球上同伦群的不变量

<正> §1.设 S~(q+1)为 q+1维球.讨论同伦群 П_r(S~(q+1))时 H.Hopf,G.W.WhiteheadP.J.Hilton 等发展了广义 Hopf 不变量,H:П_r(S~(q+1))→П_r(S~(2q+1)). (1)在同伦群 П_r(S~(q+1))中,差数 r—(q+1)比 П_r(S~(2q+1))中的差数 r—(2q+1)大.在同伦群的计算中差数小的应该先计算,所以通过 Hopf不变量利用差数较小的同伦群表达差数较

ON INVARIANTS ASSOCIATED WITH HOMOTOPY GROUPS OF SPHERES

By the reduced product,S_∞~n,of a sphere,S~n,we mean the CW-complex(?)where e~(rn)is attached to the(r-1)n-skeleton,(S_∞~n)~((r-1)n),of S_∞~n by the secondary productaccording to[1].In[1]and[7]it has been independently proved that(?)here we consider the complex(?)being attached to(S_∞~n)~((r-1)n)bythe same map as e~(rn).Then(?)Let g:S~P→(S_∞~n)~(rn)represent an element{g}of(?)we mean an injection and(?)(?)we mean a projection.If(?),there is a homotopy F:S~p×I→(S_∞~n)~(rn)∪e′~(rn)suchthat(?)and(?).This homotopy supplies anelement of(?).With this view point the anthordefines bomomorphisms(?) (1)such that Н_(ρ-1)is defined upon(?)uponК_(j+1)~(-1)(0),j+1,…,p-2.Then he proves the followingTheorem.To each element α of(?)either there is an element β of(?)such that α=Еβ,Е being the suspension homomorphism,or there is one of the2(p-1)homomorphisms in(1)denoted by G so that G is defined on α and G(α)≠0.If p=2 the anthor shows that К_1=0,meanwhile Н_1 is actually the Hopf invariantdefined by G.W.Whitehead.If p=3 the anthor shows that К_2=К_1=0 and Н_2,Н_i areexplicitly expressed as follows:(?)