On Closed Manifolds And Moment-angle Complex In Toric Topology

This dissertation consists of three chapters.In the first chapter, we discuss the classification of manifolds admitting （Z₂）^k-actions withfixed point set of constant codimension2k+2v+1up to cobordism. Let φ:（Z₂）^k×Mn→Mndenote a smooth action of the group （Z₂）^k={T₁, T₂,..., T_k|T₂ⁱ=1, T_iT_j=T_jT_i} on a closedmanifold Mⁿ. The fixed point set F of the action is the disjoin union of closed submanifolds ofMn, which are finite in number. If each component of F is of constant dimension n r, we saythat F is of constant codimension r. Let J_n,k^rbe the set of unoriented cobordism class of Mnthat admits a （Z₂）^k-action with fixed point set of constant codimension r. J_n,k^ris a subgroup ofunoriented cobordism group MOnand J_n,k^r=（?）J_n,k^r is an ideal of the unoriented cobor-dism ring MO₊=（?）. In this paper, we use the mathematical induction to constructgenerators of MO and determine the algebraic structure of J_n,k^r, where r=2^k+2v+1. So,we complete the classification of such manifolds up to cobordism.In the second chapter, we discuss the classification up to equivariant bordism of smoothmanifolds with involution whose the fixed point set is the disjoint union of an real projectivespace of even dimension and a Dold manifold. Let （M, T） be a smooth closed manifold witha smooth involution T, when the fixed point set of T on M is F=RP （2m） P （2m,2n+1）, we prove that (M^2m+4n+k+2, T) is equivariantly bordant to （P （2m, RP （2n+2））, T₀） or（RP （2m）×RP （2m）, twist）.In the third chapter, we mainly discuss the properties of topological space with torus ac-tions. In some cases, the orbit space of torus action carries a rich combinatorial structure（suchas convex polytope）. So we can study the topology of the toric space through the combinatoricsof the orbit space. On the other hand, the equivariant topology of a torus action sometimes helpsto interpret and prove the most subtle combinatorial results topologically. In this chapter, firstly,by using the combinatorics of the orbit space, we calculate the number of classes of a kind oftoric space up to equivariant homeomorphism, i.e. small covers overâ–³ⁿ¹Ã—â–³ⁿ²Ã—P（m）, whereâ–³ⁿⁱis the simplex of dimension n_i and P （m） is an m-polygon, n₁≥2, n₂≥1, m≥3.Secondly, we calculate the Euler characteristic for orbit configuration space of moment-anglecomplex of simplicial complex K in terms of the f-vector of K.