let $n e 1$ and assume the canonical model structure exists on $operatorname{Mod}(IC{n+1})$. Let $0leq k<n$, let $X:IC{n+2}toSet$ be a model, and let $f:D^kto UX$ be a map. Suppose…

Let $C_n^r$ denote the $r$th power of the cycle on $n$ vertices, and let $\Delta_d^t(C_n^r)$ denote its total $d$-cut complex. For $d\geq3$ and $n\geq(2d-1)r+1$, Shen et al.'s conj…