Subject: new Hopf listings From: Mark Hovey Date: 09 Apr 2000 07:00:34 -0400 6 new papers this time. Mark Hovey New papers uploaded to hopf between 3/4/00 and 4/9/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bendersky-DavisD/comp1 Compositions in the v1-periodic homotopy groups of spheres Martin Bendersky and Donald M. Davis mbenders@shiva.hunter.cuny.edu, dmd1@lehigh.edu 21 pages, completed March 7, 2000, submitted to Forum Mathematicum Abstract Let p_i in pi_{n+8i-1}(S^n) denote an element which suspends to a generator of the image of the stable 2-primary J-homomorphism. We determine the image of the composite p_j o p_k in v1-periodic homotopy v_1^{-1} pi_{n+8i+8j-2}(S^n). The method is to use Adams operations to compute the 1-line of an unstable homotopy spectral sequence constructed by Bendersky and Thompson. Odd-primary analogues are also obtained. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/stable-model Spectra and symmetric spectra in general model categories by Mark Hovey Wesleyan University hovey@member.ams.org April, 2000 This is a revised version. The basic idea is to automate the passage from unstable to stable homotopy theory, so that it applies in particular to the A^1 category of Voevodsky. So if we start with a model category C and a left Quillen endofunctor G of C, we want to make a new model category, the stabilization of C, where G becomes a Quillen equivalence. The simplest way to do this is with ordinary spectra. Thanks to Hirschhorn's localization technology, we can construct the stable model structure on ordinary spectra with almost no hypotheses on C and G. A new feature of this revision is that we show that, under strong smallness hypotheses on G and C, the stable equivalences coincide with the appropriate generalization of stable homotopy isomorphisms. In particular, this holds for the A^1 category. If C has a tensor product, and G is given by tensoring with a cofibrant object K, then we also can construct symmetric spectra. The localization techniques apply here as well, so we get a stable model structure of symmetric spectra without having to assume anything like the Freudenthal suspension theorem. In particular, this is a new construction of the stable model structure on simplicial symmetric spectra. Symmetric spectra form a monoidal model category, unlike ordinary spectra, but we are unable to prove that the monoid axiom holds in general. Also new to this revision is a much more careful comparison between symmetric spectra and ordinary spectra when both are defined. Symmetric spectra and ordinary spectra are not always Quillen equivalent; we need the cyclic permutation map on K tensor K tensor K to be homotopic to the identity. Under some additional technical hypotheses (which again are satisfied in the A^1 category), we construct a zigzag of Quillen equivalences between symmetric spectra and ordinary spectra. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Oliver-Segev/2dim Fixed point free actions on $Z$-acyclic 2-complexes by Bob Oliver and Yoav Segev E-mail: bob@math.univ-paris13.fr, yoavs@math.bgu.ac.il We show that a finite group has an "essential" fixed point free action on an acyclic 2-complex if and only if it is one of the simple groups in the following list: - $PSL_2(2^k)$ for $k\ge2$, - $PSL_2(q)$ for $q\equiv3,5$ (mod 8) and $q\ge5$, - $Sz(2^k)$ for odd $k\ge3$. More precisely, for any finite group $G$, and any 2-dimensional acyclic $G$-CW complex $X$ without fixed points, there is a normal subgroup $H$ in $G$ such that $G/H$ is in the above list, and such that the $G$-action on $X$ looks "essentially" like the $G/H$-action which we construct. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rezk/rezk-simpl-alg-proper Title: Every homotopy theory of simplicial algebras admits a proper model Author: Charles Rezk rezk@math.nwu.edu Abstract: We show that any closed model category of simplicial algebras over an algebraic theory is Quillen equivalent to a proper closed model category. By ``simplicial algebra'' we mean any category of algebras over a simplicial algebraic theory, which is allowed to be multi-sorted. The results have applications to the construction of localization model category structures. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Scheerer-Stanley-Tanre/Qcat Fibrewise construction applied to Lusternik-Schnirelmann category by Hans Scheerer, Donald Stanley and Daniel Tanr\'e scheerer@math.fu-berlin.de Don.Stanley@agat.univ-lille1.fr Daniel.Tanre@agat.univ-lille1.fr Abstract: In this paper a variant of Lusternik-Schnirelmann category is presented which is denoted by Qcat(X). It is obtained by applying a base-point free version of Q = Omega-infinity Sigma-infinity fibrewise to the Ganea fibrations. We prove cat(X) >= Qcat(X) >= scat(X), where scat(X) denotes Y. Rudyak's strict category weight. However, Qcat(X) approximates cat(X) better, because e.g. in the case of a rational space Qcat(X)=cat(X) and scat(X) equals the Toomer invariant. We show that Qcat(X x Y) <= Qcat(X)+Qcat(Y). The invariant Qcat is designed to measure the failure of the formula cat(X x S^r)=cat(X)+1. In fact for 2-cell complexes Qcat(X)< cat(X) if and only if cat(X x S^r) <= cat(X) for some r >= 1. We note that the paper is written in the more general context of a functor L from the category of spaces to itself satisfying certain conditions; L= Q, Omega^n Sigma^n, Sp^infinity or L_f are just particular cases. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/TanK-XuK/dicknew (This is a revised version) Dickson Invariants hit by the Steenrod Squares BY K. F. Tan and Kai Xu Abstract: Let $D_3$ be the Dickson invariant ring of $F_2[X_1,X_2,X_3]$ by GL(3,F_2)$. In this paper, we prove each element in $D_3$ is hit by the Steenrod square in $F_2[X_1,X_2,X_3]$. Our method provides a clue in attacking the question in the general case. (This paper contains some tedious computations which will be dropped in the simplified version that will be written later.) ---------------------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< 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. 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