By J.L. Dupont

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Trivial. -I x a neighbourhood is therefore of Proposition vectors, p 6 M ~x : V ~ U point G-bundle foliation. let p a principal induced diffeomorphically through -I For 0. space F is ~ = of (see the is by hence leaf every obvious I following maps leaf is tangent in x G. subbundle a foliation is EIU defines distribution e a trivialization horizontal clearly tiable U ~ around to Suppose subspace iff e in connection and of connection A 2 EIU by is defined by 6 G. 22. following ge6 are I) E 2) There a s e t of agree.

Dt. Adt. Adt. Adt. ^dt. k$r i s 3k l 0 is ip 30 ]k ]q + [ (-1)r+it. Adtjq, I%S ll 3r 10 i p n Using dt I = 0 we get ~=0 WiA~j=[(-1)s+rt t. + [ (-1)s+k+r+kt. t + [ (-1)r+l+l+st. t ] is 3r k%r is 3k l%s ii 3r • dt. Adt. Adt. ^dt. Adt. A . . A d t 10 is ip 30 3r 3q = (-1)s+rt, dt. Adt. Adt. Adt. Adt. Adt. is l 0 is Ip 30 3r 3q It follows that (£(a) AF(b)) =p'q' Iil=p IJi=q aui(a)buj(o) "(-1)r+st. ^dt. ^.. ^dt. Adt. ^ . . ,n) ^ integers . are common. times the s i g n of the into . ^ (i0 ..... is .....

There exists (see e . g . A . 32. in the to s h o w that (p+q)-cochain I (E(a) ^ E(b)) of a H(C~(S)). 36). Put = p~ [ I = ip ~ n, has one suppose let #~ and J have exactlv b £ cq(s) ® C~(S) ~ C~(S) b C cq(s). 36) : An , b iJl= q ~j(~)~I' and J = (J0 ..... Jq) Jq ~ n. in common. then ~ A~(S) . the c u p - p r o d u c t : C~(S) ~ £ S n. 0 ~ J0 <'''< in c o m m o n simplicial E • C ~(s) a 6 cP(s), E(b)~ ( i 0 , . . , i p) o 6 Sp+q. ,p+q) By T h e o r e m to a form enough b = n" the c o c h a i n the than AW ~ 6 S is e x p l i c i t l y and (0 .....

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