Hatcher 2.1.11

Hatcher, Algebraic Topology, Chapter 2, Section 1

11. Show that if A is a retract of X then the map H_n(A)\to H_n(X) induced by the inclusion A\subset X is injective.

Proof. Let A\subset X, let \iota:A\to X denote inclusion, and suppose that there exists a map r:X\to A which is a retraction. In particular, then, the composition r\iota is homotopic to the identity on A, i.e. r\iota\simeq 1_A. In terms of the induced maps, then, Theorem 2.10 ensures that (r\iota)_*=(1_A)_*:H_n(A)\to H_n(A). Utilizing the two facts on page 111 (directly beneath Proposition 2.9), it follows that (r\iota)_*=r_*\iota_* and that (1_A)_*=\mathbf{1} where \mathbf{1} denotes the identity on the level of homology. Thus, \mathbf{1} injective implies that r_*\iota_* is injective, which in turn implies that \iota_*:H_n(A)\to H_n(X) is injective. \square


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