Some Properties Of K-theory In Some Spaces And The Computation Of Equivariant K-theory

In this paper,we focus on two problems:(1)The computation of topological K-theory for common spaces and properties of product of the torus with some spaces;(2)The representation of equivariant K-theory of the torus by the algebra of Grothendieck differentials.Firstly,we describe the definition and properties of the topological K-theory in the category of vector bundles,and give the relationship between the reduced K-theory and the set of homotopic maps from the topological space X to the Classification Space BU,i.e.K(X)=[X,BU].Secondly,we summarize the computation of K*for common spaces by the homotopic lemma,the Bott periodicity theorem K0(X)≌K-2(X),and the exact sequence.That is K*(Sn)≌Z⊕Z,K*(Tn)≌Z,K*(Rn)≌Z,K*(Tn)≌Z⊕2n,and K*(CPn)≌Z⊕n+1.Then we prove that the formula of the torus in topological K-theory by the properties of the product of the n-sphere Sn and a compact space.Based on the isomorphism Kp(X)≌Kp(X\A)⊕Kp(A),the Bott periodicity theorem K0(X)≌K-2(X)and the Künneth theorem in topological K-theory,we obtain an isomorphism between the product of the torus and some spaces.If X be a compact Hausdorff space with p=0,-1,Kp(Tn)≌Kp-1(Tn-1)⊕Kp(Tn-1)and Kp(X×Tn)≌[K0(X)⊕K-1(X)]⊕…⊕[K0(X)⊕K-1(X)].And let X be a locally compact Hausdorff space of finite type with p= 0,-1,then Kp(X×Tn)≌[⊕i=1 2n-1 K0(X)]⊕[⊕i=1 2n-1 K-1(X)].Finally,we inscribe the representation theory of compact Lie groups,the algebra of Grothendieck differentials,equivariant K-theory of compact G-space,and spectral sequence.By the algebraic isomorphism φ:ΩR/Z*→KG*(G)and the algebra of Grothendieck differentials,thus we obtain the representation of equivariant K-theory of the torus.