Subject: new Hopf listings From: Mark Hovey Date: 01 Nov 1998 02:17:59 -0500 Eleven new papers this time. Note in particular the paper by Jardine--we all knew this ought to work, and it is nice to see that it does. Mark Hovey New papers uploaded to hopf between 10/6/98 and 10/26/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Abrams/abrams-cotensor3 Modules, comodules and cotensor products over Frobenius algebras Lowell Abrams 16D90; 16E30 Department of Mathematics, Hill Center, Rutgers University, New Brunswick, NJ 08903 labrams@math.rutgers.edu This is a corrected version, with significant expository improvements. We characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left A^e-modules. We show that the category of right (left) comodules over A, relative to this coproduct, is isomorphic to the category of right (left) modules. This isomorphism enables a reformulation of the cotensor product of Eilenberg and Moore as a functor of modules rather than comodules. We prove that the cotensor product M \Box N of a right A-module M and a left A-module N is isomorphic to the vector space of homomorphisms from a particular left A^e-module D to N \otimes M, viewed as a left A^e-module. Some properties of D are described. Finally, we show that when A is a symmetric algebra, the cotensor product M \Box N and its derived functors are given by the Hochschild cohomology over A of N \otimes M. This paper has been submitted to the Journal of Algebra, and copyright may be transferred. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Anton/etaleappr Title of Paper : Etale approximations and mod l cohomology of GL_n Author: Marian Florin Anton AMS Classification numbers: 19D55 Addresses of Author: Dept. of Math., Univ. of Notre Dame, Notre Dame IN 46556-5683, U.S.A. also I.M.A.R., Bucharest, Romania Email address of Author: manton1@nd.edu Text of Abstract: If A is a ring of S-integers and l a prime rational number, then the mod p cohomology of the general linear group GL(n,A) is conjectured to be isomorphic to the mod p cohomology of the etale approximation of the classifying space BGL(n,A). We show that this is not possible if n is sufficiently large. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arkowitz/Induced Title of Paper: Induced Mappings of Homology Decompositions Author: Martin Arkowitz AMS Classification numbers: 55P30: 55P45, 55S99 Address of Author: Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA E-mail address of Author: martin.arkowitz@dartmouth.edu Text of Abstract: We give conditions for a map of spaces to induce maps of the homology decompositions of the spaces which are compatible with the homology sections and dual Postnikov invariants. Several applications of this result are obtained. We show how the homotopy type of the (n+1)st homology section depends on the homotopy type of the nth homology section and the (n+1)st homology group. We prove that all homology sections of a co-H-space are co-H-spaces, all n-equivalences of the homology decomposition are co-H-maps and, under certain restrictions, all dual Postnikov invariants are co-H-maps. We give a new proof of a result of Berstein and Hilton which gives conditions for a co-H-space to be a suspension. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bokstedt-Ottosen/loop title: Homotopy orbits of free loop spaces Authors: Marcel Bokstedt and Iver Ottosen Addresses: Marcel Bokstedt Department of Mathematical Sciences Ny Munkegade DK-8000 Aarhus C Denmark Iver Ottosen Département de Mathématiques Université de Paris Nord Avenue Jean-Baptiste Clément F-93430 Villetaneuse France Emails: marcel@mi.aau.dk ottosen@math.univ-paris13.fr We construct a functor which approximates the mod two cohomology of the circle homotopy orbits of a space X when applied to the mod two cohomology of the space X itself. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Jardine/proper4 Title: A^1-local symmetric spectra Author: J.F. Jardine AMS Subj. Classification: 55P42, 55U10, 19E08 Address: Mathematics Department, Univ. of Western Ontario, London, Ont. N6A 5B7, Canada E-mail: jardine@uwo.ca This paper demonstrates the existence of a theory of symmetric spectra for the Morel-Voevodsky stable category. The main results imply the existence of a categorical model for the Morel-Voevodsky stable category which has an internal symmetric monoidal smash product. More explicitly, it is shown that there is a proper closed simplicial model category structure for the category of presheaves of symmetric T-spectra, suitably defined, on the smooth Nisnevich site of a field. The weak equivalences for this structure are stable equivalences, defined by analogy with the definitions given by Hovey, Shipley and Smith for ordinary symmetric spectra and by Jardine for presheaves of symmetric spectra, except that one suspends by the Morel-Voevodsky object T, and the underlying unstable category is the A^1-local category of simplicial presheaves. The homotopy category obtained for the category of presheaves of symmetric T-spectra is equivalent to the Morel-Voevodsky stable category. The details of the basic construction of the original proper closed simplicial model structure underlying the Morel-Voevodsky stable category are required to handle the symmetric case, and are written out in the first three sections of this paper. This paper was written in lamstex, and the dvi file requires the lamstex fonts to view or print. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/GLewis/spltspec Splitting theorems for certain equivariant spectra L. Gaunce Lewis, Jr. AMS Subject Classification: Primary 55M35, 55N91, 53P42, 55P91, 57S15 Secondary 55N20 55Q10 55Q91 55R12 permanent address: L. Gaunce Lewis, Jr. Math. Dept. Syracuse University Syracuse NY 13244 98-99 academic year address: L. Gaunce Lewis, Jr. Math. Dept. Rm 2-130 MIT Cambridge MA 02139-4307 Email: lglewis@syr.edu Note: The dvi file spltspec.dvi uses the yswab font Let $G$ be a compact Lie group, $N$ be a normal subgroup of $G$, $X$ be a $G/N$-space and $Y$ be a $G$-space. There are a number of results in the literature giving a direct sum decomposition of the group of equivariant stable homotopy classes of maps from $X$ to $Y$. Here, these results are extended to a decomposition of the group of equivariant stable homotopy classes of maps from an arbitrary finite $G/N$-CW spectrum $B$ to any $G$-spectrum $C$ carrying a geometric splitting (a new type of structure introduced here). Any naive $G$-spectrum, and any spectrum derived from such by a change of universe functor, carries a geometric splitting. Our direct sum decomposition is a consequence of the fact that, if $C$ is geometrically split and $(\mathcal{F}',\mathcal{F})$ is any reasonable pair of families of subgroups of $G$, then there is a splitting of the cofibre sequence \begin{equation*} (E\mathcal{F}_+ \smsh C)^N \lrarrow (E\mathcal{F}'_+ \smsh C)^N \lrarrow (E(\mathcal{F}',\mathcal{F}) \smsh C)^N \end{equation*} constructed from the universal spaces for the families. Both the decomposition of the stable homotopy groups and the splitting of the cofibre sequence are proven here not just for complete $G$-universes, but for arbitrary $G$-universes. Various technical results about incomplete $G$-universes that should be of independent interest are also included in this paper. These include versions of the Adams and Wirthm\"{u}ller isomorphisms for incomplete universes. Also included is a vanishing theorem for the fixed-point spectrum $(E(\mathcal{F}',\mathcal{F})\smsh C)^N$ which gives computational force to the intuition that what really matters about a $G$-universe $U$ is which orbits $G/H$ embed as $G$-spaces in $U$. This is a revised and expanded version of a paper submitted to the archives several years ago. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell/einffinal $E_{\infty}$-Algebras and $p$-Adic Homotopy Theory Michael A. Mandell Let $\FP$ denote the field with $p$ elements and $\FPbar$ its algebraic closure. We show that the singular cochain functor with coefficients in $\FPbar$ induces a contravariant equivalence between the homotopy category of connected $p$-complete nilpotent spaces of finite $p$-type and a full subcategory of the homotopy category of \einf $\FPbar$-algebras. October 30, 1998. This is a revision of the Jan 26, 1998 draft. The major changes are the following: The paper is now self-contained, and does not require the work from "The Homotopy Theory of E-infty Algebras". The necessary results have been included in this draft (pp. 5-8). Also, the proofs (pp. 28-35) are entirly different and intrinsic (do not require shifting focus to the ``linear isometries operad''). More information on the analogue of the main theorem for finite fields is included (Theorem A.2 stated on p. 35 proved pp. 37-39). The connection with Bousfield-Kan p-completion (Remark 5.1 on p. 13) and the p-pro-finite completion (Appendix B, pp. 39-43) is explained. Results on the identification of the subcategory of E-infty algebras in the image of the cochain functor is now included. Among other things, the new ``Characterization Theorem'' (p. 2) and pp. 17-23. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Nave/smith-toda Title of Paper: On the nonexistence of Smith-Toda complexes Author: Lee S. Nave AMS Classification numbers: 55N22, 55T15, 55P42 Address of Author: University of Washington Department of Mathematics, Box 354350 Seattle, WA 98195-4350 Email: nave@math.washington.edu Let p be a prime. The Smith-Toda complex V(k) is a finite spectrum whose BP-homology is isomorphic to BP_*/(p,v_1,...,v_k). For example, V(-1) is the sphere spectrum and V(0) the mod p Moore spectrum. In this paper we show that if p > 5, then V((p+3)/2) does not exist and V((p+1)/2), if it exists, is not a ring spectrum. The proof uses the new homotopy fixed point spectral sequences of Hopkins and Miller. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Pengelley-Wiliams/newlowerops SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS FOR MOD 2 HOMOLOGY AND COHOMOLOGY DAVID J. PENGELLEY AND FRANK WILLIAMS Abstract. The mod 2 Steenrod algebra A and Dyer-Lashof al- gebra R have both striking similarities and differences, arising from their common origins in "lower-indexed" algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra K, whose module actions are equivalent to, but quite dif- ferent from, those of A and R. The exact relationships emerge as "sheared algebra bijections", which also illuminate the role of the cohomology of K. As a bialgebra, K* has a particularly attractive and potentially useful structure, providing a bridge between those of A* and R*, and suggesting possible applications to the Miller spectral sequence and the A structure of Dickson algebras. New Mexico State University, Las Cruces, NM 88003 E-mail address: davidp@nmsu.edu E-mail address: frank@nmsu.edu 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/LSmith/smith.invar.biblio This is an invariant theory bibliography compiled by Larry Smith. Clarence could not get this to run through TeX, so there is no .dvi version. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/WuJ/p_local_minimal Note on the Minimal Simplicial Set Models for $p$-local $H$-spaces Jie Wu Department of Mathematics University of Pennsylvania Philadelphia, PA 19104 USA jiewu@math.upenn.edu Included file: p_local_minimal.dvi In this note, we show that there exists a commutative (but non-associative in general) multiplication on the minimal simplicial set model of any path-connected $p$-local $H$-space with $p>2$. This result is the homotopy theory analogy of the Jordan algebra in the sense that one can make a new strictly commutative multiplication from an old non-commutative multiplication when the power map $2\colon X\rTo X$ is a homotopy equivalence. We need to point out that by using the results in our previous paper "On products on minimal simplicial sets", any connected (non-associative) minimal simplicial $H$-set $X$ is nilpotent with respect to associativity in the sense that for each $X_n$ the higher associators with length sufficiently large are trivial. More precisely, the nilpotency degree (with respect to the associator length) of the $(n+1)$-st Postnikov section of $X$ is at most one bigger that the nilpotency degree of the $n$-th Postnikov section of $X$. ---------------------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.cs.wesleyan.edu/Math/Guests/Mark If this doesn't work or is missing a few issues, try http://www.lehigh.edu/~dmd1/public/www-data/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 -------