0.5. Higher homotopy groups 9 0.4. The Vietoris-Begle Theorem Another algebraic result needed much later on is the Vietoris-Begle Map- ping Theorem. It implies that a closed, surjective mapping for which all point preimages are cohomologically acyclic induces isomorphisms of Cechˇ or Alexander cohomology. See (Spanier, 1966, p. 344) for the proof. Theorem 0.4.1. Let f : X X be a closed surjective map between para- compact Hausdorff spaces, and let G be an abelian group. Assume that there is some m 0 such that ˇ q (f −1 (x) G) = 0 for all x X and q m. Then f : ˇ q (X G) ˇ q (X G) is an isomorphism for q m and a monomorphism for q = m. 0.5. Higher homotopy groups Let X be a space with basepoint x0 X. For each integer n 0 there is a group πn(X, x0), called the nth homotopy group of X with basepoint x0. The definition of πn(X, x0) closely parallels that of the fundamental group, only with the role of the unit interval taken over by the unit n-cube. For that reason, throughout this section In will be used to denote [0, 1]n. Let F n denote the set of all maps (In,∂In) (X, x0). Define two maps in F n to be equivalent if they are homotopic relative to ∂In i.e., f, f F n are equivalent if there exists a homotopy Ψt : In X between f and f with Ψt(∂In) = {x0} for all t I. As a set, πn(X, x0) is just the set of equivalence classes. Define the group operation on F n , n 1, as follows: for f, g F n their sum f + g F n is the function determined by (f + g)(t1,t2,...,tn) = f(2t1,t2,...,tn) if 0 t1 1/2 g(2t1 1,t2,...,tn) if 1/2 t1 1. Since the homotopy class [f + g] depends only on the classes [f] and [g], addition in F n induces a well-defined addition in πn(X, x0) via [f] + [g] = [f + g]. One can easily verify that this makes πn(X, x0) a group its identity element is the class of the (only possible) constant map. The 0-dimensional cube I0 is just a point and ∂I0 = ∅. With this understanding the definition of the set π0(X, x0) makes sense (it consists of the set of path components of X), but there is no group structure defined on it. Even though π0(X, x0) does not have a group structure, we still designate the equivalence class of the map I0 x0 as the identity element. This allows us to include π0(X, x0) in an important exact sequence to be described below.
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