Subject: new Hopf listings From: Mark Hovey Date: 05 Mar 1998 10:41:18 +0000 -------------- There are six new papers this time. Mark Hovey New papers uploaded to Hopf between 2/26/98 and 3/5/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Adin-Blanc/Adin_Blanc Resolutions of Associative and Lie algebras Ron Adin Bar Ilan University David Blanc University of Haifa We here describe certain explicit canonical resolutions for free associative and free (graded) Lie algebras, in the category of non-associative algebras. Both resolutions are based on the combinatorics of suitable collections of leaf-labeled trees. The Lie case was needed for the second author's description of higher homotopy operations in rational homotopy theory: it turns out that in order to describe all such higher operations, one must resolve the rational differential graded Lie algebra L_* (representing the rational homotopy type of a given space X) simplicially, by suitable free (differential) graded Lie algebras. The higher homotopy operations correspond to relations and syzygies for these free graded Lie algebras, thought of as non-associative algebras. Since we must replace all the Lie algebras by the corresponding free differential algebras in a functorial manner (to preserve the simplicial structure of the original resolution of L_* we need canonical resolutions of free Lie algebras in the category of non-associative algebras, as described in this paper. The construction is closely related to ``strongly homotopy Lie algebras'' Our main interest is indeed in the Lie case. The associative case, which is based on work of Stasheff, is included mainly as a preliminary illustration of the ideas involved, and to fix notation. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Blanc/Blanc_Cwres CW simplicial resolutions of spaces, with an application to loop spaces David Blanc University of Haifa (blanc@mathcs2.haifa.ac.il) A simplicial resolution of a space X by wedges of spheres is a simplicial space W_. such that (a) each space W_n is homotopy equivalent to a wedge of spheres, and (b) for each k>0, the augmented simplicial group \pi_k W_.->\pi_k X is acyclic. Such resolutions were first constructed by Stover, and have a number of applications. However, the Stover construction yields very large resolutions, which do not lend themselves readily to computation. We show here that in fact any space X has such resolutions, which may be constructed from purely algebraic data, consisting of an (arbitrary) simplicial resolution of \pi_* X as a \Pi-algebra ( \ -- \ that is, as a graded group with an action on the primary homotopy operations on it), and in fact every such algebraic resolution of \pi_* X is realizable topologically. Moreover, such resolutions can be given a convenient ``CW structure''. There is an analogous result for maps. As an application of such CW resolutions, we describe an obstruction theory for deciding whether a given space X is a loop space, in terms of higher homotopy operations. The present approach does not require a given $H$-space structure on $\X$, and may be adapted also to the existence of A_{n}-structures. Note: the second part of this paper contains the second part of the preprint previously posted under the title: "Loop spaces and homotopy operations". 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Blanc/Blanc_Loop Loop spaces and homotopy operations David Blanc University of Haifa (blanc@mathcs2.haifa.ac.il) The question of whether a given H-space X is, up to homotopy, a loop space has been studied from a variety of viewpoints. Here we address this question from the aspect of homotopy operations, in the classical sense of operations on homotopy groups. First, we show how an H-space structure on X can be used to define the action of the primary homotopy operations on the shifted homotopy groups \pi_{*-1} X (which are isomorphic to \pi_* Y, if X=\Omega\Y. This action will behave properly with respect to composition of operations if X is homotopy-associative, and will lift to a topological action of the monoid of all maps between spheres if and only if X is a loop space. The obstructions to having such a topological action may be formulated in the framework of an obstruction theory for realizing \Pi-algebras, which is simplified here by showing that any (suitable) \Delta-simplicial space may be made into a full simplicial space (a result which may be useful in other contexts). Note: this paper is an expanded version of the first half of the preprint previously posted under the same name. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/mon-mod This is a slightly revised version of the paper announced here last time (Monoidal model categories). The main new feature is a proof that the monoid axiom holds in any category where everything is fibrant and there is a well-behaved "unit interval" I. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey-Shipley-Smith/symm This is a revised version of the paper "Symmetric spectra" that was previously announced. The prior version had several errors in the section on topological symmetric spectra that are now cleared up. The rest of the paper has changed only minimally. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lydakis/smash_gamma Abstract for "Smash products and $\Gamma$-spaces" by Manos Lydakis Fakultaet fuer Mathematik Universitaet Bielefeld Postfach 100131 33501 Bielefeld Germany manos@math206.mathematik.uni-bielefeld.de We study a symmetric monoidal smash product of $\Gamma$-spaces, which corresponds to the smash product of spectra under the Bousfield-Friedlander equivalence between the homotopy categories of connective spectra and $\Gamma$-spaces. ---------------------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. You can also use 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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