Some Problems On Characteristic Functions Of Toric Manifolds

It is very important to study topological space with torus action,which links composition,algebra and topology closely.Small cover and torus manifolds is kind of important manifold with group action.We can understand its homology algebra and topological properties,by studying their orbital space(simple convex polytope)combination.Especially,obtained by group action of characteristic function on simple convex polytope can reconstruct small cover or torus manifolds.Namely,characteristic function determines the topological properties of the manifold.In this paper,we study two problems on the characteristic function.Firstly,we give a necessary and sufficient condition,which depends only on the characteristic function,for a toric manifold to admit a spin structure.The main results are as follows:Let the characteristic function of torus manifolds M2n be ?(Fj)=(llj,…,lnj),where lij?Z,j=1,2,…,m.toric manifold M2n has a spin-structure if and only if(?)lij?1(mod2),we can also get that the four-color theorem on a 3-simple polytope P3 is equivalent to the existence of any one 3-spin small cover or 6-spin toric manifold over it.Secondly,we discuss the problem of a lifting of characteristic function.A basic result is that small cover and toric manifolds over Pn are classified by their characteristic function.Especially,the characteristic function of small cover can be obtained by the characteristic function modulo 2 of torus manifolds.Conversely,can the characteristic function of torus manifolds be lifted by the characteristic function of small cover?When the dimension is less than or equal to 3,the Z2n-characteristic function of small cover can be naturally lifted Zn-characteristic function of torus manifolds.When the dimension is greater than 3,using dimensions less than or equal to 3,we through the inductive hypothesis,the inherited Z2n-1-characteristic of function facets(codimension-one faces)can be lifted.Namely lift each Z2n-1-vector on the facets,we can get a Zn-1-vector of dimension is n-1.Then,we can construct a Zn-vector.Finally,verify that the Zn-vector intersecting n complementary facets at a point form a set of basis,so as to show that it is a lifting from Z2n-characteristic function to Zn-characteristic function.That is,for any small cover Mn over simple convex polytope Pn there is a toric manifold M2n such that(M2n,Mn)is a conjugation pair.In other words,for any Z2n-characteristic function ?,there exists a Zn-characteristic function (?) such that ? is the mod 2 reduction of (?).