X). 3. o f t h e induced hcmomorphism The image ~ ~: I(G) - * consists of e l e m e n t s The elements in subgroup of "K0(X). p > 1. will be called it i s n o t t r u e is a torsion-group degree of f i n i t e o r d e r . im(~') Of course is given by e R~ shown, l y i n g in (2q-1)-skeletons (or of a s i m p l y connected X = S 2 n - 1 U ~ e 2n, n > 1, however, there that for o f ~' i s e q u a l t o t h e t o r s i o n complex with B~r 2 q - 1 7r = 7rlX f i n i t e , are obstructions H 2(q+i)+l(yr, Z ) , X such that K0(X) ~: S 2 n - 1 - - ~ S 2 n ' l of i > 0.
F 11G w i t h t h e u n i v e r s a l we denote by Then condition ~= EG • M, (iii) in 6 . 1 c a n b e r e s t a t e d the as f $ ~ ~- TX . i t s e l f is a f l a t m a n i f o l d . map X structure is a flat bundle. for the representation for the classifying commutativity X is f l a t if M. 6) BP ~ BGL(V) f ~ X If the G-structure is flat, f factorizes through then g. Since for GL+(V2n), H~(BGL(V), Q) ~ : X--~ B~', ~" = ~'iX and so does is generated by Pontrjagin classes and the Euler class w e will have to study these classes.