arXiv subjects
Publications and source records attributed to Haimiao Chen.
A {\it regular balanced Cayley map} (RBCM for short) on a finite group $Γ$ is an embedding of a Cayley graph on $Γ$ into a surface, with some special symmetric property. People have classified RBCM's for cyclic, dihedral, generalized quaternion, dicyclic, and semi-dihedral groups. In this paper we classify RBCM's on the group ${\rm PSL}(2,p)$ for each prime number $p>3$.
Suppose $Γ$ is a finite group acting freely on $S^{n}$ ($n\geqslant 3$ being odd) and $M$ is any closed oriented $n$-manifold. We show that, given an integer $k$, the set $°^{-1}(k)$ of based homotopy classes of mappings with degree $k$ is finite and its cardinality depends only on the congruence class of $k$ modulo $\#Γ$; moreover, $\#°^{-1}(k)$ can be expressed in terms of the Dijkgraaf-Witten invariants of $M$.
A {\em solvable} cover of a graph is a regular cover whose covering transformation group is solvable. In this paper, we show that a solvable cover of a graph can be decomposed into layers of abelian covers, and also, a lift of a given automorphism of the base graph of a solvable cover can be decomposed into layers of lifts of the automorphism in the layers of the abelian covers. This procedure is applied to classify metacyclic covers of the tetrahedron branched at face-centers.
This article proposes an effective criterion for lifting automorphisms along regular coverings of graphs, with the covering transformation group being any finite abelian group.
In this article we complete the work of enumerating typical abelian coverings of Cayley graphs, by reducing the problem to enumerating certain subgroups of finite abelian groups.
Given a finite connected simple graph $Γ$, and a subgroup $G$ of its automorphism group, a general method for finding all finite abelian regular coverings of $Γ$ that admit a lift of each element of $G$ is developed. As an application, all connected arc-transitive abelian regular coverings of the Petersen graph are classified up to isomorphism.
I give a formula for computing the number of regular $Γ$-coverings of closed orientable Seifert 3-manifolds, for a given finite group $Γ$. The number is computed using a 3d TQFT with finite gauge group, through a cut-and-glue process.
We study the twisted Lefschetz number, which is a generalization of classical Lefschetz number. A formula in terms of homology with local coefficients is given. We then propose a method to estimate the Nielsen number. As applications, an algorithm for computing the Nielsen number of a self map of a space with abelian fundamental group is discussed; and Bass' conjecture for a class of groups is proved.