Compute the Pontryagin product structure in \(H_*(L; \mathbb{Z}_p)\) where \(L\) is an infinite-dimensional lens space \(S^\infty / \mathbb{Z}_p\), for \(p\) an odd prime, using the coproduct in \(H^*(L; \mathbb{Z}_p)\).
Solution
For the infinite lens space \[ L=S^\infty/\mathbb Z_p=K(\mathbb Z_p,1), \qquad p\text{ odd}, \] one has \[ H^*(L;\mathbb Z_p)\cong \Lambda(\alpha)\otimes\mathbb Z_p[\beta], \qquad |\alpha|=1,\ |\beta|=2, \] with \(\beta\) the Bockstein of \(\alpha\).
The H-space structure comes from the abelian group structure on \(K(\mathbb Z_p,1)\). The degree-one class \(\alpha\) is primitive: \[ \Delta(\alpha)=\alpha\otimes1+1\otimes\alpha. \] Naturality of the Bockstein gives \[ \Delta(\beta)=\beta\otimes1+1\otimes\beta, \] so both algebra generators are primitive. Therefore \[ \Delta(\beta^n)=\sum_{i=0}^n\binom ni\beta^i\otimes\beta^{n-i}, \] and \[ \Delta(\alpha\beta^n) =\sum_{i=0}^n\binom ni \bigl(\alpha\beta^i\otimes\beta^{n-i} +\beta^i\otimes\alpha\beta^{n-i}\bigr). \]
Let \(x\in H_1(L;\mathbb Z_p)\) be dual to \(\alpha\), and let \(y_n\in H_{2n}(L;\mathbb Z_p)\) be dual to \(\beta^n\), with \(y_0=1\). Let \(xy_n\in H_{2n+1}\) denote the class dual to \(\alpha\beta^n\). Dualizing the coproduct gives \[ x^2=0, \qquad y_i y_j=\binom{i+j}{i}y_{i+j}, \qquad x y_i=y_i x. \] Thus the Pontryagin algebra is \[ \boxed{H_*(L;\mathbb Z_p) \cong\Lambda_{\mathbb Z_p}(x)\otimes\Gamma_{\mathbb Z_p}(y),} \] with \(|x|=1\), \(|y_n|=2n\), and the divided-power multiplication displayed above.