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Extra info for Algebraic Topology: Proceedings, University of British Columbia, Vancouver, August 1977
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 . T Y = (Tf) ¢ of tangent (f*TX), vectors that = (~f) ¢ f * ( T X ¢ T'X) .
To the d i a g o n a l some p r o p e r some Xkl there exists that with V - G(W) con- and U ~ k, kl,k 2 , h e n c e -I gk = ~W represents g : M - X g , and e are transverse. > X × X The lat- is t r a n s v e r s e X × X ; in our case w h e r e 0 E RP , is smooth in X is is a r e g u l a r mapping (y,m) (0) - w h i c h = e(y) - g(m) coincides . w i t h the p u l l b a c k Y x M of Y e> X ( g M - is a smooth s u b m a n i f o l d of X Y x M ~ Rq× M w i t h t r i v i a l i z e d (by T
E. GI+(R) reductions of the s t r u c - . oE = set of h o m o t o p y c l a s s e s of s t a b l e c o m p l e x s t r u c t u r e s on . ~E = set of h o m o t o p y c l a s s e s of s t a b l e t r i v i a l i z a t i o n s (RmxE ~ RN×B) (v) is u E oE . qE = set of o r i e n t a t i o n s E (iv) example = I . ture group (iii) . This t r i v i a l shows t h a t the e m p t y b u n d l e a unique structure (ii) E If so t h e n the i n c l u s i o n of the fibre shows that o ( R n ~ pt) ~ ~ . A n d has for all .