重乘积的计算

<正> §1.对弧连通的拓扑空间 X J.H.C.Whitehead 介绍了一种乘积,它使α∈πm(X),β∈πn(X)对应(?),这种乘积使我们有可能去了解低维同伦群的元素对高维同伦群的影响.

COMPUTATIONS OF SECONDARY PRODUCTS

According to1,2]the definition of secondary product is as follows:Let E~N=E~(r_1)×…×E~(r_n)(N=∑r_i).Let ψ_i:E~(r_i),(?)→S~(r_i),pi be a map ofdegree 1.Define a mapf:E~N→S~(r_1)×…×S~(r_n),such thatf(x_1×…×x_n)=ψ_1(x_1)×…×ψ_n(x_n),then(?)where(?)is the space obtained from S~(r_1)×…×S~(r_n)by removing its top dim-entional cell.If there is a map~(1))φ:(?),q_0→X,x_0,then φg:(?)~N,p_0→X,x_0 determines anelement ∑ of(?).Identify(?)with E~(r_1)×…×E~(r_(i-1))××yi×E~(r_(i+1)×E~(r_n), yi being a point of(?).Then φg induces a map,φi,of(?)into X.On the other hand(?).determines an elements α_i of π_(r_i)(X)(i=1,…,n).The element ∑ is denotedbyφ_1,…,φ_n]and called a secondary product~(2))of α_1,…,α_n.As an example of the existence of non-trivial secondary product,S.C.Chang proves that the secondary productφ_1,…,φ_n]is a free generator ofπ_(N-1)((?)).The purpose of this note is to determine the algebraic structureand geometrical representations of the group π_r((?)),if r<2r_1+2r_2+r_3++…+r_n-3. In the space S~lU S~mU S~n,we have the Whitehead products(6])S~l,S~m],S~m S~n],S~n S~l]and repeated Whitehead productsS~l,S~m S~n]],S~mS~n S~l]],S~n,S~l,S~m]].Between these repeated products,the Jacobi identity(-1)~(l_(n+1))S~l,S~m,S~n]]+(-1)~(m(l+1))S~m,S~n,S~l]]++(-1)~(n(m+1))S~n,S~l,S~m]]=0holds 3,5].Let(?),where(?)is attachedto(?)by Whitehead product(?).Then it is natural to askwhether there exists any relation between the secondary products(?)(?)in Y?The answer is negative.All the results are stated as follows:Theorem 1.(?),then(?)Denote the Whitehead product of S~r andS~r,φ_1,…,φ_n]]by(S~r)~2,φ_1,…,φ_n,]]and then define(?)Theorem 2.The elements of the group π_r,((?))mentioned in theorem 1 aregeometrically realized by the following mappings:(i)i_kπ_r(S~(r_k),(ii)φ_1,…,φ_n,](?)π_r(S~(Σr_i-1)),(iii)Sr_k,φ_1,…,φ_n]](?)π_r(S~(r_k+Σr_i-2),and(iv)(S~(r_1)~((p-1)),φ_1,…φ,φ_n]](?)π_r(?),k=1,…,n,p=3,4,….Theorem 3.The relation(?)holds in Y if,and only if,p_1=p_2=P_3=P_4=0.