Primary decomposition of the \(J\)-groups of complex projective and lens spaces (Q2502971)

For a finite-dimensional CW-complex \(X\), let \(J(X)\) denote the \(J\)-group of \(X\) and \(J_p(X)\) the \(p\)-summand of \(J(X)\) for a prime \(p\). In a previous paper [K-Theory 29, 27-74 (2003; Zbl 1037.55007)] the author determined \(J_p(P_n(\mathbb C))\) and \(J(L^n(p^k))\) by means of a set of generators and a complete set of relations, where \(P_n(\mathbb C)\) and \(L^n(p^k)\) denote the complex projective space of dimension \(n\) and the associated lens space, respectively. The purpose of the present paper is to determine the decomposition of these two groups as a direct sum of cyclic groups. Let \(r_n\) be the greatest integer such that \(p^{r_n} \leq n/(p-1)\). Then for \(0 \leq s \leq r_n\) and \(0 \leq j \leq r_n-s\), let \(t^s_j=[n-p^s(p^j-1)/p^{s+j}(p-1)]\) and set \(t_k=t_k^0\). Let \(\omega\) denote the realification of the reduction of the Hopf bundle over \(P_n(\mathbb C)\). Here one decides to consider only \(J_p(P_n(\mathbb C))\) for brevity. Then the main theorem of the paper cited above states that \(J_p(P_n(\mathbb C))\) is generated by \(\omega, \psi^p(\omega), \cdots, \psi^{p^{r_n}}(\omega)\) with \((r_n+1)\)-relations of the form: \(\Sigma^{r_n-s}_{j=0} \alpha^s_jp^{t^s_j} \psi^{p^{s+j}}(\omega)=0\) \((0 \leq s \leq r_n)\) (where the definition of \(\alpha^s_j \in \mathbb Z\) is omitted since it is rather difficult to be described here). In addition it is noted that the primary decomposition of \(J_p(P_n(\mathbb C))\) can be induced from this theorem for a certain type of \(n\). In the present paper the author extends this result to the general case. The result obtained is the following theorem: \(J_p(P_n({\mathbb C}))=\bigoplus^{r_n}_{k=0}{\mathbb Z}_{p^{t_k+u_k}}\) (where the definiton of \(u_k \in \mathbb Z\) is omitted for the same reason as above), and also the order of the first summand generated by \(\omega\) coincides with the \(p\)-component of the Atiyah-Todd number \(M_{n+1}\). The proof is carried out within the framework of the above-cited paper.