4 problems

Let $P$ be a lattice polytope, let $k[P]$ be its polytopal algebra, and let $f:k[P]\to k[P]$ be a codimension $1$ retraction. Write $f^{\alpha}=\alpha\circ f\circ\alpha^{-1}$ for c…

Let $k[P]$ be a polytopal algebra, and let $f:k[P]\to k[P]$ be a retraction, meaning an idempotent homogeneous endomorphism. Image conjecture. The image of every retraction in…

A treelike continuum is a continuum whose every subcontinuum is unicoherent and whose nondegenerate subcontinua are arc-like. A map $f$ is strictly locally 1-to-1 if it is locally…

Let a codimension-1 retraction mean the type of retraction whose associated variety is described in the cited result for polygons. In arbitrary dimensions of the polytope, the resu…