29 problems

For a closed oriented or non-orientable manifold product with an Arf invariant normal map, the resulting surgery obstruction modulo the ordinary Arf–Kervaire invariant is called a…

Stable Carlsson conjecture. There exists a constant $C$ such that, for every prime $p>C$,

Toral rank and Carlsson conjectures.

Let $\mathcal{T}$ be a triangulation of a closed and connected $d$-dimensional manifold, with $d>0$, $f_0$ vertices, and $f_d$ facets. Spreer–Tobin's sharp vertex bound conjecture.…

Let $(V,\partial V)$ be a manifold with $\dim(V)\geq 3$. Let $X\hookrightarrow B\to V$ be a bundle with structure group $\operatorname{Homeo}(X)$, where $X$ is a manifold with a po…

Finiteness conjecture. There exist $k_{0}\in\mathbb{N}$ such that

Let $N$ be a closed connected orientable $n$-manifold with $H_1(N)$ torsion-free, $n\geq 4$, and $n$ even, and let $N_0$ denote a punctured version of $N$. For an embedding…

Let $M$ and $N$ be compact connected $C^p$ manifolds, and let $\Diff^p_c(M)_0$ and $\Diff^p_c(N)_0$ denote their groups of compactly supported $C^p$ diffeomorphisms diffeotopic to…

Let $X$ and $Y$ be closed $(n{-}1)$-connected $m$-manifolds, with $n>1$ and $m\leq 5n-2$. Let … be an isomorphism of cohomology algebras. Write $\mathcal{T}_b$ and $\mathcal{T}_c$…

Let $M$ be a simply connected manifold that is not homeomorphic to $S^n$, and let $N$ be a manifold of dimension $n+k<2n$. The group $\operatorname{Homeo}_0(M)$ acts nontrivially o…

Manifold-diagram representation conjecture. Let $C$ be a presented associative $n$-category. The mapping from labelled singular cubes to colored cubes descends to an injection of s…

Let $\mathbb{F}$ be a field. A simplicial complex $X$ is $\mathbb{F}$-tight if it is connected and, for every induced subcomplex $Y$ of $X$, the inclusion-induced map … is injectiv…

Let $K_1$ and $K_2$ be generalized homology spheres of dimension $n-1$, with $m_1$ and $m_2$ vertices, respectively. Let … The operation $\mathcal{G}$ on manifolds is the gyration,…

Let $(M^{2k+1},F^{2k})$ be a tidy pair, with $i:F\hookrightarrow M$ the inclusion and $i_*:L^2H_k(F,\partial F)\to L^2H_k(M,\partial M)$ the induced map. Tidy-pair vanishing conjec…

Let $(M,\partial M)$ be an $n$-manifold with contractible components that admits a geometric group action, and let $L^2H_i(M)$ denote its $L^2$-homology. Manifold cocompact action…

Let $M$ be a closed simply connected Riemannian manifold with non-negative sectional curvature. A singular Riemannian foliation of codimension 1 is a singular Riemannian foliation…

Let $M$ be a compact oriented manifold, let $S(M)$ be its structure group, and let $S_0(M)$ be the subgroup generated by elements $[(f,M')]-[(\psi\circ f,M')]$, where $f:M'\to M$ i…

Let $M$ be a compact manifold, and let $\mathrm{Homeo}_{0}(M)$ denote the identity component of its homeomorphism group. A group morphism is trivial if it maps every element to the…

Let $M$ and $N$ be closed aspherical manifolds with isomorphic fundamental groups. Their universal covers are contractible open manifolds. The Weak Borel Conjecture. The universal…

Let $M$ be a closed aspherical manifold of dimension greater than $1$, and let $\widetilde{M}$ denote its universal cover. A space is strongly connected at infinity when its end ha…

Let $M$ be a manifold. Say that $M$ splits when it is divided by every divisor, and let $\beta_1$ be the first Betti number of $M$ with coefficients in $\mathbb{Z}/2\mathbb{Z}$. Th…

Let $M$ be an acyclic compact orientable manifold, let $\pi(M)$ denote its fundamental group, and let $k[\pi(M)]$ be the corresponding group algebra. Circle-bundle superpotential c…

Let $M$ be an acyclic oriented compact manifold of dimension $d$, and let $k[\pi_1(M)]$ denote its fundamental group algebra. Superpotential conjecture. If $M$ is an acyclic orient…

Positivity conjecture. For every compact $2$-dimensional surface $S$, this quadratic function has no kernel: if

Let $p\colon E\to B$ be a Serre fibration, where $B$ is a locally arcwise connected metric space, and suppose that every fiber of $p$ is homeomorphic to a fixed manifold $M^n$ of d…