Subject: new Hopf listings From: Mark Hovey Date: 29 Jan 2000 21:35:27 -0500 This seems a good time to remind you that if you have submitted a paper to Hopf and it does not appear on this list, it is NOT because Clarence has rejected it. Hopf is not an automated archive, so sometimes it takes a while for papers to be moved into the appropriate spot. On the other hand, it is always possible that Clarence or I have made a mistake, so it doesn't hurt to send e-mail reminding us. 5 new papers this time. Mark Hovey New papers uploaded to hopf between 1/25/00 and 1/29/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Levi/homotopy-ext On spaces of self homotopy equivalences of p-completed classifying spaces of finite groups and homotopy group extensions By Carles Broto and Ran Levi Fix a prime p. A mod-p homotopy group extension of a group $\pi$ by a group G is a fibration with base space $B\pi^\wedge_p$ and fibre $BG^\wedge_p$. In this paper we study homotopy group extensions for finite groups. We observe that there is a strong analogy between homotopy group extensions and ordinary group extensions. The study involves investigating the space of self homotopy equivalences of a p-completed classifying space. In particular we show that under the appropriate assumption on $G$, the identity component of this space is homotopy equivalent to $BZ(G)$, the classifying space of the centre of $G$. We proceed by studying the group of components. We show that this group maps into a group of natural equivalences of a certain functor with kernel and cokernel, which are computable in terms of the first and second derived functors of the inverse limit for a certain diagram of abelian groups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Cohen-Levi/simp-models Combinatorial models for iterated loop spaces By: Fred Cohen and Ran Levi The objective of this paper is to provide free simplicial group models for the functors $\Omega^n X$ and $\Omega^n\Sigma^{n+k}X$. The models are based on classical constructions in simplicial homotopy theory. Specifically, Milnor's functor F, Kan's loops group functor G and the Moore loop space construction $\Omega$ are used to produce these models. The models are given in terms of free groups with specific generators and the formulas defining the simplicial operators are given. The utility of these models is that in them certain maps can be written explicitly in a relatively easy way. To illustrate this a null homotopy of the commutator map on a double loop space is given. Similar ideas are used to give a model for pointed mapping spaces out of a Riemann surface. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Cohen-Levi/stunted-proj ON THE HOMOTOPY TYPE OF INFINITE STUNTED PROJECTIVE SPACES By: Fred Cohen and Ran Levi Let $X_n$ denote the infinite stunted projective space ${\Bbb R}P^\infty/{\Bbb R}P^{n-1}$. In this note we study the homotopy type of this family of spaces. In particular we show that for $n=2 $ and 4, the space $X_n$ splits after looping once and for $n=3$ after looping four times and passing to connected covers. In each case the factors are loop spaces on naturally occuring finite complexes. These result generalise to higher values of $n$, but in those cases without a splitting result. The splittings enable us to carry out a calculation of low dimensional homotopy and loop space homology for these spaces, which complements a computer calculation of Sergeraert and Smirnov. A number of interesting related facts and questions is also discussed. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McGibbon-Strom/numphant Numerical invariants of phantom maps C. A. McGibbon and Jeffrey Strom Wayne State University and Dartmouth College Two numerical homotopy invariants of phantom maps, the Gray index G(f) and the essential category weight E(f), are studied. The possible values of these invariants are determined. In certain cases bounds on these values are given in terms of rational homotopy data. Examples are provided showing that the Gray index can take any positive finite value. For certain cases it is shown that every essential phantom f: X --> Y has finite Gray index. However it is also shown that there exist spaces, e. g. CP^\infty, which are the domains of essential phantoms with infinite index. The same type of analysis is carried out on the essential category weight of a phantom map. If the loop space of X is homotopy equivalent to a finite complex, then every phantom f: X --> Y has E(f) = \infty. However, in certain other cases it is shown that E(f) is strictly less than the rational Lusternik-Schnirelmann category of the domain. A homotopy classification of phantoms f: K(Z, n)--> S^m is given along with the values of E(f). The invariants G and E provide decreasing filtrations on the set of homotopy classes of phantoms from X to Y. A third filtration on this set is introduced for certain special targets. When the rational cohomology of the domain X is finitely generated, this filtration enables one to reduce the search for essential phantoms (into finite type targets) to a finite list of spheres. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/TanK-XuK/dickson 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. Clarence has explicit instructions for the form of this abstract: see http://hopf.math.purdue.edu/pub/new-html/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.