Date: Tue, 6 Oct 1998 04:39:47 -0400 From: Mark Hovey Sender: hovey@picard.math.wesleyan.edu Subject: new Hopf listings Six new papers this time, all by Smiths. The ones by Larry are only available as .ps files. Mark Hovey New papers uploaded to hopf between 9/26/98 and 10/6/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/JRSmith/m-homotop Title: m-structures determine integral homotopy type Author: Justin R. Smith (jsmith@mcs.drexel.edu) Comments: 26 pages. LaTeX2e with XYPic, version 3.3. Uses XYPic fonts. MSC-class: 55R91 (Primary) 18G30 (Secondary) This paper proves that the functor $\mathscr{C}(*)$ that sends pointed, simply-connected CW-complexes to their chain-complexes equipped with diagonals and iterated higher diagonals, determines their integral homotopy type --- even inducing an equivalence of categories between the category of CW-complexes up to homotopy equivalence and a certain category of chain-complexes equipped with higher diagonals. Consequently, $\mathscr{C}(*)$ is an algebraic model for integral homotopy types similar to Quillen's model of rational homotopy types. For finite CW complexes, our model is finitely generated. Our result implies that the geometrically induced diagonal map with all ``higher diagonal'' maps (like those used to define Steenrod operations) collectively determine integral homotopy type. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/LSmith/feshbach %%%%%%%%%%%%%% %% feshbach.txt : summary of contents of feshbach.ps %%%%%%%%%%%%%% Let $\rho : G \hookrightarrow \GL(n, \F)$ be a representation of a finite group $G$ over the field $\F$\/, and let $\F[V]$ be the algebra of polynomial functions on the vector space $V = \F^n$\/. The group $G$ acts on $V$ and hence also on $\F[V]$ via the representation $\rho$ and we denote by $\F[V]^G$ the subalgebra of $G$-invariant polynimials in $\F[V]$\/. (We refer to \cite{poly} for basic facts about the invariant theory of finite groups.) The {\bf transfer homomorphism} \[ \Tr^G : \F[V] \longrightarrow \F[V]^G \] is defined by \[ \Tr^G(f) = \sum_{g \in G} gf\comma\qquad \forall f \in \F[V]\period \] If the characteristic of $\F$ is relatively prime to the order of $G$ then the transfer map is surjective. By contrast, if the characteristic of $\F$ divides the order of $G$ the image of the transfer, $\Im(\Tr^G)$\/, is an ideal of height strictly less than $n$ \cite{Feshtwo}, \cite{kuhnigk} \cite{survey}. We begin this note by reproving this result as well as M. Feshbach's unpublished description of the radical of $\Im(\Tr^G)$\/. We then go on to describe the variety defined by the extended ideal $(\Im(\Tr^G))^e \subset \F[V]$\/. It turns out that this variety has a particularly elegant description, namely it is the union of the fixed point sets of the elements of order $p$ in $G$\/, where $p$ is the characteristic of $\F$\/. As an application we show that the Dickson polynomial of least degree $\dick_{n, n-1}$ is always a nonzero divisor in the quotient ring $\F[V]/\Im(\Tr^G)$\/, so this quotient ring has positive homological codimension (or depth). 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/LSmith/koszul %%%%%%%%%%%%%%%%%%%%%% %% koszul.txt Invariant Theory and the Koszul Complex (Toulouse mars 1997) %% summary of the contents of koszul.ps %%%%%%%%%%%%%%%%%%%%%% Let $\rho : G \hookrightarrow \GL(n,\ \F)$ be a representation of a finite group over the field $\F$ of characteristic $p$\/, and $h_1\commadots h_m \in \F[V]^G$ ivariant polynomials that form a regular sequence in $\F[V]$\/. In this note we introduce a tool to study the problem of whether they form a regular sequence in $\F[V]^G$\/. Examples show they need not. We define the cohomology of $G$ with coefficients in the Koszul complex \[ (\KK,\ \partial) = (\F[V] \otimes E(s^{-1}h_1\commadots s^{-1}h_n),\ \partial (s^{-1} h_i) = h_i\ :\ i=1\commadots n)\mkern 3mu, \] which we denote by $H^*(G;\ (\KK,\ \partial))$\/, and use it to study the homological codimension of rings of invariants of permutation representations of the cyclic group of order $p$\/, for $p \neq 0$\/, and to answer the above question in this case. If you enjoyed the Eilenberg-Moore spectral sequence then you may also find this paper interesting. It leads to the following problem in the cohomology of groups. Let $\rho : \Z/p \hookrightarrow \GL(n, \F)$ be a representation of the cyclic group of order $p$\/, where $p \in \N$ is an odd prime. Then $H^1(\Z/p; \F[V])$ and $H^2(G; \F[V])$ are both modules over the Dickson algebra $\D^*(n)$\/. Are they isomorphic? Perhaps stably? Do they at least have the ame projective dimension? They certainly have the same Poincar\'e series. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/LSmith/regular %%%%%%%%%%%%%%%%%%% %% regular.txt : summary of the contents of regular.ps %%%%%%%%%%%%%%%%%%% Let $G$ be a finite group and $\rho : G \hra \GL(n, \C)$ a complex representation. Barbara Schmid has shown that the algebra of invariant polynomial functions $\C[V]^G$ on the vector space $V = \C^n$ is generated by homogeneous polynomials of degree at most $\beta$\/, where $\beta$ is the largest degree of a generator in a minimal generating set for $\C[\reg_\C(G)]^G$\/, and $\reg_\C(G)$ is the complex regular representation of $G$\/. In this note we give a new proof of this result, and at the same time extend it to fields $\F$ whose characteristic $p$ is larger than $|G|$\/, the order of the group $G$\/. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/LSmith/squeeze %%%%%%%%%%%%%%%%%%%%%% % squeeze.txt : summary of contents of squeeze.ps %%%%%%%%%%%%%%%%%%%%%% Let $\rho : G \hookrightarrow \GL(n,\ \F)$ be a faithful representation of the finite group $G$ over the field $\F$\/. In 1916 E. Noether proved that for $\F$ of characteristic zero the ring of invariants $\F[V]^G$ is generated as an algebra by the invariant polynomials of degree at most $|G|$\/. This result has been generalized to the case where the characteristic of $\F$ is greater than $|G|$\/, or when the characteristic of $\F$ is prime to the order of $G$ and the group $G$ is solvable. In this note we prove that if Noether's bound fails in the nonmodular case, then it fails for a finite nonabelian simple group. We then show how yet another reworking of Noether's argument leads to a proof that Noether's bound holds in the nonmodular case for the alternating groups. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/LSmith/wreath %%%%%%%%%%%%%%%%%%% %% wreath.txt : summary of contents of wreath.ps %%%%%%%%%%%%%%%%%%% Let $\rho : G \hra \GL(n,\ \F)$ be a faithful representation of the finite group $G$ over the field $\F$\/. In 1916 E. Noether proved that for $\F$ of characteristic zero the ring of invariants $\F[V]^G$ is generated as an algebra by the invariant polynomials of degree at most $|G|$\/. This result has been generalized to the case where the characteristic of $\F$ is greater than $|G|$\/, or when the characteristic of $\F$ is prime to the order of $G$ and the group $G$ is solvable. The result for fields whose characteristic is greater than the order of the group follows directly from the fact that in this case the ring of invariants is generated by orbit Chern classes. However this can fail in the more general nonmodular situation; e.g., for the quaternion subgroup of $\GL(2, \F_3). In this note we show how to rework Noether's proof to yield a more more refined notion of orbit Chern classes (here called the {\bf fine} orbit Chern classes). This leads to a very general result where Noether's bound holds, that includes both the previously mentioned theorems, and in particular, shows that Noether's bound holds for the alternating groups in the nonmodular case. We also illustrate with the example of the quaternion group that these new, fine, orbit Chern classes are sufficient to generate the ring of invariants in cases where orbit Chern classes fail to do so. An interesting group representational problem that arises in this connection is the following. If $G$ is a finite group of order $n$ and $\reg_G : G \longrightarrow \Sigma_n$ its' regular representation, then the image of $G$ in $\Sigma_n$ is not {\it arbitrary}\/, in the sense that there is always a proper subgroup of $\Sigma_n$\/, depending only on the orders of the groups in a chain of maximal subgroups of $G$\/, that contains the image. Is there a {\it smallest } such subgroup? If so how does it depend on $G$? What is its structure? This note contains a very crude solution which is an iterated wreath product of symmetric groups. This factorization of $\reg_G$ certainly has applications to the study of the Chern classes of the regular representation of $G$\/. ---------------------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. 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/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. ------- End of forwarded message -------