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 $G$ 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 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 $\Gamma$ 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…

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 $\ell$ 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…