論雅各必恒等式

<正> §1.設X是一個拓撲空間,其中任何二點可以用弧聯結。以x_o∈X為參考點,那麼可以定義π_r(X,x_o)(r=1,2,…)。這種羣的元素全體,成為一個集E。在E中有魏德海乘積2],即α,β為E中一點,那麼α,β]也是E中一點。關於魏德海乘積的重覆使用問題,就文獻而諭,首先在W.S.Massey6]

ON JACOBI IDENTITY

Let X be an arcwise connected topological space and α, β, γ be elements of the homotopy groups π_ρ(X, x_o), π_q(X, x_o), π_r(X,x_o) respectively, where r ≥q≥p≥2. By β, γ] we mean the Whitehead product and α,β, γ]] the repeated Whitehead product. The use of the method which the author has developed in 1] leads to the following consequence: (-1)~(r(p+1)) α, β, γ] ] + (-1)~(p(q+1)) β, γ,α]]+(-1)~(q(r+1)) γ, α,β] ] = 0,namely, the Jacobi identity.