Homotopy groups of a map?

by user101010   Last Updated August 01, 2020 11:20 AM - source

I am reading the chapter about surgery theory in Wall's "Differential Topology" and he starts talking about homotopy groups of a map $f : M \to X$, denoted $\pi_r(f)$. What is the definition of $\pi_r(f)$? It looks like it should vanish if (and only if?) $f$ is a homotopy equivalence. Further, it looks to me like there is suppose to be a map $\partial : \pi_r(f) \to \pi_{r-1}(M)$ and I would like to know how that goes. A reference would also be great.

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