41 problems

Let $r\geq 2$ be an integer. An almost $r$-embedding is a function from a simplicial complex to $\mathbb{R}^d$ that does not identify $r$ points lying in pairwise disjoint faces. T…

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

Let $N$ be a closed $n$-manifold, and let two embeddings $N\to S^{n+2}$ be given. Say that two such embeddings are almost concordant when they are related by the almost-concordance…

Consider PL embeddings of the torus $S^1\times S^1$ into $\mathbb{R}^4$, up to PL almost concordance. Three-element conjecture. The set of PL embeddings … up to PL almost concordan…

Let $N$ be a smooth manifold, and consider the Almost Embeddings Theorem (b), the cited results corresponding to [Sk02, Theorem 2.3.q], and the injectivity assertion corresponding…

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…

Shastri's conjecture. The algebraic trefoil knot embedding is non-rectifiable over $\mathbb{C}$.

Let a reservoir computer be driven by a generic continuously differentiable ($C^1$) observation of a smooth dynamical system on an $m$-dimensional compact manifold. Reservoir-compu…

Hirsch conjecture. There is an immersion of $M$ in $\mathbb{R}^n$ which is regularly homotopic to an embedding. This is a reformulation of the Hirsch conjecture, which concerns emb…

Let $M$ and $N$ be smooth closed manifolds of dimensions $p$ and $q$, respectively. Write … A $(\mathbb{Z}/2)^2$-equivariant map is understood with respect to the involutions excha…

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…

Browder-Levine uniqueness conjecture. If

Let $(X,T)$ be a topological dynamical system, let $a$ be a positive real parameter, let $\operatorname{mdim}(X,T)$ denote its mean dimension, let $\operatorname{dim}$ denote topol…

Let $(X,T)$ be a topological dynamical system, let $a$ be a positive real parameter, let $\operatorname{mdim}(X,T)$ denote its mean dimension, let $\operatorname{dim}$ denote topol…

Let $N^n_m$ be the subset of $\mathbb{R}^n$ consisting of points with at most $m$ rational coordinates. Chigogidze's conjecture. Every $m$-dimensional subset of $\mathbb{R}^n$ can…

Topological classification conjecture. Isovariance and the tying condition are sufficient for a full classification of embeddings in codimension $3$; no additional conditions are r…

The Prem Conjecture. Smoothly $k$-realizable general-position smooth maps $N^n\to\mathbb{R}^m$ are smooth $k$-prems, at least in the metastable range

Let $s,d,r\ge2$ be integers satisfying … and let $K$ be a finite $s$-dimensional simplicial complex. An almost $r$-embedding is a map $f:K\to\mathbb R^d$ such that…

Let $D=D^{s_1}\sqcup\ldots\sqcup D^{s_r}$ be a disjoint union of $r$ disks, let $f:D\to B^d$ be a proper PL map such that…

Let $n,k\geq 1$ be integers, let $\Delta_n^{(k)}$ denote the $k$-skeleton of the $n$-simplex, and let $M$ be a $2k$-dimensional manifold. An almost-embedding of $\Delta_n^{(k)}$ in…

Let $n,k\geq 1$ be integers, let $\Delta_n^{(k)}$ denote the $k$-skeleton of the $n$-simplex, and let $M$ be a compact, $(k-1)$-connected $2k$-manifold. Write $b_k(M)$ for its $k$t…

Dimension-restriction conjecture. The restriction $m\geq 3(n+1)/2$ can be weakened if one uses the configuration space of $r$-tuples of distinct points, and methods of this paper w…

An embedded projective plane in $4$-space is a smoothly embedded non-orientable surface whose non-orientable genus $g$ is the number of projective-plane connected-summands; its nor…

PL embedding conjecture for contractible $2$-complexes. Every contractible $2$-complex PL-embeds in