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Download Curvature and Characteristic Classes by J.L. Dupont PDF

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By J.L. Dupont

Publication by way of Dupont, J.L.

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31) IIl~d~ I ^ hi(~o)+(-1)IIi II1~I^dhi(~c). 22) (-1)kh (i 0 ..... 32) Sk+1(d~) ~ + d(Sk~) a = 111 [ IIl[~i^( [ (-1)Jh ~j = E(I(~))o-~o-0

88 has G ~ pt. By therefore = z~6 0 agree connection 8 in on 8 E I U s N U 6. 21 from the we ~ can cover : U ~ U flat connection in M E by with open sets × G such that U × G. , particular ~(U and it f o l l o w s the let Show is t h e (x,g0g) induced the f r a m e projection. be e (U ~ go N U B) of N UB) in such m a p of L i e bundle The geB for Then the h o r i z o n t a l x g, g 6 G. some g o 6 G, the : ~n H-bundle that with ~ T equal to to G go" homomorphism connection ~'8 E = ~, tangent group Since o G.

34 ~(6(a®b)) = #~(da®b + (-I) p a ® 6b) = l(E(6a) hE(b)) + (-I)PI(E(a) h E ( ~ b ) ) = l(dE(a) h E(b) + (-I)PE(a) h d E ( d b ) ) = l(d(E(a) hE(b)) = BI(E(a) hE(b)) = ~(a®b). This ends the proof. Remark. 35). 37) is an average of the A l e x a n d e r - W h i t n e y c u p - p r o d u c t over the p e r m u t a t i o n s given by (I,J) in order to m a k ~ the p r o d u c t ~raded c o m m u t a t i v e on the c o c h a i n level. In fact the A - W - p r o d u c t is not graded c o m m u t a t i v e on the cochain level as since h is graded commutative.

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