14 problems

Let $f$ be a map from a closed $(m-1)$-manifold to a closed $m$-manifold, where $m\ge 3$, and suppose that its singular set $S(f)$ is $0$-dimensional. Approximation-by-embeddings c…

Let the two $n$-spheres in the linking described immediately before the claim be joined by a tube, producing an embedding … Let $\alpha$ denote the alpha-invariant used in the pape…

Borromean-rings conjecture. The embedding $f$ can be obtained from these Borromean rings by “wedging” the two copies of $S^n$.

Let $N$ be a $1$-connected $6$-manifold, and let $\bar W_3(N)$ denote its normal Stiefel–Whitney class. Six-manifold embedding conjecture. There exists such an $N$ with … that does…

Let $S^1\times S^1$ be the torus, and consider smooth embeddings into $\mathbb R^4$, with equivalence given by PL isotopy. Torus embedding conjecture. Every smooth embedding … is P…

Let $n$ and $d$ be parameters in Theorem 2.6.a, whose embedding dimension is $2n-d$. Dimension conjecture. The dimension $2n-d$ in Theorem 2.6.a cannot be significantly decreased f…

Let $n$ be a positive integer, and let $\alpha(n)$ denote the number of units in the dyadic expansion of $n$. Embedding-dimension conjecture. Every closed $n$-manifold should embed…

Let $(\mathsf{M},g)$ be a smooth, closed non-orientable manifold. Its orientable double covering is the two-sheeted covering manifold associated with the orientation bundle of…

Low-codimension extension conjecture. The theorem should also hold for $m=n$ and for $m=n+1$.

Homology-equivalence embedding conjecture. The set $\mathop{\mathrm{Emb}}_{\mathrm{CAT}}^m(N)$ should be unchanged under homology equivalence, in the PL category for $m\ge n+3$ and…

Let $N$ be a closed connected orientable $n$-manifold, with $n\ge 4$. Let $N_0$ denote $N$ minus the interior of a codimension-$0$ open ball, let $H_1(N)$ denote the first homology…

Product lower-bound conjecture. The invariant should satisfy

Quotient monotonicity conjecture. One should have

Totally skew embedding upper-bound conjecture. For every such manifold,