Note that I changed the theme again.

I’ve been spending a lot of time on the basic definition of the homotopy groups. Basically, in summary, we have defined for a pointed space ${(X, x_0)}$, the group

$\displaystyle \pi_n(X, x_0)$

of pointed homotopy classes ${(S^n, s_0) \rightarrow (X, x_0)}$. Alternatively, this is the group of pointed homotopy classes of maps ${f: (I^n, \partial I^n) \rightarrow (X, x_0)}$; this is because ${I^n/\partial I^n}$ is homeomorphic to the sphere. This is a functor on the category of pointed topological spaces. If ${f: (X, x_0) \rightarrow (Y, y_0)}$, then there is an induced homomorphism ${f_*: \pi_n(X, x_0) \rightarrow \pi_n(Y, y_0)}$.

That’s a short summary of what I’ve discussed in these past few weeks; I won’t keep repeating it.

But now we want to fit these groups into exact sequences. This can be done directly in an ad hoc manner. Another way is to use the Puppe sequence.

1. General nonsense

Recall that a sequence of pointed sets

$\displaystyle (A, a_0) \stackrel{f}{\rightarrow} (B, b_0) \stackrel{g}{\rightarrow} (C, c_0)$

(with maps preserving the basepoints) is called exact if

$\displaystyle f(A) = g^{-1}(c_0).$

In other words, the “kernel” at one end (i.e. the set mapping into the basepoint of ${C}$) is the image of ${A}$. (more…)