Opérations d'Adams en K-homologie et applications

Abstract

The author studies the K homologyK\sb \ast({\bf C}P\sp \infty) as a Pontrjagin ring; it is theHopf algebra corresponding to the ``multiplicative'' formalgroup. Subject to suitable localisation, the operationψ\sp q can be considered as a map of spectra, and so inducesa map of K homology; its effect on K\sb 0({\bf C}P\sp \infty) is computed.\par There is an application to thestable homotopy π\sb \ast\sp S({\bf C}P\sp \infty) modulotorsion; it is polynomial on one obvious generator of degree 2,and the image of π\sb {2n}\sp S({\bf C }P\sp \infty) inH\sb {2n}({\bf C}P\sp \infty) is the multiples of n!. (Thiscan also be seen by using the Chern character.) There is apartial result for π\sb \ast\sp S(B\text U) modulotorsion.\par The author continues with a similar study forK\sb \ast({\bf C}P\sp \infty;{\bf Z}/l), where l is an oddprime. He thus obtains (\text{Ad}\sb q)\sb \ast({\bf C}P\sp \infty;{\bf Z}/l), where\text{Ad}\sb q is the spectrum corresponding to the ``fibre ofψ\sp q 1''. The interesting case is that in which q is amultiplicative generator \text{mod}\,l\sp 2, and in this casethe algebra has a presentation (over the coefficient ring) withgenerators Y\sb i\ (i\geq 0) and relationsY\sb i\sp l=αY\sb i, Y\sb iY\sb j=0 if ieq j,δY\sb i=0 where α,δ are suitable elements ofthe coefficient ring. This leads to a similar description of(K{\bf F}\sb q)\sb \ast({\bf C}P\sp \infty;{\bf Z}/l).\par Abrief section shows how the results are modified if one replaces{\bf C}P\sp \infty=BS\sp 1 by B{\bf Z}/l.\par The authorplans to give elsewhere applications to π\sb \ast\sp S({\bf C}P\sp \infty;{\bf Z}/l)