3 problems

Let $G$ be a graph drawn on a surface, and call two edges independent if they have no common endpoint. Surface Hanani–Tutte conjecture. If $G$ has a drawing on a surface in which n…

The genus $\mathrm{g}(G)$ of a graph $G$ is the minimum $g$ such that $G$ has an embedding on the orientable surface $M_g$ of genus $g$. Its $\mathbb{Z}_2$-genus $\mathrm{g}_0(G)$…

Let $M_1$ and $M_2$ be orientable manifolds of dimensions $n_1$ and $n_2$, possibly with boundaries, with $n_1+n_2=n$, PL embedded in general position in $\mathbb{R}^n$. Let $X$ be…