Connective Im(J)–theory for cyclic groups

Abstract

797Connective Im.J /-theory for cyclic groups KARLHEINZ KNAPPWe study connective Im.J /-theory for the classifying space B‫=ޚ‬p a of a finite cyclic p -group and compute the Im.J /-cohomology groups completely. We also compute the Im.J /-homology groups, with the exception of a finite range of dimensions. 55N35, 19L64; 19D99, 19L20 IntroductionIm.J /-theory A is a generalized homology theory appearing both in homotopy theory and algebraic K -theory. It is (sometimes) a fairly good first approximation to stable homotopy capturing v 1 -periodic phenomena, and is closely related to the classical J -homomorphism and the e -invariant. Let p be an odd prime, k be an integer which generates .‫=ޚ‬p 2 / and k be the (stable) Adams operation in p -local complex K -theory. Then the spectrum of nonconnected Im.J /-theory Ad may be defined by the cofiber sequence: Knapp [13] and Crabb and Knapp [5]. Connective Im.J /-theory A is then defined as the ( 1)-connected cover of Ad. More importantly, A appears as the p -localization of the K -theory of a finite field. If k is chosen as a prime power, thenwhere K‫ކ‬ k is the algebraic K -theory spectrum of the finite field ‫ކ‬ k .

J-Groups of Lens Spaces modulo Odd Prime Powers

“…It is a p-group being a quotient of the p-group KO(L). From Lemma 4, we deduce the following two interesting results which are also stated, for example, in [7], [8] and [3], respectively.…”