237 problems

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…

Let $G$ be a group, $H$ an abelian group, and $R$ a commutative ring with unit group $R^*$. Let $B_G$ be the classifying space of $G$, and for each cohomology class…

Homotopy hypothesis for snugroupoids. There are functors

Lück's homotopy-invariance conjecture. $L^2$-Reidemeister torsion is a homotopy invariant.

Let ${\mathcal A}$ be a central hyperplane arrangement with connected underlying matroid, and let $C({\mathcal A})$ denote its complement. Homotopy-type conjecture. The homotopy ty…

Localize all spaces at $3$. Let $B_2(27,35)$ denote the space occurring in the proposed spherical resolution, and let $\alpha_2$ be the element generating…

Three-component connectivity conjecture. The map $l_3$ is $(3n-\max\{2p_1+p_2+p_3,p_1+2p_2+p_3,p_1+p_2+2p_3\}-5)$-connected.

Connectivity conjecture. The map $l_2$ is $(2n-\max\{2p+q,2q+p\}-3)$-connected.

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…

Strong Barratt exponent conjecture. The double loop map

Barratt's exponent conjecture. The image of the induced map

Let $p$ be a prime, let $\operatorname{eo}_{p-1}$ be the proposed connective form of $EO_{p-1}$, and let $a_2,\dots,a_{p-1}$ be exterior generators with … Let…

Let $p$ be any prime, let $\alpha,l\in\mathbb{N}$, and let $n,r\in\mathbb{Z}$ with $n\ge 2p^{\alpha}-1$. The theorem gives a lower bound for the $p$-adic valuation of the relevant…

Let ${\mathcal L{\it ie}}$ be the Lie operad, and let ${\mathcal B}_{{\mathcal L{\it ie}}}$ be the operad of natural operations on the cohomology of Lie algebras. Lie-operad conjec…

Let $X$ be a connected $p$-compact group with maximal torus $T_X$. Its maximal torus normalizer is represented by the loop space $NT_X$. Maximal torus normalizer conjecture. The gr…

Proposed conjecture. For $n\equiv 6\pmod 8$ with $n\geq 14$,

Let $G$ be a compact connected Lie group, let $X$ be a $p$-complete space, and let $\mathcal{A}_p$ denote the mod-$p$ Steenrod algebra. Suppose that … as algebras over…

Let $PrCat$ be the model category of Segal precategories, let $QCat$ be the model category of quasi-categories, and let $\phi:PrCat\to QCat$ and $\psi:QCat\to PrCat$ be the adjoint…

Let $k$ be a field and let $F\in LAff_k^{\sim,ffqc}$ be a schematic homotopy type. Let $i(F)$ be its image among stacks over $Hk$, and let $LParf(i(F))$ be the associated $k$-tenso…

Let $k$ be a field and let $(X\otimes k)^{sch}\in LAff_k^{\sim,ffqc}$ be the schematic homotopy type associated with a connected finite CW complex $X$, and let $LParf(X,k)$ be its…

Let $X$ be a simply connected polyGEM of finite type at $2$, and let $\widetilde{\operatorname{H}}^*X$ denote its reduced cohomology with coefficients in $\mathbb{F}_2$. Grodal's c…