By By G. H. Hardy, P. V. Seshu Aiyan, B. M. Wilson, ed.

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**Example text**

General and cohomology X , up to h o m o t o p y , n fibration by the usual process. L2' t h e H - s p a c e ( j n ) . : Iri(X) ~- lri(X n) to a f i b r a t i o n Such Postnikov obvious possibility i > n, Ef Kahn, Stasheff]. {Xl,k 2,X 2 .... -~ X n P n X n _ l -~ . . lri(Xn) = 0 f o r ~E c a s e of a m o r e kn ¢ H n + l {Xn - l ' • wn(X))' c a l l e d k - i n v a r i a n t s , t o g e t h e r that as For of H 2 ( Z , 2 ; Z ) , t h e n H . ( S 1 × K ( Z , 2)) a b o v e is a s p e c i a l the k-invariants a sequence to h 2 -~ {b - b ° ) .

And all the of t h e s e . will now be represented branches edge is subdivided needlessly. ~kk x 1. . x a s a u n i o n of c u b e s C(T) (= i n p u t s = e d g e s w h i c h c a n b e r e m o v e d our present The cube C(T) purposes we will assume will have parameters no indexed by t h e e d g e s of t h e t r e e w h i c h a r e n o t b r a n c h e s . Definition II. 17. WA with n-branches. (n, I) is the union with identifications of C(T) over all trees T h e identifications are that a face t.

More generally, fibration the skeleta to m a k e qi+l Let of until Now form an isomorph- 42 Theorem map, 9. 5. [Zabrodsky]. then X{PI) If f : X - * X 0 is a rational equivalence and an H - admits a multiplication so that fl' f2 are H - m a p s . Proof: If f : X - ~ X' is an H - m a p with f~ : H r ( X '; Z P being an i s o m o r p h i s m for r < n, then ~ ~ K e r f',~ I Hn(X' , Z )-~ Mr(x; Z ) P ) is represented by P an H - m a p , for f'~ ~ = 0 is represented b y an H - m a p being represented by an H - m a p lie in H n ( x ' ~ and the obstructions to X' ; Z ) w h i c h is m a p p e d iso- P morphically resented trivial to HncK ~ X; Zp) by an H-map one) showing will admit Corollary H-maps, The homotopy showing so as to map into a homotopy f$¢~ i s a n H - m a p .