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Register a new userOn affinely 3-regular maps and trapezoids
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We show that any embedding $mathbb{R}^d to mathbb{R}^{2d+rho(d)-1}$ inscribes a trapezoid or maps three points to a line, where $rho(d)$ denotes the Radon-Hurwitz function. This strengthens earlier results on the nonexistence of affinely $3$-regular maps for infinitely many dimensions $d$ by further constraining four coplanar points to be the vertices of a trapezoid.
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In this paper, we prove that any closed orientable 3-manifold $M$ other than $#^k S^1times S^2$ and $S^3$ satisfies the following properties: (1) For any compact orientable 4-manifold $N$ bounded by $M$, the inclusion does not induce an isomorphism on their fundamental groups $pi_1$. (2) For any map $f:Mto N$ from $M$ to a closed orientable 4-manifold $N$, $f$ does not induce an isomorphism on $pi_1$. Relevant results on higher dimensional manifolds are also obtained.
The Hurwitz problem asks which ramification data are realizable, that is appear as the ramification type of a covering. We use dessins denfant to show that families of genus 1 regular ramification data with small changes are realizable with the exception of four families which were recently shown to be nonrealizable. A similar description holds in the case of genus 0 ramification data.
A vertex-transitive map $X$ is a map on a closed surface on which the automorphism group ${rm Aut}(X)$ acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. We show that there are eleven types of semi-equivelar maps on the torus. Three of these are equivelar maps. It is known that two of the three types of equivelar maps on the torus are always vertex-transitive. We show that this is true for the remaining one type of equivelar map and one other type of semi-equivelar maps, namely, if $X$ is a semi-equivelar map of type $[6^3]$ or $[3^3, 4^2]$ then $X$ is vertex-transitive. We also show, by presenting examples, that this result is not true for the remaining seven types of semi-equivelar maps. There are ten types of semi-equivelar maps on the Klein bottle. We present examples in each of the ten types which are not vertex-transitive.
We give a generalization of Thurstons Bounded Image Theorem for skinning maps, which applies to pared 3-manifolds with incompressible boundary that are not necessarily acylindrical. Along the way we study properties of divergent sequences in the deformation space of such a manifold, establishing the existence of compact cores satisfying a certain notion of uniform geometry.
As a generalization of Davis-Januszkiewicz theory, there is an essential link between locally standard $(Z_2)^n$-actions (or $T^n$-actions) actions and nice manifolds with corners, so that a class of nicely behaved equivariant cut-and-paste operations on locally standard actions can be carried out in step on nice manifolds with corners. Based upon this, we investigate what kinds of closed manifolds admit locally standard $(Z_2)^n$-actions; especially for the 3-dimensional case. Suppose $M$ is an orientable closed connected 3-manifold. When $H_1(M;Z_2)=0$, it is shown that $M$ admits a locally standard $(Z_2)^3$-action if and only if $M$ is homeomorphic to a connected sum of 8 copies of some $Z_2$-homology sphere $N$, and if further assuming $M$ is irreducible, then $M$ must be homeomorphic to $S^3$. In addition, the argument is extended to rational homology 3-sphere $M$ with $H_1(M;Z_2) cong Z_2$ and an additional assumption that the $(Z_2)^3$-action has a fixed point.
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