We define a collapsing or trivialization procedure for bundles over X which yields a bundle over X / A for a closed subset A of X. With this construction we are able to give alternative descriptions of \(K(X,A) = \tilde K(X/A).\). For a finite CW-pair (X, A) we can define an exact sequence K(A) ← K(X) ← K(X, A) ← K(S(A)) ← K(S(X)), using an appropriate “coboundary operator.” With this sequence we see that in some sense the K-cofunctor can be used to define a cohomology theory.
KeywordsExact Sequence Vector Bundle Group Morphism Cohomology Theory Natural Morphism
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