51 problems

Let $(M,\tau)$ be a finite von Neumann algebra with separable predual, and let $R$ be the hyperfinite $\mathrm{II}_1$-factor over ${\mathbbm k}\in\{{\mathbb C},{\mathbb R}\}$. Writ…

Metric cotype growth conjecture. Every Banach space with Rademacher cotype $q<\infty$ satisfies the metric-cotype inequality with an even integer $m\lesssim_X n^{1/q}$, without ass…

Conjectured failure of regular-neighborhood invariance. The property fails in each of the following cases: (e) $M=S^{2k}$ and $K=\Sigma(S^{k-1}\sqcup S^{k-1})$, where $f$ and $g$ c…

Let $KT^m_{p,q}$ denote the group of knotted tori $T^{p,q}=S^p\times S^q$ in $\mathbb{R}^m$ up to the equivalence used in the paper, let $V_{m-q,p+1}$ be the Stiefel manifold of…

Let $X$ be the polyhedral object and let the Fibering Lemma 7.1 refer to the fibering assertion established there for PL embeddings. The topological fibering conjecture. The Fiberi…

For $0<p<l$, let $KT^{3l+p}_{p,2l-1,AD}$ and $KT^{3l+p}_{p,2l-1,PL}$ denote the almost smooth and PL knotted-torus classification groups. The PL forgetful-map conjecture. The forge…

Let $D^{p+q}$ be a disk in $S^p\times S^q$, let $\pi^{m-1}_{eq}$ denote the indicated equivariant homotopy group, and let $\Pi^{m-1}_{p-1,q}$ be the comparison group used in the pa…

Let $\bar\alpha'\tau$ be the map from the Restriction Lemma 5.2, and let $\beta$ and $w_{l,p}$ denote the associated invariants and element. The restriction-map non-surjectivity co…

Let … be the exact sequence in the smooth case of Main Theorem 1.4. The even-l splitting conjecture. This exact sequence splits also when $l$ is even. The source explains that prov…

Let $\beta^{3l+p-1}(T^{p,2l-1}_+)$ denote the beta invariant in the indicated dimension. The beta-invariant noninjectivity conjecture. The map … is not injective; consequently, the…

Let $T^{p,q}=S^p\times S^q$, let $R(f(S^p\vee S^q))$ denote the relevant regular neighborhood of the wedge image under an embedding $f$, and set … Suppose that $f,g:T^{p,q}\to S^m$…

Let $N$ be a connected PL $n$-manifold and let $\alpha$ denote the Haefliger–Wu invariant of a PL embedding $N\to\mathbb{R}^m$. The refined Haefliger–Wu conjecture. The invariant…

Suppose that $1\le p\le l-2$, and let $KT^{3l+p}_{p,2l-1,AD}$ denote almost smooth embeddings of $T^{p,2l-1}$ in $\mathbb{R}^{3l+p}$ up to almost smooth isotopy. Let…

For each integer $k$, let $KT^{6k+4}_{1,4k+1}$ denote the corresponding group of knotted tori and let $\pi_{4k+1}(V_{2k+3,2})$ be the indicated homotopy group of the Stiefel manifo…

Let $KT^7_{1,3,DIFF}$ denote the set of smooth embeddings $S^1\times S^3\to\mathbb{R}^7$ up to smooth isotopy. The smooth knotted-torus classification conjecture. The set…

Let $N$ be a non-planar tree-like continuum, and let $\widetilde N$ denote its deleted product with the involution exchanging the two factors. Tree-like continuum conjecture. There…

Connected-sum classification conjecture. The embeddings $f\#pt$ and $f\#qt$ are isotopic if and only if

Compression-theorem characterization conjecture. The 3-manifolds with this realization property are exactly the 3-manifolds covered by Compression Theorem (2).

Alexander-dual formula conjecture. The Alexander dual of $[f|_{\partial B^3}]$ equals $\pm BH(f')$.

Codimension-two compression conjecture. Any two such embeddings are isotopic; equivalently, the Codimension Two Compression Theorem should hold without its assumptions.

Compression uniqueness conjecture. Any two embeddings $N\to\mathbb R^4$ whose images lie in $\mathbb R^3\subset\mathbb R^4$ are isotopic; equivalently, only the standard embedding…

Let $1.13b$ denote the smooth approximation theorem discussed in the source: in the range $m>\frac{3(n+1)}2$, smoothly $\delta$-isotopic smooth embeddings are smoothly…

Massey–Rolfsen conjecture. Theorem still holds if one component of the link $L$ is not embedded (but mapped into $\mathbb{R}^4$ disjointly from the other components).

Hypothesis (O) conjecture. An embedding satisfies Hypothesis $(O)$, or equivalently the stronger condition that every pair of points of $\mathscr{O}$ at mutual distance $4$ is oppo…

Ovoidal embedding conjecture. Every symmetric embedding is ovoidal.