Research On The Structure Space

In this paper we study two kinds of generalized configuration spaces-orbit configuration spaces and graphic configuration spaces. However, up to now, there are no available result about non-free orbit configuration spaces. Toric topology provides us some nicely behaved equivariant manifolds, such as small covers and quasi-toric manifolds, each of which admits a non-free group action. We study the orbit configuration spaces given by these equivariant manifolds. Furthermore, we can calculate the Euler number of those orbit configuration spaces and also determine their homotopy type in a spacial case. As an application, we calculate the homology group of some examples. In the study of graphic configuration spaces, we give the relationship between the Euler number of graphic configuration spaces and the chromatic polynomial of graph. We also generalize the chromatic polynomial to chromatic ring, and prove that the cohomology ring of graphic configuration spaces of Euclidian spaces is our chromatic ring.