# Curvature and Characteristic Classes by J.L. Dupont By J.L. Dupont

Ebook by way of Dupont, J.L.

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Extra info for Curvature and Characteristic Classes

Sample text

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V n} : F(V) v 6 V vector weaker: from over a vector to be a given to t h e s e V : Vp ~ V q . T something of for bundle "parallel" seems a curve concept So and is a d i f f e r e n t i a b l e v 6 V Therefore p,q this the a differentiable vector isomorphism bundle associate translate T M of corresponding along let will v t 6 Vy(t), V the ~ M bundle points from manifold. a real is p o s s i b l e : [a,b] "connection" space of translation suppose parallel Given require What parallel y(b) we tangent a trivial requirement.

W h i t n e y ) . an isomorphism I : A*(S) ~ C*(S) in h o m o l o g y . 18) I o E = id, E : C*(S) ~ A~(S) ~ Ak-1(S), ~ o I, and n a t u r a l k = 1,2 .... , such chain that E 0 ~ = d o E E o I - id = Sk+ I o d + d o s k, k =0,1,... For the p r o o f we f i r s t usual Ap c ~p+1 is the s t a n d a r d = canonical basis coordinates respect {e0, .... ,tp). 2. 21) and t h e r e f o r e satisfy i,j by the is star s h a p e d w i t h (of. E x e r c i s e (j)d~ + d h ( j ) ~ For Ap j = 0 ..... p, ~ 6 A0(&P) .