More Exercises for Algebraic Topology
نویسنده
چکیده
1. Given a map f :X→Y , show that there exists a map g :Y→X with gf ≃ 1 iff X is a retract of the mapping cylinder Mf . 2. (a) Suppose a CW complex X is the union of a finite number of subcomplexes Xi and that a subcomplex A of X is the union of subcomplexes Ai ⊂ Xi . Show that if each Xi deformation retracts onto Ai and each intersection of a subcollection of the Xi ’s deformation retracts onto the corresponding intersection of Ai ’s, then X deformation retracts onto A . [By an induction argument the problem reduces to the case of two Xi ’s and Ai ’s. In this special case show that the inclusions A֓A∪ (X1 ∩ X2)֓ X are homotopy equivalences.]
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