Determination of the \(J\)-groups of complex projective and lens spaces (Q1417757)

Let \(J(X)\) denote the \(J\)-group of \(X\) and for any prime \(p\) let \(J_p(X)\) denote the \(p\)-summand of \(J(X)\). Let \(P_n(\mathbb{C})\) and \(L^n(p^k)\) denote the complex projective space of dimension \(n\) and the lens space of dimension \(2n+1\). The purpose of this paper is to determine \(J_p(P_n(\mathbb{C}))\) and \(J(L^n(p^k))\) for a prime \(p\). Let \(\omega \in \widetilde{KO}(P_n(\mathbb{C}))\) denote the reduced bundle of the realification of the complex Hopf bundle over \(P_n(\mathbb{C})\) and also denote by \(\omega\) the pull-back of \(\omega\) to \(\widetilde{KO}(L^n(p^k))\). Here \(L^n(p^k)\) is considered to be a subcomplex of \(P_n(\mathbb{C})\) in a natural way. The author begins with showing that \(J_p(P_n(\mathbb{C}))\) is generated by the images of the Adams operations \(\psi^{p^i}\) acting on \(\omega\) for \(0 \leq i \leq r_n\), where \(r_n\) denotes the greatest integer such that \(p^{r_n} \leq n/(p-1)\). The major part of this paper is devoted to the determination of a complete set of relations between these generators. To seek the relations the author introduces a new sequence of integers, called \(s\)-admissible \(\alpha\)- and \(\beta\)-sequences for \(0 \leq s \leq r_n\). In Section 6 it is shown that each \(s\)-admissible sequence gives rise to a relation in \(J_p(P_n(\mathbb{C}))\) between \(\psi^{p^s}(\omega), \psi^{p^{s+1}}(\omega), \cdots \) and so one can obtain relations in \(J_p(P_n(\mathbb{C}))\) corresponding to the \(\alpha\)- and \(\beta\)-sequences. One also has an example of computation in which the author writes down the \(\alpha\)- and \(\beta\)-relations for \(p=2\) and \(n=164\). Let \(G(p, n, k)\) denote the subgroup of \(J(L^n(p^k))\) generated by the images of the Adams operations \(\psi^{p^i}\) acting on \(\omega\). Then one finds that this is a quotient of \(J_p(P_n(\mathbb{C}))\) with the relation \(\psi^{p^k}(\omega)=0\), so that the complete set of relations on \(G(p, n, k)\) is given by either the \(\alpha\)- or \(\beta\)-relations with \(\psi^{p^k}(\omega), \psi^{p^{k+1}}, \cdots \). From the author's prior results [Pac. J. Math. 223--241(1999; Zbl 1013.55004)] it follows that \(G(p, n, k)\) is isomorphic to \(J_p(P_n(\mathbb{C}))\) for \(k \geq r_n+1\). Hence what is discussed here is the case \(1 \leq k \leq r_n\) which is dealt with in Section 7. One of the main tools for the proof is the \(K\)-theory characteristic class \(\rho^k\) associated to the Adams operation \(\psi^k\). This plays an important role in investigating whether or not \(J_p(x)\) is zero for \(x \in \widetilde{KO}(X)\). The proof consists of numerous calculations, but using the new ideas of singular exponents and indices the author seems to succeed in obtaining the relations in a most simple and computable form. Finally in Section 8 the author studies the group \(J(P_n(\mathbb{C}))\) from a purely algebraic viewpoint by means of truncated weak \(D\)-series introduced by the author in [J. Algebra 164, 468--480 (1994; Zbl 0843.13006)].