Two apparently contradictory statements in Strom's "Modern Classical Homotopy Theory"

Asked Apr 27 '17 at 15:31

Active Apr 27 '17 at 15:31

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I am struggling to see where this apparent contradiction is not a contradiction: these examples and exercise all come from Strom's book "Modern classical algebraic topology":

of course the map $[S^1, (S^1\vee S^1)\amalg (S^1\vee S^1)] \to \langle S^1, (S^1\vee S^1)\amalg (S^1\vee S^1)\rangle$ is not surjective, since $\pi_0(Y)$ here is not a point. But since $S^1$ is a CW complex, it is well-pointed, and hence should induce an injection according to 5.144.

asked Apr 27 '17 at 15:31

fosco

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What is the definition of an exact sequence $* \to A \to B$ of pointed sets? Does that imply that $A\to B$ is injective? – Justin Young Apr 27 '17 at 15:51
Ah! I learned something I never thought about. – fosco Apr 27 '17 at 16:01

Two apparently contradictory statements in Strom's "Modern Classical Homotopy Theory"

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