A Note on Solid Coloring of Pure Simplicial Complexes

We establish a simple generalization of a known result in the plane. The simplices in any pure simplicial complex in R^d may be colored with d+1 colors so that no two simplices that share a (d-1)-facet have the same color. In R^2 this says that any planar map all of whose faces are triangles may be 3-colored, and in R^3 it says that tetrahedra in a collection may be "solid 4-colored" so that no two glued face-to-face receive the same color.Comment: 11 pages, 6 figure

Branched Coverings, Triangulations, and 3-Manifolds

A canonical branched covering over each sufficiently good simplicial complex is constructed. Its structure depends on the combinatorial type of the complex. In this way, each closed orientable 3-manifold arises as a branched covering over the 3-sphere from some triangulation of S^3. This result is related to a theorem of Hilden and Montesinos. The branched coverings introduced admit a rich theory in which the group of projectivities plays a central role.Comment: v2: several changes to the text body; minor correction

Manifolds of isospectral arrow matrices

An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space $M_{St_n,\lambda}$ of Hermitian arrow $(n+1)\times (n+1)$-matrices with fixed simple spectrum $\lambda$. We prove that this space is a smooth $2n$-manifold, and its smooth structure is independent on the spectrum. Next, this manifold carries the locally standard torus action: we describe the topology and combinatorics of its orbit space. If $n\geqslant 3$, the orbit space $M_{St_n,\lambda}/T^n$ is not a polytope, hence this manifold is not quasitoric. However, there is a natural permutation action on $M_{St_n,\lambda}$ which induces the combined action of a semidirect product $T^n\rtimes\Sigma_n$. The orbit space of this large action is a simple polytope. The structure of this polytope is described in the paper. In case $n=3$, the space $M_{St_3,\lambda}/T^3$ is a solid torus with boundary subdivided into hexagons in a regular way. This description allows to compute the cohomology ring and equivariant cohomology ring of the 6-dimensional manifold $M_{St_3,\lambda}$ using the general theory developed by the first author. This theory is also applied to a certain $6$-dimensional manifold called the twin of $M_{St_3,\lambda}$. The twin carries a half-dimensional torus action and has nontrivial tangent and normal bundles.Comment: 29 pages, 8 figure