Hatcher-Algebraic-Topology · Problem 4.1.9

Homotopy Theory

Mathematics — Graph Theory · long-exact-sequence-pi0

Problem

Suppose we define π0(X,A,x0)\pi_0(X, A, x_0) to be the quotient set π0(X,x0)/i∗(π0(A,x0))\pi_0(X, x_0)/i_*\bigl(\pi_0(A, x_0)\bigr), so that the long exact sequence of homotopy groups for the pair (X,A)(X, A) extends to ⋯→π0(A,x0)→i∗π0(X,x0)→π0(X,A,x0)→0\cdots \to \pi_0(A, x_0) \xrightarrow{i_*} \pi_0(X, x_0) \to \pi_0(X, A, x_0) \to 0. (a) Show that with this extension, the five-lemma holds for the map of long exact sequences induced by a map (X,A,x0)→(Y,B,y0)(X, A, x_0) \to (Y, B, y_0), in the following form: One of the maps between the two sequences is a bijection if the four surrounding maps are bijections for all choices of x0x_0. (b) Show that the long exact sequence of a triple (X,A,B,x0)(X, A, B, x_0) can be extended only to the term π0(A,B,x0)\pi_0(A, B, x_0) in general, and that the five-lemma holds for this extension.

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