Smith运算与Steenrod运算的关系

<正> 设K为一复合形,I_p为法p整数域,p为质数.所谓 Steenrod 冪■系 Steenrod 从 K 的 p 次乘冪 K~p=■考虑巡迴变换 t(x_1,…,x_p)==(x_p,x_1,…,x_(p-1)),x∶∈|K|下的作用而导道得.另一面,从K~p在t下的作用,根据Smith 的理论([2],[3]),可以自然地引进一组准同构.

ON THE RELATIONS BETWEEN SMITH OPERATIONS AND STEENROD POWERS

The Steenrod powers(p a prime and I_p the field of mod p integers)St_(p)~k=St~k:H~r(K,I_p)→H~(r+k)(K,I_p)in a complex K were discovered by Steenrod~[1]from the consideration of theproduct complex(?)under the cyclic transformation t(x_1,…,x_p)=(x_p,x_1,…,X_(p-l)),X_i ∈|K|.On the other hand,by Smith's theory,[2,3]withrespect to K~p under t,we may introduce in a natural manner,a system ofhomomorphisms.(?)The question of the relations between Smith operetions Sm_k and Steenrodpowers St~k arises naturally.The author discovered~[4]that these two systemsof oPerations are actually equivalent,the one being determined by the other.This furnishes a more natural and simple definition of Steenrod powers andmakes it directly connected with the theory of Smith.The previous proofof the author depends on the intrinsic axiomatic theory of Steenrod powersof Thom.The present paper aims at givfng a direct proof independent ofthe work of Thorn.