63 problems

Exact-sequence conjecture. If $(n,k)=(5,1)$ or $(6,2)$, then there are exact sequences of sets

A double disk bundle decomposition writes a smooth manifold as the union of two disk bundles glued along their common boundary by a diffeomorphism. The conjecture concerns closed,…

Let $N$ be a $k$-connected closed $n$-manifold, with $n$ even, $m=(3n+1-k)/2\ge n+3$, and with $H_{k+1}(N)$ free. Write $\widetilde N$ for the universal cover and let…

Let $p(d)$ denote the universal sesquilinear pairing obtained by gluing $d$-manifolds with a common boundary, with positivity meaning that $\langle x,x\rangle\neq 0$ whenever…

For an integer $n$, the triangulation conjecture in dimension $n$ is the assertion that every closed topological $n$-manifold is homeomorphic to a simplicial complex. Triangulation…

Mathai's converse conjecture. If $\pi_1(M)$ is torsion-free, then $\rho_{(2)}$ is a homotopy invariant.

A Morse parameterization is the simultaneous continuous parameterization of the members of a family of manifolds, generalizing Whitney's continuous choice of a point in curves. Shc…

Let $\kappa^h_j$ denote the surgery-theoretic obstruction maps discussed in the paper. Oozing conjecture. The original oozing conjecture asserts that $\kappa^h_j=0$ for every…

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 $M$ and $N$ be open, one-ended manifolds of dimension greater than $3$. Their end sum is formed using a proper ray in each manifold, and the resulting manifold may depend on th…

Let $d>0$ be odd, and let $\mathcal{T}$ be a triangulation in the remaining case where $f_d$ is odd and $f_d<d$. Spreer–Tobin's small odd-facet vertex maximum conjecture. The true…

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 $d$ be an even integer, and let $\mathcal{T}$ be a (generalised) triangulation of a $d$-manifold $\mathcal{M}$ with $f$-vector $(f_0,f_1,\ldots,f_d)$. Spreer–Tobin's vertex bou…

Let $n\geq 4$. A bilipschitz bundle is a fiber bundle whose structure group lies in $\operatorname{bilipschitz}(X)$. The high-dimensional non-compactification conjecture. There exi…

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

Asymptotic mod 2 Kühnel conjecture. There is a constant $c_k>0$ such that, whenever $[n]^{*k+1}$ has a ${\mathbb Z}_2$-embedding into a closed $2k$-manifold $M$, one has

Tame tangle presentation conjecture. Every smooth manifold can be presented as a tame tangle.

Let $K$ be a $k$-complex and let $M$ be a PL $2k$-manifold. A map $h:K\to M$ is a ${\mathbb Z}_2$-embedding if the mod-$2$ intersection number of the images of every pair of non-ad…

Let $M$ and $N$ be smooth, compact manifolds. For each positive integer $n$, let $M^n$ and $N^n$ carry the action of the permutation group $\Sigma_n$ by permuting factors; write…

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…

The paper studies embeddings of punctured $n$-manifolds $N_0$ in $\mathbb{R}^{2n-1}$ and associates to each embedding $f$ a Seifert linking form $L(f)$, an integer-valued symmetric…

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 $\mathcal{T}_c$…