Section 13.4 Exercises

Let \(n \geq 2\text{.}\) Exhibit a fibration \(F \to E \to B\) where, up to weak homotopy equivalence, \(F\) is a \(K(G, n-1)\text{,}\) \(B\) is a \(K(G, n)\) and \(E\) is contractible. Hint: Remark 10.27.

Show that if \(X\) is a \(K(G, n)\) and \(Y\) is a \(K(H, n)\text{,}\) then \(X \times Y\) is a \(K(G \times H, n)\text{.}\) Deduce that the torus \((S^1)^k\) is a \(K(\mathbb{Z}^k, 1)\text{.}\)

Let \(\mathbb{Z}/m \subseteq S^1\) act on \(S^\infty = \bigcup_k S^{2k+1} \subseteq \mathbb{C}^\infty\) by scalar multiplication. Show that the quotient \(L^\infty_m = S^\infty/(\mathbb{Z}/m)\text{,}\) the infinite lens space, is a \(K(\mathbb{Z}/m, 1)\text{.}\)

Using Proposition 13.13 and the homology of \(\mathbb{R}P^\infty\text{,}\) show that every CW complex which is a \(K(\mathbb{Z}/2, 1)\) is infinite-dimensional.

Let \(G\) be an abelian group and \(n \geq 1\text{.}\) Choose a free resolution \(0 \to F_1 \xrightarrow{d} F_0 \to G \to 0\text{,}\) with bases \(\{x_\beta\}\) of \(F_1\) and \(\{y_\alpha\}\) of \(F_0\text{.}\) (Such a resolution exists since subgroups of free abelian groups are free.) Construct a CW complex \(M = \bigl(\bigvee_\alpha S^n_\alpha\bigr) \cup \bigcup_\beta e^{n+1}_\beta\) whose cellular chain complex is \(F_1 \xrightarrow{d} F_0\) in degrees \(n+1\) and \(n\text{,}\) and deduce that \(M\) is a Moore space \(M(G, n)\text{.}\) Show that \(M(\mathbb{Z}, n)\) can be taken to be \(S^n\text{,}\) and deduce that Moore spaces are in general not Eilenberg–MacLane spaces.