Subject: new Hopf listings Date: 29 Jun 2002 18:54:27 -0400 From: Mark Hovey Hola from Barcelona! Sorry for the lack of updates recently. There are 7 new papers listed here (and there are a few more that have been submitted and should be announced soon), from Baker-May, Bruner-Ha-Hung, Gaudens-Schwartz, Fausk-Hu-May, Hu-Kriz-May, and 2 from May. Mark Hovey New papers appearing on hopf between 05/01/02 and 06/29/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Baker-May/CoresMay30 Title: Minimal atomic complexes Authors: A.J. Baker and J.P. May Classification: 55P15 55P42 (55P60) Address: Math. Dept., University of Glasgow, Glasgow G12 8QW, Scotland. E-mail: a.baker@maths.gla.ac.u Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: may@math.uchicago.edu Hu, Kriz and May recently reexamined ideas implicit in Priddy's elegant homotopy theoretic construction of the Brown-Peterson spectrum at a prime p. They discussed May's notions of nuclear complexes and of cores of spaces, spectra, and commutative S-algebras. Their most striking conclusions, due to Hu and Kriz, were negative: cores are not unique up to equivalence, and BP is not a core of MU considered as a commutative S-algebra, although it is a core of MU considered as a p-local spectrum. We investigate these ideas further, obtaining much more positive conclusions. We show that nuclear complexes have several non-obviously equivalent characterizations. Up to equivalence, they are precisely the irreducible complexes, the minimal atomic complexes, and the Hurewicz complexes with trivial mod p Hurewicz homomorphism above the Hurewicz dimension, which we call complexes with no mod p detectable homotopy. Unlike the notion of a nuclear complex, these other notions are all invariant under equivalence. This simple and conceptual criterion for a complex to be minimal atomic allows us to prove that many familiar spectra, such as ko, $eo_2$, and BoP at the prime 2, all $BP$ at any prime p, and the indecomposable wedge summands of the suspension spectra of $CP^\infty$ and $HP^\infty$ at any prime p are minimal atomic. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bruner-Ha-Hung/alg-trans Title: On behavior of the algebraic transfer Authors: Robert R. Bruner, Le Minh Ha, and Nguyen H. V. Hung MSC-class: 55P47, 55Q45, 55S10, 55T15 Paper: math.AT/0205170 Addresses: Robert R. Bruner Department of Mathematics Wayne State University Detroit, MI 48202 USA rrb@math.wayne.edu Le Minh Ha IHES, F-91440, Bures-sur-Yvette France lha@ihes.fr Nguyen H. V. Hung Department of Mathematics Wayne State University Detroit, MI 48202 USA nhvhung@math.wayne.edu Abstract: Let V be a mod 2 vector space of rank k. W. Singer defined a transfer homomorphism from the GL(k,2) coinvariants of the primitives in the homology of BV to the cohomology of the Steenrod algebra, as an algebraic version of the geometric transfer from the stable homotopy of BV to the stable homotopy of spheres. It has been shown that the algebraic transfer is highly nontrivial and, more precisely, that it is an isomorphism for k=1, 2, or 3. However, Singer showed that it is not an epimorphism for k=5. In this paper, we prove that it also fails to be an epimorphism when k=4. Precisely, it does not detect the non zero elements in the g family, in stems 20, 44, 92, and in general, 12*2^s - 4, for each s > 0. The transfer still fails to be an epimorphism even after inverting Sq^0, thereby giving a negative answer to a prediction by Minami. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk-Hu-May/FormalFeb16 Title: Isomorphisms between left and right adjoints authors: H. Fausk, P. Hu, and J.P. May Classification: 14A99, 18F99, 55P91 (18D10, 55U30) Address: Dept. Math., Northwestern University, Evanston, IL 60208-2730, USA. E-mail: fausk@math.northwestern.edu Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: poh@math.uchicago.edu Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: may@math.uchicago.edu There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being isomorphic to a shift of the left adjoint specified by an appropriate ``dualizing object''. Typically the left adjoint is well understood while the right adjoint is more mysterious, and the result identifies the right adjoint in familiar terms. We give a categorical discussion of such results. One essential point is to differentiate between the classical framework that arises in algebraic geometry and a deceptively similar, but genuinely different, framework that arises in algebraic topology. Another is to make clear which parts of the proofs of such results are formal. The analysis significantly simplifies the proofs of particular cases, as we illustrate in a sequel discussing applications to equivariant stable homotopy theory. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Gaudens-Schwartz/GS Title Sur les sous-modules instables des alg\`ebres instables Authors G\'erald Gaudens et Lionel Schwartz gerald.gaudens@math.univ-nantes.fr Département de Mathématiques 2, rue de le Houssinière - BP 92208 44322 NANTES Cédex 3 FRANCE schwartz@math.univ-paris13.fr UMR 7539 du CNRS Institut Galil\'ee Universit\'e Paris 13 Av. J. B. Cl\'ement 93430 Villetaneuse FRANCE 55S10 Cet article fait suite \`a une pr\'epublication de Laurent Piriou et du second auteur. Il contient des r\'esultats reli\'es \`a la conjecture de finitude, plus pr\'ecisement \`a la structure du treillis des sous-modules instables d'une alg\`ebre instable r\'eduite. Le premier r\'esultat, d\^u au second auteur, montre que les sous-modules instables de l'alg\`ebre de Dickson sont, soit l'alg\`ebre toute enti\`ere, soit petits vis \`a vis de l'alg\`ebre. Le second r\'esultat, d\^u au premier auteur, montre que la s\'erie des socles d'une alg\`ebre instable connexe r\'eduite non-triviale est infinie, ceci avait \'et\'e conjectur\'e par le second auteur dans [13].Un outil important, d\^u au second auteur, est la construction et l'action de certaines op\'erations de Steenrod sur des classes appartenant \`a des alg\`ebres instables. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hu-Kriz-May/99April1 Title: Cores of spaces, spectra, and $E_{\infty}$ ring spectra Authors: P. Hu, I. Kriz, and J.P. May Classification: 55P15, 55P42, 55P43, 55S12 Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: pohu@math.uchicago.edu Address: Dept. Math., University of Michigan, Ann Arbor, MI 48109-1107, USA E-mail: ikriz@math.lsa.umich.edu Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: may@math.uchicago.edu In a paper that has attracted little notice, Priddy showed that the Brown-Peterson spectrum at a prime p can be constructed from the p-local sphere spectrum S by successively killing its odd dimensional homotopy groups. This seems to be an isolated curiosity, but it is not. For any space or spectrum Y that is p-local and (n_0-1)-connected and has $\pi_{n_0}(Y)$ cyclic, there is a p-local, $(n_0-1)$-connected ``nuclear'' CW complex or CW spectrum X and a map $f: X\to Y$ that induces an isomorphism on $\pi_{n_0}$ and a monomorphism on all homotopy groups. Nuclear complexes are atomic: a self-map that induces an isomorphism on $\pi_{n_0}$ must be an equivalence. The construction of X from Y is neither functorial nor even unique up to equivalence, but it is there. Applied to the localization of MU at p, the construction yields BP. {Appeared: Homology, homotopy, and applications 3(2001), 341--354} 6. http://hopf.math.purdue.edu/cgi-bin/generate?/May/97April1 Title: Idempotents and Landweber exactness in brave new algebra Author: J.P. May Classification: 55N20, 55N91, 55P43 Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: may@math.uchicago.edu We explain how idempotents in homotopy groups give rise to splittings of homotopy categories of modules over commutative $S$-algebras, and we observe that there are naturally occurring equivariant examples involving idempotents in Burnside rings. We then give a version of the Landweber exact functor theorem that applies to $MU$-modules. {Appeared in Homology, homotopy, and applications 3(2001), 355--359} 7. http://hopf.math.purdue.edu/cgi-bin/generate?/May/WirthRev Title: The Wirthmuller isomorphism revisited author: J.P. May Classification: 55P91, 55U30 Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA. E-mail: may@math.uchicago.edu We show how the formal Wirthmuller isomorphism theorem proven in "Isomorphisms between left and right adjoints", by Fausk, Hu, and May, simplifies the proof of the Wirthmuller isomorphism in equivariant stable homotopy theory. Other examples from equivariant stable homotopy theory show that the hypotheses of the formal Wirthmuller and formal Grothendieck isomorphism theorems in the cited paper cannot be weakened. ---------------------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 (Netscape or Internet Explorer) to the URL listed. The general Hopf archive URL is http://hopf.math.purdue.edu There are links to 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/new-html/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.