On the mod p J-homology of complex projective space (Q1122828)

Let p be an odd prime and \(J_*\) be the generalized homology theory defined by the cofibration \[ J\to k_{(p)}\to^{\Theta}\Sigma^{2(p- 1)}k_{(p)} \] of spectra. Here \(k_{(p)}\) is the spectrum of p-local connected K-theory, \(\Theta\) the operation with \(u^{p-1}\Theta =\psi^{\iota -1,u}\) generates \(k_ 2(*)\), \(\iota\) generates \(({\mathbb{Z}}/p^ 2)^*\) and \(\psi^{\iota}\) is the stable Adams operation. Denote by \(P=P_{\infty}{\mathbb{C}}\) the infinite complex projective space. Since \(J_*\) is multiplicative and P an H-space, \(J_*(P)\) and \(J_*(P,{\mathbb{F}}_ p)\) are rings. For applications to \(\pi^ s_*(P)\) a good understanding of \(J_*(P)\) is desirable. By the work of \textit{K. Knapp} [Some applications of K-theory to framed bordism, Habilitationsschrift, Bonn (1979)] one knows \(J_{2n}(P)={\mathbb{Z}}_{(p)}\) and the order and the number of cyclic summands in the finite group \(J_{2n-1}(P)\). Hence \(J_*(P,{\mathbb{F}}_ p)\) is known additively. The ring structure of the even dimensional part of \(J_*(P,{\mathbb{F}}_ p)\) is known by the work of \textit{L. Schwartz} [Bull. Math. Soc. Fr. 109, 237-257 (1981; Zbl 0472.55013)]. To complete the description of \(J_*(P,{\mathbb{F}}_ p)\) as a ring, one needs generators for \(J_{2n-1}(P;{\mathbb{F}}_ p).\) The main result of this paper is a construction of an explicit \({\mathbb{F}}_ p\)-basis of the odd dimensional part of \(J_*(P,{\mathbb{F}}_ p)\). The author uses connective Morava K-theory \(k_*(1)\) to compute in \(J_*(P;{\mathbb{F}}_ p)\) and the knowledge of the map induced by multiplication with p in the H-space structure of P to obtain information on \(\Theta (\beta_ n)\) for the usual generators \(\beta_ n\in k_*(1)(P)\). Then an involved inductive method is set up to describe cok(\(\Theta)\) and to work out the numbers \(j_ i\) such that \(\{\beta_ 1,...,\beta_{s_ r}\}\) map to a basis of \(J_{2n-1}(P;{\mathbb{F}}_ p)\). An explicit basis for \(J_{2n}(P;{\mathbb{F}}_ p)\) is also given, reproving the results of Schwartz [loc. cit.].