93 problems

Juan-Pineda–Leary conjecture. A classifying space ${\mathcal E}_{VC}(G)$ cannot be finite unless $G$ is virtually cyclic.

Let $G$ be a compact Lie group and let $BG$ be its classifying space. Kono–Yagita's even-degree conjecture. The Brown–Peterson cohomology $BP^*(BG)$ is concentrated in even degrees…

Let $G$ be an algebraic group whose classifying space $BG$ has cohomology concentrated in even degrees. Write $CH^*BG$ for its Chow ring and $MU^*BG/MU^{<0}(pt).MU^*BG$ for the ind…

Let $G$ be a complex linear algebraic group, let $CH^i(BG)_{(p)}$ be the $p$-localized Chow group of its classifying space, and let $BP^*(BG)$ denote its Brown–Peterson cohomology.…

Let $G$ be a finite discrete group, and let $B(2,G)$ denote the classifying space associated with the space of commuting pairs in $G$. Asphericity conjecture. The space $B(2,G)$ sh…

Let $p$ be a prime, and let $e$ and $k$ be positive integers. Consider groups of the form $\mathbb Z_{p^e}\times\mathbb Z_{p^k}$, together with their $ku$-toral and complex bordism…

Totaro's conjecture. This homomorphism is an isomorphism.

Transferred Euler class generation conjecture. The Chow ring $CH^*BG$ is generated by transferred Euler classes.

Kontsevich's conjecture. The space

Let $G$ be the fundamental group of a closed hyperbolic manifold $M$ of dimension $d$. Write $obreakoperatorname{hdim}^G(underline{underline{E}}G)$ for the equivariant homotopy di…

Let $\Gamma$ be a discrete subgroup of a Lie group that is not virtually cyclic. Write $vcd(\Gamma)$ for its virtual cohomological dimension, and let $B_{\mathcal{VC}}\Gamma$ denot…

Segal conjecture. When $G'$ is the trivial group, the homomorphism $\alpha$ is a completion with respect to a certain ideal. The conjecture is a foundational statement relating Bur…

Nonvanishing conjecture. The class $[c_k^\delta]$ is a nonzero multiple of the generator in degree $2k$.

Let $\Gamma$ be a discrete group whose classifying space $B\Gamma$ is a finite CW complex of odd dimension, and let $E\Gamma$ be its universal cover. Let $\phi_{E\Gamma}$ denote th…

Let $W$ be a Coxeter group, let $G_W$ be its associated Artin group, and let … be the orbit configuration space, where $I$ is the Tits cone in the real reflection representation…

Let $\Gamma$ be a Coxeter graph, and let $\mathfrak{cl}^f(\Gamma)$ denote the spherical clique number of $\Gamma$, namely the number of vertices in a maximal spherical clique. A co…

Let $G$ be a group. Say that $G$ is of type $\Phi_\mathbb{Z}$ if every $\mathbb{Z}G$-module $M$ has finite projective dimension over $\mathbb{Z}G$ if and only if, for every finite…

Let $M$ be an ungroup-like abelian monoid, let $X$ be a topological space, and define $\mathrm{SP}(X,M)$ to be the free strictly commutative topological monoid generated by $X$-man…

Let $G$ be a finite-dimensional Lie group, and write $G^\delta$ for its underlying discrete group. Milnor's conjecture. The map induced by the identity $G^\delta\to G$, … is an iso…

Let $\mathcal{F}$ be a saturated fusion system. A classifying space associated to $\mathcal{F}$ is a space encoding its fusion-theoretic structure, denoted by $B\mathcal{F}$. Gener…

Let $K$ be a simply connected finite simplicial complex, and let $M$ be a finite monoid. Fiedorowicz's conjecture. Every homotopy type of a simply connected finite simplicial compl…

Let $A_\Gamma$ be an Artin group with Salvetti complex $\mathrm{Sal}(\Gamma)$. The Salvetti complex is the CW complex associated to the Artin presentation, with higher-dimensional…

Totaro's conjecture. Under these hypotheses, this map is an isomorphism. The conjecture compares the Chow ring with Brown–Peterson cohomology after localization. No resolution is s…

Let $X_{\infty}=\varinjlim_nX_n$ be the cofibrant $E_2$-algebra with contractible components, let $\alpha,\alpha'\colon X_{\infty}\to\bigsqcup_nBU(n)$ be $E_2$-maps preserving comp…

Let $(\mathrm{W}[\Gamma],S)$ be a Coxeter system with $S$ finite, and let $N(\Gamma)$ be the quotient of the complement of the complexification of the reflection arrangement by the…