Subject: new Hopf listings From: Mark Hovey Date: 17 Jun 1998 02:50:42 -0400 Its getting so I can't go out of town anymore! Twelve new papers on hopf, and one on xxx. Mark Hovey New papers uploaded to hopf and xxx between 5/21/98 and 6/17/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Abrams/abrams-cotensor 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 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 comodules over A, relative to this coproduct, is isomorphic to the category of right 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 right A^e-module D to M \otimes N, viewed as a right A^e-module. Some of the properties of D are investigated, and some sample calculations are given. Finally, we show that when A is commutative or semisimple, the cotensor product M \Box N and its derived functors are given by the Hochschild cohomology over A of M \otimes N. 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/Arone/SuspIterates Iterates of the suspension map and Mitchell's finite spectra with $A_k$-free cohomology Greg Arone arone@math.uchicago.edu AMS classification: 55P40, 55P42, 55P65 We study certain cross-effects of the unstable homotopy of spheres. These cross-effects were constructed by Weiss in the context of ``Orthogonal calculus''. We show that Mithchell's finite spectra with $A_k$-free cohomology arise naturally as stabilizations of Weiss' cross-effects. Furthermore, we find that after a suitable Bousfield localization, our cross-effects, which capture meaningful information about the unstable homotopy of spheres, are homotopy equivalent to the infinite loop spaces associated with Mitchell's spectra. This last result is a partial generalization of a previous result of Mahowald and the author. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Arone-Mahowald/ArMahowald The Goodwillie Tower of the identity functor and the unstable periodic homotopy of spheres AMS Classifiaction: 55P47, 55Q40, 55S12 Greg Arone arone@math.uchicago.edu Mark Mahowald mark@math.nwu.edu We investigate Goodwillie's ``Taylor tower'' of the identity functor from spaces to spaces. More specifically, we reformulate Johnson's description of the Goodwillie derivatives of the identity, and prove that when evaluated at an odd-dimensional sphere, the only layers in the tower that are not contractible are those indexed by a prime power. Furthermore, in the case of a sphere the tower is finite in $v_k$-pe- riodic homotopy. It has $k+1$ stages if the sphere is odd dimensional, and $2(k+1)$ stages if the sphere is even-dimensional. This is a revised version of a previously uploaded preprint. The paper has been accepted for publication, and is now in its final form. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Levi/htbg ON THE HOMOTOPY TYPE OF BG FOR CERTAIN FINITE 2-GROUPS G Carles Broto and Ran Levi We consider the homotopy type of classifying spaces $BG$, where $G$ is a finite $p$-group and study the question, whether or not the mod $p$ cohomology of $BG$, as an algebra over the Steenrod algebra together with the associated Bockstein spectral sequence determine the homotopy type of $BG$. This article is devoted to producing some families of finite 2-groups, where cohomological information determines the homotopy type of $BG$. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Levi/snk LOOP STRUCTURES ON HOMOTOPY FIBRES OF SELF MAPS OF A SPHERE By Carles Broto and Ran Levi Let $S^{2n-1}\{k\}$ denote the fibre of the degree $k$ map on the sphere $S^{2n-1}$. If $k=p^r$, where $p$ is an odd prime and $n$ divides $p-1$ then $S^{2n-1}\{k\}$ is known to be a loop space. It is also known that $S^3\{2^r\}$ is a loop space for $r\geq 3$. In this paper we study the possible loop structures on this family of spaces for all primes $p$. In particular we show that $S^3\{4\}$ is not a loop space. Our main result is that whenever $S^{2n-1}\{p^r\}$ i a loop space, the loop structure is unique up to homotopy. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ishiguro/toral Toral groups and classifying spaces of $p$--compact groups Kenshi Ishiguro (kenshi@ssat.fukuoka-u.ac.jp) Fukuoka University, Fukuoka 814-0180, Japan We show converses to some known results for the classifying spaces of $p$--toral groups or $p$--compact toral group. Suppose $G$ is a compact Lie group. The following results are included. (A) If there is a positive integer $k$ such that the $n$--th homotopy groups of $(BG)\p$ are zero for all $n \ge k$, then $(BG)\p$ is the classifying space of a $p$--compact toral group. (B) If the canonical map $Rep(G, K) @>>> [BG, BK]$ is bijective for any compact connected Lie group $K$, then $G$ is a $p$--toral group. We will also discuss the conditions of a compact Lie group that its loop space of the $p$--completed classifying space be a $p$--compact group. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/athom The Inverse Invariant Theory Problem and Steenrod Operations Mara D Neusel AMS Classification: 55S10 Steenrod Algebra, 13A50 Invariant Theory, 55XX Algebraic Topology AG Invariantentheorie, Germany University of Minnesota, School of Mathematics, 127 Vincent Hall, 206 Church Street S.E., Minneapolis MN 55455, USA mdn@sunrise.uni-math.gwdg.de maramara@steenrod.mast.queensu.ca neusel@math.umn.edu This is a pure postscript file. This paper is devoted to the study of inverse invariant theory and its relationship with the $\steenrod$--invariant prime spectrum of an unstable algebra over the Steenrod algebra. We will show that this spectrum is a chain saturated poset. Moreover we will prove the existence of Thom classes, detect a fractal of the Dickson algebra in any unstable algebra and give a counterexample to the Reverse Landweber--Stong Conjecture. Along the way to these results we will generalize the famous Adams--Wilkerson theorems to arbitrary Galois fields, have a closer look at fields and their extensions over the Steenrod algebra, and generalize some results about the unstable part of a module over the Steenrod algebra. [ARCHIVE NOTE: available in .ps.gz and .pdf formats only due to use of custom postscript fonts. I don't advise this format because it limits the onscreen viewing and search possibilities. ] 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rudyak/AnalyticApps (No abstract on archive. This paper is an updated version of a paper previously on the archive). 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rudyak/ArnoldConj (This is also an update of a previously announced paper). ON STRICT CATEGORY WEIGHT AND THE ARNOLD CONJECTURE Yuli B. Rudyak Rudyak and Oprea proved the Arnold conjecture for symplectic manifolds $(M,\omega)$ with $\pi_2(M)=0$. The proof used surgery and cobordism theory. Here we give a purely cohomological proof of this result. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rudyak/CategoryWeight (And so is this one) CATEGORY WEIGHT: NEW IDEAS CONCERNING LUSTERNIK--SCHNIRELMANN CATEGORY Yuli B. Rudyak The concept of category weight was introduced by Fadell--Husseini and developed by Rudyak and Strom. Here we give a survey, some further development and applications of category weight. 11. http://xxx.lanl.gov/dvi/math.AT/9806021 From: Peter Saveliev Date: Thu, 4 Jun 1998 20:37:08 GMT (20kb) Title: A Lefschetz type coincidence theorem Authors: Peter Saveliev Comments: 20 pages Subj-class: Algebraic Topology MSC-class: 55M20, 55H25 \\ A Lefschetz-type coincidence theorem for two maps f,g:X->Y from an arbitrary topological space X to a manifold Y is given: I(f,g)=L(f,g), the coincidence index is equal to the Lefschetz number. It follows that if L(f,g) is not equal to zero then there is an x in X such that f(x)=g(x). In particular, the theorem contains some well-known coincidence results for (i) X,Y manifolds and (ii) f with acyclic fibers. \\ ( http://xxx.lanl.gov/abs/math/9806021 , 20kb) 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Turner/looking-glass Simplicial Commutative F_p-Algebras Through the Looking-Glass of F_p-Local Spaces 1991 Mathematics Subject Classification. Primary: 13D03, 18G30, 18G55; Secondary: 55P60, 55P99, 55S05 James M. Turner 1395 Mathematical Sciences Building Purdue University West Lafayette, IN 47907-1395 jmt@ziplink.net Submitted to the Proceedings in Honor of Michael J. Boardman We propose a dictionary approach to studying the homotopy theory of simplicial augmented commutative F_p-algebras using the homotopy theory of connected F_p-local spaces as our guide. We indicate how standard topological tools translate to the setting of simplicial algebras. We further indicate how theorems translate as well. For example, we recall a theorem of P. Goerss giving an algebraic version of the Hilton-Milnor theorem which fits in our framework. We next propose how a theorem of J.-P. Serre on F_p-local spaces with bounded homotopy groups translates into our algebraic setting and relate it to a conjecture of D. Quillen on the vanishing of Andr\'e-Quillen homology. We also describe what a simplicial algebra version of a theorem of D. Kan and W. Thurston should look like. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Turner/vanishingnew On Simplicial Commutative Algebras with Vanishing Andr\'e-Quillen Homology 1991 Mathematics Subject Classification. Primary: 13D03, 18G30, 18G55; Secondary: 13D40 James M. Turner 1395 Mathematical Sciences Building Purdue University West Lafayette, IN 47907-1395 jmt@ziplink.net October 1997; Revised: June 12, 1998 In this paper, we study the Andr\'e-Quillen homology of simplicial commutative F-algebras, where F is a field of positive characteristic, with certain vanishing properties. We will show, under certain conditions on $\pi_0$ and the vanishing of the homotopy groups, that the vanishing of Andr\'e-Quillen homology implies that the simplicial commutative F-algebra in question is a homology complete intersection. As a consequence, we resolve a conjecture of D. Quillen in the case of commutative Noetherian F-algebras. ---------------------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 general xxx archive URL is http://xxx.lanl.gov. More useful is the front end developed by Greg Kuperberg: http://front.math.ucdavis.edu 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. 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. For instructions on uploading papers to xxx, see http://front.math.ucdavis.edu 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 ------- ------- End of forwarded message -------