新同伦不变数量 Ⅰ.一个短正合序列

<正> 自从Poincaré引入同调概念以来,人们一直想用依附于空间的各种代数结构来区分空间本身.这种代数结构,经过许多年和许多人的工作,越来越精细,越来越富有成果.如从同调论引伸出来的广义同调论,特别是其中的K理论,显示了很大的威力.但由于新代数结构的深入细致,不免也愈益复杂.

ON NEW HOMOTOPY INVARIANTS Ⅰ.A SHORT EXACT SEQUENCES

Let K be an N-dimensional CW-complex, n = N- 1.We consider the groups H~n(K, Z) and Coker Sq~2, where Sq~2:H~(n-1)(K, Z) → H~(n+1)(K, Z_2) is the Steenrod square.Denote the p-primary component of G by G_((p)) and m_G = {g|mg = 0}. Assume Then On 2~lkH~n(k,Z),we have cohomology operations Each operation T~((1))(k)has the property: Definition.Define and call them the T~((1))(k) torsions of K.Obviously, the T~((1)) torsions are all homotopy numerical invariants.Using these invariants, we can list the generators and their order of cohomotopy groupπ~n(K)_((2)) as in theorem 4.The results can be generalized to obtain the generators and their order of cohomotopy group π~m(K)_((p)), for odd p and 2p- 3≤N-m<4p- 5, cf. theorem 8.