Subject: new Hopf listings From: Mark Hovey Date: 07 Nov 1998 04:58:45 -0500 Seven new papers this time. I made a typo last time: it was Pengelley-Williams, not Pengelley-Wiliams. Mark Hovey New papers uploaded to hopf between 10/26/98 and 11/7/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arkowitz-Scheerer/Beralgebra (No title or author list in this abstract) Let the space X be a 1-connected cogroup. If R is a ring, then a cohomology (flat) product H^{p+1}(X;R) X H^{q+1}(X;R) --->H^{p+q+1}(X;R) was defined by Arkowitz. If we set A^p(X;R)=H^{p+1}(X;R) for p>0 and A^0(X;R)=R,then A*(X;R) is a graded algebra. Berstein has defined a coalgebra B_*(X;K) and dual algebra B*(X;K) when X is a cogroup and K is a field. Our main result is that A*(X;K) and B*(X;K) are isomorphic algebras if X has finite type over K. It follows that the conilpotency class of X is bounded below by the length of the longest product in the algebras B*(X;K). 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Christensen-Hovey/all-or-nothi ng Phantom maps and chromatic phantom maps J. Daniel Christensen and Mark Hovey jdc@math.jhu.edu and hovey@member.ams.org Keywords: Phantom map, chromatic phantom map, n-phantom map, cohomotopy, stable homotopy, spectrum, n-finite type. Abstract: In the first part, we determine conditions on spectra X and Y under which either every map from X to Y is phantom, or no nonzero maps are. We also address the question of whether such all or nothing behaviour is preserved when X is replaced with V smash X for V finite. In the second part, we introduce chromatic phantom maps. A map is n-phantom if it is null when restricted to finite spectra of type at least n. We define divisibility and finite type conditions which are suitable for studying n-phantom maps. We show that the duality functor W_{n-1} defined by Mahowald and Rezk is the analog of Brown-Comenetz duality for chromatic phantom maps, and give conditions under which the natural map Y --> W_{n-1}^2 Y is an isomorphism. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/wyoming TITEL: Localizations over the Steenrod Algebra. The lost Chapter AUTHOR: Mara D. Neusel EMAIL: mdn@sunrise.uni-math.gwdg.de mneusel@cfgauss.uni-math.gwdg.de neusel@math.umn.edu maramara@steenrod.mast.queensu.ca AMS CODE: 55S10 Steenrod Algebra, 13BXX Ring Extensions and Related Topics, 55XX Algebraic Topology, 13XX Commutative Rings and Algebras KEY WORDS: Steenrod Algebra, Unstable Algebras over the Steenrod Algebras, Unstable Part, Localizations, Noetherianess, Integral Closure, Dickson Algebra ABSTRACT: Let H be an unstable algebra over the Steenrod algebra, and let S\subset \H be a multiplicatively closed subset. The localization at S, i.e. S^{-1}H, inherits an action of the Steenrod algebra from H, which is, however, in general no longer unstable. In this note we consider the following three statements. (1) H is Noetherian, (2) the integral closure, \overline{H_{S^{-1}H}}, of H in the localization with respect to S is Noetherian, (3) \overline{H_{S^{-1}H}}= Un(S^{-1}H). where Un(-) denotes the unstable part. If the set S contains only (nonzero) non zero divisors and the algebras are reduced then (1) is equivalent to (2). If S contains zero divisors, then only (1) \Rightarrow (2) remains true, to show the converse is false we construct a counter example. The implication (2) \Rightarrow (3) is always true, while its converse (3) \Rightarrow (2) needs a weird bunch of technical assumptions to remain true. However, none of them can be removed: we illustrate this also with examples. Finally, as a technical tool, we characterize Delta-finite algebras. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rezk/rezk-ho-models "A model for the homotopy theory of homotopy theory" Charles Rezk (Primary 55U35; Secondary 18G30) Department of Mathematics Northwestern University Evanston, IL 60208 rezk@math.nwu.edu November 3, 1998 We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory'', or more precisely that the category of such models has a well-behaved internal hom-object. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rezk/rezk-sharp-maps "Fibrations and homotopy colimits of simplicial sheaves" Charles Rezk (Primary 18G30; Secondary 18B25, 55R99) Department of Mathematics Northwestern University Evanston, IL 60208 rezk@math.nwu.edu November 3, 1998 We show that homotopy pullbacks of sheaves of simplicial sets over a Grothendieck topology distribute over homotopy colimits; this generalizes a result of Puppe about topological spaces. In addition, we show that inverse image functors between categories of simplicial sheaves preserve homotopy pullback squares. The method we use introduces the notion of a sharp map, which is analogous to the notion of a quasi-fibration of spaces, and seems to be of independent interest. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rodriguez-Scevenels/homology TITLE: "Homology equivalences inducing an epimorphism on the fundamental group" AUTHORS: Jose L. Rodriguez Universitat Autonoma de Barcelona E--08193 Bellaterra, Spain jlrodri@mat.uab.es http://mat.uab.es/jlrodri Dirk Scevenels Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B--3001 Heverlee, Belgium dirk.scevenels@wis.kuleuven.ac.be ABSTRACT: Quillen's plus construction is a topological construction that kills the maximal perfect subgroup of the fundamental group of a space without changing the integral homology of the space. In this paper we show that there is a topological construction that, while leaving the integral homology of a space unaltered, kills even the intersection of the transfinite lower central series of its fundamental group. Moreover, we show that this is the maximal subgroup that can be factored out of the fundamental group without changing the integral homology of a space. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strom/EPhant Essential Category Weight and Phantom Maps Jeffrey A. Strom Wayne State University strom@math.wayne.edu This purpose of this paper is to study the relationship between maps with infinite essential category weight and phantom maps. The essential category weight of a map f: X --> Y is the least N such that fg is nullhomotopic whenever g: Z --> X is a map with cat(Z) < N + 1. We write E(f) > N - 1 in this case. A map has infinite essential category weight ( E(f) = \infty ) if there is no such N. The appendix to this paper contains a brief summary of the main results on essential category weight. It is not hard to see that any map with E(f) = \infty is a phantom map. We give examples to show that the reverse is not always true: there are phantom maps f with E(f) = 1. We also show that in some cases, all the phantom maps f: X --> Y have E(f) = \infty. We are able to adapt many of the results of the theory of phantom maps to give us results about maps with E(f) = \infty. Finally, we use the connections between essential category weight and phantom maps to answer a question (asked by McGibbon) about phantom maps, ---------------------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. 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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 -------