7r-+1 with F free and R a normal subgroup of F.
GLn(IR)-bundle) . Thus there is a one-one correspondence between rank n complex (resp. real) vector bundles over B and principal GLn(C)-bundles (resp. GLn(R)-bundles) over B. 4. 4. CLASSIFYING SPACES 37 over B' with total space E' = J (x, y) E B' x E I f (x) = p(x)} and with p' : E' -+ B' given by p'(x, y) = X. It is easy to see that is a fibre bundle with the same fibre as . If is a principal G-bundle then so is EXERCISES. 1. If E - B and E' -> B are n and m dimensional vector bundles, show that the pullback Ell E' I I E -B constructs an (n + m)-dimensional vector bundle E" -* B, called the Whitney sum of the two bundles.
Thus for a topological group Milnor's EG may be thought of as a sort of continuous bar resolution. 12. (i) If N is a closed normal subgroup of G, then there is a fibration BG - B(GIN) with fibre BN. (ii) If H is a closed subgroup of G there is a fibration BH -f BG with fibre the coset space G/H. PROOF. (i) Let E(GIN) and EG be contractible spaces on which GIN and G act freely. Then N also acts freely on EG, and we may take for BN the quotient space (EG)/N. Thus GIN acts on BN and we may take for BG the space (E(G/N) x EG)/G = (E(G/N) x BN)/(G/N).