.

6(z) a Relative to this global section of corresponding connection/form ~ Q any connection ~ on Q has a such that Vx(Z) = ~ ( X ) Z , V X E X . 4) Lemma. ~ £ AI(M) satisfies connection form relative to proof. Let connection. Let X E r(E). ~ Z d8 = 8 - ~ Iff of a basic connection on be the connection form relative t o Then ~' E AI(M) Then, if ~ is the Q. Z of a basic is another such iff ~lr(z) -~'ir(z). d8 m e -~, 1 - ~ ~ ( X ) = (e • ~)(X,Z) = d~(X,Z) = - ~I e([x,z]) + ~I x(e(z)) = - ~ I e ( I x , z]) = - ~ e1( ~ ( x ) z ) - ~1 z(e(x)) =- ½ ~(x), 61 w h e r e we h a v e u s e d t h e s t a n d a r d derivative relatlve of a l-form.

Proof. We use singular cohomology. class a E Hk(BGLq;R) such t h a t slngular homology, there is a finite ~q-Structures. (*). I) The "exotic" which by now are standard in the theory of is zero if k > 2q. If the assertion is false, choose \$ q Hk(BGLq;R), k > 2q, such that ~ = Bv*(\$) # O. ~(O) @ 0 . polyhedron Then there is a homology By t h e d e f i n i t i o n P~ of a homology c l a s s E Hk(P;~) P and a c o n t i n u o u s map s : P--~ BU q (the "geometric realization" of o ) s d e t e r m i n e s a homotopy c l a s s of T -structures q we can thicken as BFq.