 By Brian Osserman

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COVERING SPACES. 12. If p : E -> B is a fibre bundle in which the fibres are discrete sets, then E is called a covering space of B. 7, there is an action of 7rl(B, bo) on F = p 1(bo). If E and B are path connected then the homotopy sequence (0 =)7rn(F, eo) -+ xn(E, eo) - 7rn(B, bo) -* 7rn-1(F, eo) (= 0) (n > 2) (0 =)7r1(F, eo) - 7ri(E, eo) - 7r1(B, bo) - 7ro(F, eo)(= F) - 7ro(E, eo)(= 0) shows that 7rn(E, eo) = 7rn(B, bo) for n > 2, 7r1 (B, bo) acts transitively on F, and 71 (E, eo) is the stabiliser of the point eo E F.

K-THEORY 45 other way, from K(B) to [B; BU x Z], taking a generator [E] of dimension n to the map given by B -> BU(n) ---+ BU and the constant map B - Z with image n. If f : B' - B is a map of compact spaces, the pullback of vector bundles gives a map f* : K(B) -> K(B'). If B has a basepoint x, then the maps {x} y B -* {x} gives us maps Z = K({x}) - K(B) - K({x}) = Z, whose composite is the identity. Thus if we write K(B) for the kernel of K(B) - Z then K(B) = K(B) ® Z. Note that every element of k(B) is represented by an actual vector bundle.

There are also obvious notions of homotopy classes of pointed maps [X, xo; Y, yo], where x0 and yo are zero-simplices, and of maps between pairs [X, A, x0; Y, B, yo]. 1. The adjunction between the singular simplices functor and the topological realisation functor passes down to an equivalence of categories between CW-complexes and homotopy classes of maps, and Kan complexes and homotopy classes of simplicial maps. PROOF. See May  Chapter III §16, or Bousfield and Kan  Chapter VIII §4. This theorem says that we can do homotopy theory in the category of simplicial sets, without reference to topological spaces.