Subject: new Hopf listings From: Mark Hovey Date: 20 Jan 1999 01:37:02 -0500 Two new papers and a revision this time. Mark Hovey New papers uploaded to hopf between 1/7/99 and 1/20/99: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bisson-Joyal/Luminy Q-Rings and the Homology of the Symmetric Group by Terry Bisson and Andre Joyal ABSTRACT The goal of this paper is to study the rich algebraic structure supported by the homology mod 2 of the symmetric groups. We propose to organise the algebra of homology operations around a single concept, that of Q-ring. We are guided by an analogy with the representation theory of the symmetric groups and the concept of $\lambda$-ring. We show that $H_*\Sigma_*$ is the free Q-ring on one generator. It is a Hopf algebra generated by its subgroup ${\cal K}$ of primitive elements. This subgroup is an algebra (for the composition of operations) that we call the Kudo-Araki algebra. It is closely related to the Dyer-Lashof algebra but is better behaved: the dual coalgebra is directly representing the substitution of Ore polynomials. Many results on the homology of $E_\infty$-spaces can be expressed in the language of Q-rings. We formulate the Nishida relations by using a Q-ring structure on a semidirect extension ${\cal A}$ of Milnor's dual of the Steenrod algebra. We show that the Nishida relations lead to a commutation operator between ${\cal K}$ and ${\cal A}$. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Goerss/hopfring Title: Hopf rings, Dieudonn\'e modules, and $E_\ast \Omega^2S^3$. Author: Paul Goerss AMS Classification Nos: 55205, 55N20, 57T05, 16W30 Department of Mathematics Northwestern University Evanston IL 60208 pogerss@math.nwu.edu (01/19/99) This is a highly revised version of the original submission (7/27/99). P.G. Abstract: Hopf algebras over the prime field with $p$ elements is an abelian category which is equivalent, by work of Schoeller, to a category of graded modules, known as Dieudonn\'e modules. Graded ring objects in Hopf algebras are called Hopf rings, and they arise in the study of unstable cohomology operations for extraordinary cohomology theories. The central point of this paper is that Hopf rings can be studied by looking at the associated ring object in Dieudonn\'e modules. They can also be computed there, and because of the relationship between Brown-Gitler spectra and Dieudonn\'e modules, calculating the Hopf ring for a homology theory $E_\ast$ comes down to computing $E_\ast\Omega^2S^3$ -- which Ravenel has done for $E = BP$. The are two major algebraic difficulties encountered in this approach. The first is to decide what a ring object is in the category of Dieudonn\'e modules, as there is no obvious symmetric monoidal pairing associated to a tensor product of modules. The second is to show that Hopf rings pass to rings in Dieudonn\'e modules. This involves studying universal examples, and here we pick up an idea suggested by Bousfield: torsion-free Hopf algebras over the $p$-adic integers with some additional structure, such as a self-Hopf-algebra map that reduces to the Frobenius, can be easily classified. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/TaylorW/incompatibility_taylor / Classical results in algebraic topology establish that certain well-known spaces A (e.g. n-spheres for n not 1,3 7) are incompatible with axioms T defining group theory or H-space theory. This means one cannot model the equations T on A using continuous operations. Here we provide a generalization that is best possible on the algebraic side. That is, for all T except those that are trivially weak, we prove that these same spaces A are incompatible with T. The method is a blending of traditional techniques of algebraic topology (e.g. calculations for co-operations in the cohomology ring) with a precise understanding of the logic of equations. ---------------------Instructions----------------------------- To subscribe or unsubscribe to this list, send a message to Don Davis at dmd1@lehigh.edu with your e-mail address and name. Please make sure he is using the correct e-mail address for you. To see past issues of this mailing list, point your WWW browser to http://www.math.wesleyan.edu/~mhovey/archive/ If this doesn't work or is missing a few issues, try http://www.lehigh.edu/~dmd1/algtop.html which also has the other messages sent to Don's list. To get the papers listed above, point your WWW client (Mosaic, Netscape) to the URL listed. The general Hopf archive URL is http://hopf.math.purdue.edu There are links to conference announcements, Purdue seminars, and other math related things on this page as well. The largest archive of math preprints is at http://xxx.lanl.gov There is an algebraic topology section in this archive. The most useful way to browse it or submit papers to it is via the front end developed by Greg Kuperberg: http://front.math.ucdavis.edu To get the announcements of new papers in the algebraic topology section at xxx, send e-mail to math@xxx.lanl.gov with subject line "subscribe" (without quotes), and with the body of the message "add AT" (without quotes). You can also access Hopf through ftp. Ftp to hopf.math.purdue.edu, and login as ftp. Then cd to pub. Files are organized by author name, so papers by me are in pub/Hovey. If you want to download a file using ftp, you must type binary before you type get . To put a paper of yours on the archive, cd to /pub/incoming. Transfer the dvi file using binary, by first typing binary then put You should also transfer an abstract as well. Clarence has explicit instructions for the form of this abstract: see http://hopf.math.purdue.edu/pub/submissions.html In particular, your abstract is meant to be read by humans, so should be as readable as possible. I reserve the right to edit unreadable abstracts. You should then e-mail Clarence at wilker@math.purdue.edu telling him what you have uploaded. I am solely responsible for these messages---don't send complaints about them to Clarence. Thanks to Clarence for creating and maintaining the archive. ------- End of forwarded message -------