8 problems

Immersion-dimension conjecture. Every $n$-dimensional manifold can be immersed in

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…

Generic approximation conjecture. Any proper immersion $\beta$ admits a $C^\infty$-approximation $\tilde\beta$ such that $\tilde\beta$ is $k$-normal for all $k\in[2,n+1]$, and each…

Let $M$ be a smooth closed manifold of dimension $n$, and let $F_2(M)$ denote its ordered configuration space of two distinct points. The unordered configuration space is…

Let $\operatorname{Imm}(X)$ denote the immersion dimension of a space $X$. For every integer $a\geq2$, consider the real projective space $\operatorname{\mathbb{R}P}^{2^a+2^{a+1}}$…

Monotonicity conjecture. For every integer $j$ with $2\leq j\leq r(2^a+2^{a+1})$,

Let $P$ and $N$ be manifolds of dimensions $p$ and $n$, respectively, and let $f\colon P\to N$ be an immersion equipped with a lift $f_{j-1}\in E_{j-1}(P,N)$. For $j\geq 2$, let…

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