5 new papers this time, from Ando-Hopkins-Strickland, Christensen-Dwyer-Isaksen, Dwyer-Greenlees-Iyengar, Kitchloo-Laures-Wilson, and McClure-Smith (a new version of a previously announced paper). Mark Hovey New papers appearing on hopf between 04/03/02 and 05/01/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Ando-Hopkins-Strickland/sigma-hinfty-4.26 Title: The sigma orientation is an H-infinity map Authors: Matthew Ando Michael J. Hopkins Neil P. Strickland AMS subject classification: 55N34 arXiv number: math.AT/0204053 Adresses: Department of Mathematics, University of Illinois at Urbana-Champaign mando@math.uiuc.edu Department of Mathematics, Massachusetts Institute of Technology mjh@math.uiuc.edu Department of Pure Mathematics, University of Sheffield N.P.Strickland@sheffield.ac.uk Abstract: In "Elliptic spectra, the Witten genus, and the Theorem of the cube" (Invent. Math. 146 (2001)), the authors constructed a natural map from the Thom spectrum MU<6> to any elliptic spectrum, called the "sigma orientation". MU<6> is an H-infinity ring spectrum, and in this paper we show that if E is a K(2)-local H-infinity elliptic spectrum, then the sigma orientation is a map of H-infinity spectra. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Christensen-Dwyer-Isaksen/DC Obstruction theory in model categories J. Daniel Christensen, William G. Dwyer and Daniel C. Isaksen MSC: 55S35, 55U35, 18G55 (primary); 18G30, 55P42 (secondary) Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 jdc@uwo.ca Department of Mathematics University of Notre Dame South Bend, IN 46556 dwyer.1@nd.edu Department of Mathematics University of Notre Dame South Bend, IN 46556 isaksen.1@nd.edu Keywords: obstruction theory, closed model category, simplicial set, spectrum Working in an arbitrary pointed proper model category, we describe the cofibrations that have an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Greenlees-Iyengar/duality Duality in Algebra and Topology W. G. Dwyer, J. P. C. Greenlees, and S. Iyengar We take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in a topological setting. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can be extended to differential graded algebras or more generally to structured ring spectra. This framework allows us to view all of the following dualities o Poincare duality for manifolds o Gorenstein duality for commutative rings o Benson-Carlson duality for cohomology rings of finite groups o Poincar duality for groups o Gross-Hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon. We give a new formula for the Brown-Comenetz dual of the sphere spectrum; this turns out to be one instance of a general construction that in another setting gives the dualizing module of a Gorenstein ring. We also prove the local cohomology theorem for p-compact groups and reprove it for compact Lie groups. The key observation is that the cochain algebra on BG has a simple duality property which extends Poincare duality. Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556. USA, dwyer.1@nd.edu Department of Pure Mathematics, Hick Building, Sheffield S3 7RH. UK, j.greenlees@sheffield.ac.uk 202 Mathematical Sciences Building, University of Missouri, Columbia, MO 65211. USA, iyengar@math.missouri.edu 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Laures-Wilson/kitchloo-laures-wilson The Morava K-theory of spaces related to BO Nitu Kitchloo Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 nitu@math.jhu.edu Gerd Laures Mathematisches Institut der Universitaet Heidelberg Im Neuenheimer Feld 288 D-69120} Heidelberg, Germany gerd@laures.de W. Stephen Wilson Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 wsw@math.jhu.edu Abstract: We calculate the (p=2) Morava K-theory of all of the spaces in the connective Omega spectra for ZxBO, BO, BSO, and BSpin. This leads to a description of the (p=2) BP cohomology of many of these spaces. Of particular interest is the space BO<8> and its relationship to BSpin. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/McClure-SmithJH/McClureSmith2 (This is the same abstract as before, but a new version of the paper itself) Multivariable cochain operations and little $n$-cubes. James E. McClure and Jeffrey H. Smith 18D50, 55P48, 16E40 math.QA/0106024 Department of Mathematics, Purdue University, West Lafayette, IN 47907--1395 mcclure@math.purdue.edu jhs@math.purdue.edu In this paper we construct a small $E_\infty$ chain operad $\S$ which acts naturally on the normalized cochains $S^*X$ of a topological space. We also construct, for each $n$, a suboperad $\S_n$ which is quasi-isomorphic to the normalized singular chains of the little $n$-cubes operad. The case $n=2$ leads to a substantial simplification of our earlier proof of Deligne's Hochschild cohomology conjecture.