By Roy Douglas (auth.), Peter Hoffman, Renzo A. Piccinini, Denis Sjerve (eds.)
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Extra info for Algebraic Topology: Proceedings, University of British Columbia, Vancouver, August 1977
1 Definition. e. bundle f : E ~ E' maps f : p-1(b) with , where a natural Homotopy > map over , for all are are on fibres, : B - B' is induced by f) . E ~Ej BI,B 2 u E ~E ~-I(Bj) Additivity , and (where R×E ~ B) RxB ~ B) properties. q : RxE ~ E induces bijections If E is a v e c t o r bundle are open subsets of B , and if uI I EI n E2 = u2 I EI 0 E2 such that uj = ulEj "restriction" ulEj over then . - Here etc. stands for . (ADD). B = % ® =I Bj Sets The p r o j e c t i o n (MV). are such that exists > E .
17') tive s t r u c t u r e a' sign seems to be (-I) pq+p) (dE 1) x (dE 2) map. equivalence for every v e c t o r - b u n d l e there. can be equipped diagonal ~) structure. then cases (examples (if (up to sign; to make the c o n s t r u c t i o n structure resp. 17) u , by ~(E) ~ ~(E N) give the n e c e s s a r y the reader - This E = TM e (T'X) M . 15 Remark. , as desired. b E 2) . on is a ~E . 20) is b i j e c t i v e , fore assume s(1). (compare that the in p a r t i c u l a r , q(E) ~ ~(R stability x E) multiplication with .
E. As before, RP,R q . = 0 • o n l y dealt w i t h closed. , and with . It follows that Induced homomorphisms der a r b i t r a r y xkl(X,A) : M - Vk bounds contains of (i) is even simpler. 15) gk ; by d i r e c t e d n e s s the zero element So far w e ' v e O(Xk,Ak) . Ak2 that ~ > k . 18) comes x k E O ( X k , A k) x k l ( X , A u) = 0 then (cf. (~ n o r m a l bundle); thus R p × (Y × M) X tubular is r e p r e s e n t e d 54 as an o p e n subset of R q x M . It is t h e r e f o r e d-structured hence Y x M is d - s t r u c t u r e d .
Algebraic Topology: Proceedings, University of British Columbia, Vancouver, August 1977 by Roy Douglas (auth.), Peter Hoffman, Renzo A. Piccinini, Denis Sjerve (eds.)