Apply the Leray–Hirsch theorem to the bundle \(S^1 \to S^\infty/\mathbb{Z}_p \to \mathbb{CP}^\infty\) to compute \(H^*(K(\mathbb{Z}_p, 1); \mathbb{Z}_p)\) from \(H^*(\mathbb{CP}^\infty; \mathbb{Z}_p)\).
Solution
Consider \[ S^1\longrightarrow S^\infty/\mathbb Z_p\xrightarrow{q}\mathbb{CP}^\infty. \] The inclusion of a fiber induces on fundamental groups the reduction map \[ \mathbb Z=\pi_1(S^1)\longrightarrow\pi_1(S^\infty/\mathbb Z_p)=\mathbb Z_p. \] Therefore the generator \[ a\in H^1(S^\infty/\mathbb Z_p;\mathbb Z_p) \] restricts to the generator of \(H^1(S^1;\mathbb Z_p)\). Leray–Hirsch gives \[ H^*(K(\mathbb Z_p,1);\mathbb Z_p) \cong H^*(\mathbb{CP}^\infty;\mathbb Z_p)\{1,a\} \] as a module. If \(b=q^*c\), where \(c\) generates \(H^2(\mathbb{CP}^\infty;\mathbb Z_p)\), then \[ H^*(K(\mathbb Z_p,1);\mathbb Z_p) \cong \mathbb Z_p[b]\{1,a\}, \qquad |a|=1,\ |b|=2. \] For odd \(p\), graded commutativity gives \(a^2=-a^2\), hence \(a^2=0\), so \[ \boxed{H^*(K(\mathbb Z_p,1);\mathbb Z_p) \cong \Lambda(a)\otimes\mathbb Z_p[b]\qquad(p\text{ odd}).} \] The Bockstein associated to \(0\to\mathbb Z_p\to\mathbb Z_{p^2}\to\mathbb Z_p\to0\) satisfies \(\beta(a)=b\). For \(p=2\), this Bockstein is \(Sq^1\) on degree-one classes, so \(b=a^2\). Hence \[ \boxed{H^*(K(\mathbb Z_2,1);\mathbb Z_2)\cong\mathbb Z_2[a],\quad |a|=1.} \]