On Classification Of Closed Manifolds In Toric Topology And Related Subjects

This dissertation consists of three parts.The first part is to calculate the number of equivariant homeomorphism classes andequivariant cobordism classes of small covers. A small cover is a smooth closed manifoldMn with a locally standard (Z2)n-action such that its orbit space is a simple convex poly-tope. When the simple convex polytope is the product of a polygon with a simplex, theproduct of simplices, the product of 2-cube with a polygon or the product of the polar of thecyclic polytope C3(6) with a simplex, we calculate the number of equivariant homeomor-phism classes of small covers over it. When the dimension of each simplex is equal to 1or when the number of simplices is at most 3, we also consider determining the number ofequivariant cobordism classes of small covers over these products of simplices.The second part is to study two kinds of colorings of simplicial posets and related prop-erties. We introduce two notions of k-linear coloring and regular k-coloring of a simplicialposet and show that every k-linear colored simplicial poset S includes a representative sub-poset such that the representative subposet is a deformation retract of S. Moreover, we provethat regular coloring properties of a simplicial poset lead to splitting properties of a bundleover the associated Davis-Januszkiewicz space.The third part is to study equivariant cohomology rings of G-equivariantly formal man-ifolds. For G = Z/p (where p is an odd prime) or S~1, G-equivariantly formal manifolds are aclass of connected oriented (with respect to Z/p or Q) closed G-manifolds with a non-emptyfinite fixed point set, each of which is totally non-homologous to zero. We give explicit de-scriptions of the mod p equivariant cohomology rings of Z/p-equivariantly formal manifoldsand rational equivariant cohomology rings of S~1-equivariantly formal manifolds at any evendimension in terms of algebra. In fact, there aren't odd dimensional G-equivariantly formalmanifolds (G = Z/p or S~1).