Homotopy Theory by Sze-Tsen Hu

By Sze-Tsen Hu

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Since f that h,, = N f 1 A and Iz, {I(" 22 I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S Now, consider two given maps f : X + Y and g : X +Z. Since X is a subspace of the mapping cylinder M8 of g, the partial mapping cylinder Mf(M,) of f over k ? g is defined. 3. There exists a map h : 2 + Y such that hg N f if the range Y of Mf(M,) is a retract of Mf(kfg). Proof. Suficiency. Assume that Y is a retract of Mf(M,). 1) there exists a map F : k f g --f Y such that F I X N f . 1), this implies the existence of a map h : 2 + Y such that hg N f .

Any two maps of a binormal space into a solid space are homotopic. 3. A binormal solid space is contractible to a point. C. AR’s and ANR’s I n classical algebraic topology, attention is confined to very well-behaved spaees, namely polyhedra. One class of spaces whose members retain many of the desirable properties of polyhedra is the absolute neighborhood retracts. A subset A of a space X is said to be a neighborhood retract of X, if it is a retract of some open subspace U of X . A metrizable space Y is said to be an absolute retract (abbreviated AR), whenever a topological image of Y as a closed subset 2, of any metrizable space Z is necessarily a retract of Z.

If Y is an ANR, then every closed subspace A of an arbitrary metrizable space X has the H E P in X with respect to Y . [H-W; p. 861. 2. If Y is a compact ANR, then every closed subspace A of an arbitrary binormal space X has the H E P in X with respect to Y . As an application of these theorems, prove that every contractible ANR is an AR. 0. Closed A N R subspaces in an A N R Let X be an ANR, A a closed subspace of X , and T denote the subspace ( X x 0) U ( A x I ) of the product space X x I . Prove that the following statements are equivalent, [Hu 31 : (1) A has the AHEP in X .

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