Suppose \((X,\alpha)\) and \((X',\alpha')\) both represent the same functor on based CW complexes. Let \(f\colon X\to X'\) and \(g\colon X'\to X\) classify \(\alpha\) and \(\alpha'\text{,}\) respectively. Show directly, using naturality and uniqueness of classifying maps, that \(gf\simeq\mathrm{id}_X\) and \(fg\simeq\mathrm{id}_{X'}\text{.}\)
Section 15.5 Exercises
Exercise 15.18. Uniqueness of representing spaces.
Exercise 15.19. Homotopy groups of the representing spaces.
Let \(X_n\) represent \(E^n\text{.}\) Show that, for \(i\geq1\text{,}\)
\begin{equation*}
\pi_i(X_n)\cong E^n(S^i)\cong E^{n-i}(S^0).
\end{equation*}
Deduce that the representing spaces for reduced ordinary cohomology with coefficients in \(G\) have the homotopy groups of \(K(G,n)\) for \(n\geq1\text{.}\)
