Revisiting the image of J

23 February 2009
Neil Strickland
I'll discuss my ongoing attempt to modernise the theory of the image of J.<br /> Some features<br /> that I would like to have are as follows:<br /> <br /> 1) Most of the spectra involved in the story should be E_\infty (or strictly<br /> commutative)<br />     ring spectra, and most of the maps involved should respect this structure.  New<br />     machinery for dealing with E_\infty rings should be used systematically.<br /> <br /> 2) As far as possible the constructions used should not depend on arbitrary choices<br />      or on gratuitous localisation.<br /> <br /> 3) The Bernoulli numbers should enter via their primary definition as coefficients of a<br />      certain power series.<br /> <br /> 4) The image of J spectrum should be defined as the Bousfield localisation of S^0 with<br />     respect to KO, and other properties or descriptions should be deduced from this one.<br /> <br /> 5) There should be a clear conceptual explanation for the parallel appearance of<br />     Bernoulli numbers in the homotopy groups of J, K(Z) and in spectra related to<br />     surgery theory.<br /> <br />