Algebraic Topology: Proceedings, University of British by Roy Douglas (auth.), Peter Hoffman, Renzo A. Piccinini,

By Roy Douglas (auth.), Peter Hoffman, Renzo A. Piccinini, Denis Sjerve (eds.)

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Extra resources for Algebraic Topology: Proceedings, University of British Columbia, Vancouver, August 1977

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The total space of the W h i t n e y - s u m E ~ T'X We shall not c a r r y out the d e t a i l s of this e q u i v a l e n c e , not n e e d e d here. a vec- . 10 Definition. nifolds where A continuous is o - s t r u c t u r e d T'X up to ( ~ : E ~ X) T'X w tured X ignore X ~ pt) , i,e. Rn x X ~ X (corresponding zero-section where E' . If f : Y ~ X hood f(Y) in If where along f : Y ~ X w f = ker(Tf the fibre". g. = map : E - B • we of X ~ R n) d(~ this of are struc- structuring structure is a v e c t o r b u n d l e : X ~ E) ~ d(~,:E '~ X) follows and V because , ~*(T'E) = is an o p e n n e i g h b o r - : Y ~ X) = d ( f fibre bundle X by structuring canonical : E ~ X is t h e mappings vector-bundle has : Y ~ V) then "bundle .

G u is c a l l e d the c o m p o s i t e The f o r g e t - f u n c t o r maps b e t w e e n remarkable properties. = u element and ~ ~ smooth m a n i f o l d s F o r instance, where I X E ~(id X) maps ~ q(idx)) maps . 20 implies that . Its ob- are s t r u c t u r e d u E ~(f) ~ : ~ is a s s o c i a t i v e of the group of s t r u c t u r e d its m o r p h i s m s is as above continuous , ~ ou (= neutral are smooth manifolds, defined. e. composition structure. 10) cE = @ , EI = E , E2 = B . We still TY @ f*T'X .

E. -7 i J B t h e n the p r o o f b e c o m e s and y If Rn × B an H-map). E 2 = E' > easy BO (using the fact that is the inverse b u n d l e of E = EI structure then sn(1) on EI ¢ E2 = (Rn and R n × B . 22 Corollary. structure is an H - s p a c e | and we have the c a n o n i c a l h e n c e on) Z u' For every s t r u c t u r e E ~(E') on the inverse b u n d l e sn(1) E ~(Rn~B) ture. 23 Corollary. t. u' = is c a l l e d the inverse u~-~u' is a n a t u r a l functor d' d(id:B ~ B 1 is a unique .

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