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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 [180] Chapter III §16, or Bousfield and Kan [50] Chapter VIII §4. This theorem says that we can do homotopy theory in the category of simplicial sets, without reference to topological spaces.

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