By Armand Borel, Edward Halpern
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Extra info for Topics in the Homology Theory of Fibre Bundles
Sample text
Then the image of H~(E,K) Proof. ,x m) w i t h x i u n d e r f~ is g e n e r a t e d by p r i m i t i v e proved that H~(X,K) are dual to the x i. Since f commutes = we have v ul, a E H~(E,K). shows that the image of f ~ is stable under the v-product with any u i. The t h e o r e m will then follow if we show that any s u b a l g e b r a A c H*(X) w h i c h v-product with all u i is g e n e r a t e d by p r i m i t i v e Let P denote the subspace basis additively so that the first k basis elements guarantees A m is stable u n d e r elements.
Of x o. if we take x o = e then p~ and a ~ are the identity Theorem element product of the choice if it has a product is an element x § e 9 x are h o m o t o p i c moreover independent = x 9 y, a c o n t i n u o u s Since X is a s s u m e d to be arcwise of spaces w i t h A - e s s e n t i a l space X is an H-space explicitly, Hopf a l g e b r a over Z then odd. in X; then h is an A - e s s e n t i a l x o 9 x induce a u t o m o r p h i s m s logical generated of H-spaces. Let X be a t o p o l o g i c a l connected for a Hopf a l g e b r a H ~ over Z without | H*(X,Kp) 2).
5. The P o n t r ~ a s i n product ([2],[3],[5],[7]). Let H ~ denote a Hopf a l g e b r a (of finite type) dual space to H i and define H~ = [ H i . The duality over a field Kp. Let H i be the b e t w e e n H ~ and H ~ is e x p r e s s e d as usual by