240 problems

Let $D\colon\mathrm{Span}(C,E)\to\mathrm{Cat}$ be a six-functor formalism, and let $P$ and $I$ denote respectively the classes of $D$-proper and $D$-acute{e}tale morphisms. The con…

For every simply connected finite CW complex $X$, if $X$ is rationally hyperbolic, then there exists a finite set of primes $P_X$ such that, for every prime $p\notin P_X$ and every…

Let $K$ be a shifted simplicial complex, let $(CX_i,X_i,x_i)$ be based pairs indexed by the vertices of $K$, and write $Z(K;(\underline{CX},\underline{X}))$ for the associated poly…

Let $X$ be a smooth closed manifold. Write $\mathrm{Diff}(X)$ for its diffeomorphism group, $B\mathrm{Diff}(X)$ for its classifying space, and $\pi_k$ and $H_k(-;\mathbb{Z})$ for h…

Finite-test homotopy equivalence conjecture. The map $f$ is a homotopy equivalence.

Let $N$ and $L$ be closed, simply connected manifolds of the same dimension. The cotangent bundle $T^*N$ carries its natural symplectic structure, and consider the space of Lagrang…

This entry tracks Graph Hall algebra cohomology realization conjecture. The realization links combinatorial Hall algebras with topological invariants and yields a new graph invaria…

Singer's conjecture. The homomorphism

Let $M$ and $N$ be closed $d$-manifolds with $d \geq 5$ and the same normal $2$-type. Assume that both $M$ and $N$ admit metrics of positive scalar curvature. Write…

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…

For $n\ge 1$, let $S^n$ be the unit $n$-sphere and let $\mathrm{\check{C}}(S^n;r)$ denote its Čech complex at scale $r\in(0,\pi)$. Write $\operatorname{conn}(K)$ for the connectivi…

Pattern-preserving weak equivalence conjecture. The embedding $\mathcal I$ is a weak homotopy equivalence.

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…

Moore's conjecture. The following are equivalent:

Loop-space conjecture. The space $\operatorname{\mathcal{R}^\text{Cores}}(\operatorname{S}^n)$ equipped with $\mu$ is an associative $H$-space and is homotopy equivalent to a loop…

Homotopy Conjecture. There exists a surjective map $F:P[T]\twoheadrightarrow J$ such that

Let $X$ be a space, let $NORM(X)$ be the site of normal coverings of $X$, and let $OPEN(X)$ be the standard site. Let $S$ be a set, write $S_{\#}$ for the associated constant coshe…

Let an $n$-cube of spaces be a diagram indexed by the powerset of $[n]=\{0,1,\ldots,n\}$, and call it absolutely cartesian when every homotopy functor sends it to a homotopy cartes…

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…

Unitality–homotopy unitality equivalence. An $A_\infty$-morphism is unital if and only if it admits a homotopy unital structure; equivalently, any unital $A_\infty$-morphism admits…

Metric thickening finite-support homotopy equivalence conjecture. The inclusion induces a homotopy equivalence

Homotopy equivalence conjecture. The maps $l^\pm_V(p)$ are homotopy equivalences of ${\cal I}$-filtered spaces.

Generalized pushout conjecture. There is a notion of pushout of ${\cal C}$-categories such that the loop space functor from spaces to ${\cal C}$-groupoids takes pushouts to pushout…

May's generalized Seifert–van Kampen conjecture. The loop space functor from spaces to grouplike $\widehat{{\cal C}}$-spaces takes pushouts of connected spaces to pushouts of…

Effective categorification conjecture. For any $A_{\infty}$-operad ${\cal C}$, there is an operation $Cat _{{\cal C}}$ from $\widehat{{\cal C}}$-precats to $\widehat{{\cal C}}$-cat…