33 problems

Let $G$ be a Lie group. Regard $BG$ as the classifying space of the underlying discrete topological group, and let $BG^{\rm top}$ be the classifying space with its usual topology.…

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…

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 $M$ 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 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…

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…

Equivariant monoid realization conjecture. Any $G$-connected based $G$-space $X$ is weakly equivalent to

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.…

Soit $W$ un groupe de Coxeter affine agissant par réflexions sur $\mathbb{R}^n$. On note $Y_W$ l'espace de configuration de type $W$, obtenu comme le complémentaire dans…

Contact Haefliger–Thurston conjecture. The natural map induces a homology isomorphism through degree $n$ and a surjection on homology in degree $n+1$.

Haefliger–Thurston conjecture. The map $\eta$ is a homology isomorphism in degrees at most $\operatorname{dim}(M)$ and is surjective on homology in degree $\operatorname{dim}(M)+1$…

Classifying-map conjecture. Under the relevant zigzag of equivalences, the map $N\alpha_*$ is a classifying map for $p:E\to B$: its homotopy class coincides with…

The hammock-space conjecture. The space $\widehat{A}$ is weakly equivalent to $\emph{Bhaut} A$:

Let $W$ be an arbitrary closed manifold on which $SU(2)$ acts smoothly and non-trivially. Let $s\colon BSU(2) \to B\operatorname{Diff}(W)$ be the induced map on classifying spaces,…

Let $G$) be a residually finite finitely presented group. For a finitely presented group $G$, its deficiency $\delta(G)$ is the maximum of $|X|-|R|$ over all presentations…

Let $\Gamma$ be a group and let $B\Gamma$ denote its classifying space. Assume that $B\Gamma$ is finitely dominated, and denote its finiteness obstruction by $\sigma(B\Gamma)$. Hsi…

Let $G$ be a connected compact Lie group of dimension $d$, let $BG$ be its classifying space, and let $LBG$ denote the free loop space of $BG$. Write $S_*(G)$ for the singular chai…

Let $G$ be a knot group, let $H\leq G$ be a peripheral subgroup, and let $U\trianglelefteq G$ be a normal subgroup of finite index. Write $E_\mathcal{G}G/U$ for the associated clas…

Let $L$ be the nerve of an Artin group $A_L$, and let $\mathcal{S}(L)$ be the poset of simplices of $L$ together with the empty simplex. The poset of groups over $\mathcal{S}(L)$ h…

Let $G$) be a group, and let ${\underline{\underline E}}G=E_{\mathcal{VCY}}(G)$ denote a $G$-CW-model for the classifying space for the family of virtually cyclic subgroups. The mo…

Let $\ell$ be an odd prime, let $\mathrm{G}=\operatorname{SL}_{\ell^2}/\mu_\ell$ over $\mathds{C}$, and let $c_2$ generate $\operatorname{H}^4(\mathrm{BG},\mathds{Z})$. Integral Ch…

Let $G$ be a non-commutative compact connected Lie group, let $B_nG$ denote the $n$-th stage of the classifying-space construction, and let $\iota_n\colon B_nG\to BG$ be the canoni…

Nucinkis's conjecture. Every group of finite $\operatorname{\mathfrak{F}}$-cohomological dimension admits a finite-dimensional model for…