Happy New Year! 10 new papers this time, from Anton, DavisDaniel, Harper (2) (that is John E. Harper of Notre Dame, not John Harper of Rochester), Hovey-Lockridge (2), Neusel (2), and Yagita (2). Mark Hovey New papers appearing on hopf between 11/29/07 and 1/18/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Anton/homologicalSymbols Title: Homological symbols and the Quillen Conjecture Author(s): Marian F. Anton Abstract: We formulate a "correct" version of the Quillen conjecture on the cohomology of linear groups by defining an unstable form of Milnor K-theory and show that this version can be solved by a finite process. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/fibrantmodel4 Title: Explicit fibrant replacement for discrete G-spectra Author: Daniel G. Davis Abstract: If C is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is well-known that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when G is a profinite group, the fibrant objects in the model category of discrete G-spectra are also difficult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in C, under certain finiteness conditions. Similarly, in this paper, we show that if G has finite virtual cohomological dimension and X is a discrete G-spectrum, then there is an explicit fibrant model for X. Also, we give several applications of this concrete model related to closed subgroups of G. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Harper/modules-operads-monoidal Title: Homotopy theory of modules over operads and non-Sigma operads in monoidal model categories Author: John E. Harper Author's mailing address: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA Comments: 49 pages, uses xy-pic; we have compiled the .tex file without using the better looking dvips,ps options in xy-pic, so the .dvi file should be device independent, but the diagrams may appear jagged etc. Abstract: This paper studies the existence of model category structures on modules and algebras over operads in monoidal model categories. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Harper/modules-spectra Title: Homotopy theory of modules over operads in symmetric spectra Author: John E. Harper Author's mailing address: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA Comments: 21 pages, uses xy-pic; we have compiled the .tex file without using the better looking dvips,ps options in xy-pic, so the .dvi file should be device independent, but the diagrams may appear jagged etc. Abstract: This paper establishes model category structures on modules and algebras over operads in symmetric spectra, and studies when a morphism of operads induces a Quillen equivalence between corresponding categories of modules (resp. algebras) over operads. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge/gen-gen-hyp The ghost dimension of a ring Mark Hovey Wesleyan University Keir Lockridge Wake Forest University We introduce the concept of the ghost dimension of a ring R. This is the longest nontrivial chain of maps in the derived category emanating from a perfect complex such that each map is zero on homology. We show that the ghost dimension of R is less than or equal to the weak dimension of R, with equality if R is coherent or has weak dimension 1. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge/triproj Triangulations of Projective Modules Mark Hovey Wesleyan University Keir Lockridge Wake Forest University We show that the category of projective modules over a graded commutative ring admits a triangulation with respect to module suspension if and only if the ring is a finite product of graded fields and exterior algebras on one generator over a graded field (with a unit in the appropriate degree). We also classify the ungraded commutative rings for which the category of projective modules admits a triangulation with respect to the identity suspension. Applications to two analogues of the generating hypothesis in algebraic topology are given, and we translate our results into the setting of modules over a symmetric ring spectrum or $S$-algebra, where semisimple and von Neumann regular ring spectra are defined and discussed. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/hilbert Title: On the Hilbert Ideal Author: Mara D. Neusel Abstract: We prove the Hilbert number conjecture. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/schmid Title: Degree Bounds and the Regular Representation Author: Mara D. Neusel Abstract: This is a revised version of the paper with the same name posted during last summer. We prove Schmid's inequality in the general case, and Killius' conjecture for permutation representations. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/coniveaufilt Title: Coniveau filtration of cohomology of group Author: Nobuaki Yagita Abstract: We consider natural filtrations of mod p cohomology of a classifying space BG for a compact Lie group G, such that the reduced power operation preserves the filtration but the Bockstein opration descends the filtration degree one. An example of such filtrations is defined by the image from the motivic cohomology. For example, when BG=BO(n), this filtration coincides the coniveau filtration defined by Grothendieck. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/torsorEE Title: Note on Chow rings of nontrivial G-torsors over a field. Author: Nobuaki Yagita Abstract: Let G(k) be a split reductive group over a field k corresponding to a compact Lie group G. Let E be a nontrivial G(k)-torsor over a field k. In this paper we study the Chow ring of nontrivial G(k)-torsors E. For example when (G,p)=(F_4,3), we see that the positive degree of the mod 3 Chow ring of E is zero. ---------------- ---------------------------------------------- 4 new papers this time, from Blanc, Harper (again, this is John E. Harper of Notre Dame, not John Harper of Rochester), Kuhn, and Sati-Schreiber-Stasheff. Mark Hovey New papers appearing on hopf between 1/18/07 and 3/3/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc/quil Title: Generalized Andre-Quillen Cohomology Author: David Blanc Address: Dept. of Mathematics, U. Haifa, Haifa, Israel Abstract: We explain how the approach of Andre and Quillen to defining cohomology and homology as suitable derived functors extends to generalized (co)homology theories, and how this identification may be used to study the relationship between them. As a side benefit, we clarify exactly what assumptions on an (algebraic) category are needed in order for the approach of Beck and Andre-Quillen to work. We also show how the description may be applied to construct universal coefficient and reverse Adams spectral sequences. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Harper/QuillenHomology Title: Bar constructions and Quillen homology of modules over operads Author: John E. Harper Author's mailing address: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA Comments: 33 pages, uses xy-pic; compiled the .tex file without using the dvips,ps options in xy-pic, to ensure .dvi is device independent, but diagrams may now appear jagged, etc. Abstract: This paper shows that Quillen derived homology of modules and algebras over an operad, for symmetric sequences of symmetric spectra and unbounded chain complexes, can be calculated using simplicial bar constructions, modulo cofibrancy conditions. Working with several model category structures, a homotopical proof is given, after showing that certain homotopy colimits in modules and algebras over an operad can be easily understood. The key result here, which is at the heart of this paper, is showing that the forgetful functor commutes with certain homotopy colimits. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/telescopic Title: A guide to telescopic functors Author: Nicholas J. Kuhn Address: University of Virginia, Charlottesville, VA 22904 Abstract: In the mid 1980's, Pete Bousfield and I constructed certain p--local `telescopic' functors Phi_n from spaces to spectra, for each prime p and each positive integer n. These have striking properties that relate the chromatic approach to homotopy theory to infinite loopspace theory: roughly put, the spectrum Phi_n(Z) captures the v_n periodic homotopy of a space Z. Recently there have been a variety of new uses of these functors, suggesting that they have a central role to play in calculations of periodic phenomena. Here I offer a guide to their construction, characterization, application, and computation. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Sati-Schreiber-Stasheff/LCon Title: L-infinity algebra connections and applications to String- and Chern-Simons n-transport Authors: Hisham Sati, Urs Schreiber and Jim Stasheff Abstract: We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L-infinity -algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their parallel transport and quantization. It is known that over a D-brane the Kalb-Ramond background field of the string restricts to a 2-bundle with connection (a gerbe) which can be seen as the obstruction to lifting the PU(H)-bundle on the D-brane to a U(H)-bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1) -> U(H) -> PU(H) to higher categorical central extensions, like the String-extension B U(1) -> String(G) -> G. Here the obstruction to the lift is a 3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a String-structure. We discuss how to describe this obstruction problem in terms of Lie n-algebras and their corresponding categorified Cartan-Ehresmann connections. Generalizations even beyond String- extensions are then straightforward. For G = Spin(n) the next step is "Fivebrane structures'' whose existence is obstructed by certain generalized Chern-Simons 7-bundles classified by the second Pontrjagin class. -------------------- ---------------------------------- My semester has ended, my daughter has chosen a college, and I finally have some time to deal with Hopf. Sorry for the long delay. 7 new papers this time, from Blanc-Johnson-Turner, Broto-Moller-Oliver, Carlson-Chebolu-Minac, Neusel-Sezer, Serikbaev-Bitibaeva-Yerzhanov-Myrazukulov, Yagita (2). Mark Hovey New papers appearing on hopf between 3/3/07 and 5/15/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Johnson-Turner/lgss Title: Local-to-global spectral sequences for the cohomology of diagrams Authors: David Blanc, Mark W. Johnson, and James M. Turner Address: Department of Mathematics, University of Haifa, 31905 Haifa, Israel Department of Mathematics, Penn State Altoona, Altoona, PA 16601, USA Department of Mathematics, Calvin College, Grand Rapids, MI 49546, USA Abstract: The cohomology of diagrams arises in various areas of mathematics, such as deformation theory, classifying diagrams of groups, and in homotopy theory, in the context of the rectification of homotopy-commutative diagrams, and thus in the study of higher homotopy and cohomology operations. For this purpose we construct ``local-to-global'' spectral sequences for the cohomology of a diagram, which can be used to compute the cohomology of the full diagram in terms of smaller pieces. We also explain why such a local-to-global approach is relevant to higher operations. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Moller-Oliver/bmo1 Authors: C. Broto, J. M. M\o{}ller, and B. Oliver Title: Equivalences between fusion systems of finite groups of Lie type Subject class: Primary 20D06, Secondary 55R37, 20D20 keywords: groups of Lie type, fusion systems, classifying spaces, p-completion Abstract: We prove, for certain pairs $G,G'$ of finite groups of Lie type, that the $p$-fusion systems $F_p(G)$ and $F_p(G')$ are equivalent. In other words, there is an isomorphism between a Sylow $p$-subgroup of $G$ and one of $G'$ which preserves $p$-fusion. This occurs, for example, when $G=\Gamma(q)$ and $G'=\Gamma(q')$ for a simple Lie ``type'' $\Gamma$, and $q$ and $q'$ are prime powers, both prime to $p$, which generate the same closed subgroup of $p$-adic units. Our proof uses homotopy theoretic properties of the $p$-completed classifying spaces of $G$ and $G'$, and we know of no purely algebraic proof of this result. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Carlson-Chebolu-Minac/fgt Finite generation of Tate cohomology Jon F. Carlson Department of Mathematics University of Georgia Athens, GA 30602, USA Sunil K. Chebolu Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada Jan Minac Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada Abstract: Let G be a finite group and let k be a field of characteristic p. If M is a finitely generated indecomposable non-projective kG-module, we conjecture that the Tate cohomology of G with coefficients in M is finitely generated over the Tate cohomology ring of G if and only if the support variety V_G(M) of M is equal to the entire maximal ideal spectrum V_G(k). We prove various results all of which support this conjecture. It is also shown that all finitely generated kG-modules over a group G have finitely generated Tate cohomology if and only if G has periodic cohomology. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/separating TITLE: Characterizing Separating Invariants AUTHORS: Mara D.~Neusel and M\"uf\.it Sezer ABSTRACT: We study separating algebras for rings of invariants of finite groups. We give an algebraic characterization for these. Furthermore, we describe a particularly nice separating subalgebra for rings of invariants of p-groups in characteristic p. This leads to a characterization of subalgebras such that their p-root and integral closure is equal to the ring of invariants. Finally, we present separating sets for invariants rings of nonmodular representations of abelian groups whose size depends only on the degree of the representation. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Serikbaev-Bitibaeva-Yerzhanov-Myrazukulov/flow [Your moderator was in a quandary over this paper, which is clearly not remotely algebraic topology, but decided to err on the side of openness] Integrable isotropic geometrical flows and Heisenberg ferromagnets N.S.Serikbaev, Zh.M.Bitibaeva, K.K.Yerzhanov, R.Myrzakulov* Department of General and Theoretical Physics, Eurasian National University, Astana, 010008, Kazakhstan Abstract Geometrical Flows (GF) play an important role in modern mathematics and physics. In this letter we have considered some integrable isotropic GF Ricci Flows (RF) and mean curvature flows (MCF) ~ which are related with integrable Heisenberg ferromagnets. In 2+1 dimensions, these GF have a singularity at t = t0. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/motsplitG Title: Note on the mod p motivic cohomology of algebraic groups. Author: Nobuaki Yagita AMS classification numbers: 55P35, 57T25. Adress of Author: Faculty of Education, Ibaraki University, Ibaraki, Japan. Abstract: Let G_k be a split reductive group over a field k of ch(k)=0 corresponding to a compact Lie group G. In this paper, we show that the mod p motivic cohomology is isomorphic to the tensor product of the usual mod p cohomology H^*(G;Z/p) and the motivic cohomology H^{*,*'}(Spec(k);Z/p), when G=SO_n,G_2,F_4,E_6. We also give an example of nonsplit case (G=G_2,p=2,k=R) which does not hold the above isomorphism. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/realchow Title: Note on motivic cohomology of anisotropic real quadrics. Author: Nobuaki Yagita AMS classification numbers: 55P35, 57T25. Adress of Author: Faculty of Education, Ibaraki University, Ibaraki, Japan. Abstract: In this paper, we compute the mod 2 motivic cohomology H^{*,*'}(X;Z/2) for the anisotropic quadric X over R the field of real numbers. ----------- ------------------------------------------------------------ Hmm, I seem to be losing momentum on these Hopf announcements. 9 new papers this time, from Behrens-Davis, Gillespie-Hovey, Gonzalez-Landweber, Hovey (2), Hovey-Lockridge, Kashiwabara, Kuhn, and Monico-Neusel. Mark Hovey New papers appearing on hopf between 5/15/08 and 9/6/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens-Davis/funcspec15 Title of Paper: The homotopy fixed point spectra of profinite Galois extensions Authors: Mark Behrens, Daniel G. Davis AMS Classification numbers: 55N20, 55P43 ArXiv ID: math.AT/0808.1092 Abstract: Let E be a k-local profinite G-Galois extension of an E_\infty-ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes's Galois correspondence extends to the profinite setting. We show the function spectrum F_A((E^hH)_k, (E^hK)_k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]])^hK)_k where H and K are closed subgroups of G. Applications to Morava E-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived functor of fixed points. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Gillespie-Hovey/gorenstein Gorenstein model structures and generalized derived categories James Gillespie and Mark Hovey In a previous paper, the second author introduced the Gorenstein projective and Gorenstein injective model structures on $R$-Mod, the category of $R$-modules, where $R$ is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce; the Gorenstein flat model structure. The homotopy category with respect to each of these is called the stable homotopy category of $R$. Here we show that if such a ring $R$ has finite global dimension, the graded ring $R[x]/(x^2)$ is Gorenstein and the three associated Gorenstein model structures on $R[x]/(x^2)$-Mod, the category of graded $R[x]/(x^2)$-modules, are nothing more than the usual projective, injective and flat model structures on Ch($R$), the category of chain complexes of $R$-modules. Although these correspondences only recover these model structures on Ch($R$) when $R$ has finite global dimension, we can set $R = \Z$ and use general techniques from model category theory to lift the projective model structure from Ch($\Z$) to Ch($R$) for an arbitrary ring $R$. This shows that homological algebra is a special case of Gorenstein homological algebra. Moreover, this method of constructing and lifting model structures carries through when $\Z[x]/(x^2)$ is replaced by many other graded Gorenstein rings (or Hopf algebras, which lead to monoidal model structures). This gives us a natural way to generalize both chain complexes over a ring $R$ as well as the derived category of $R$ and we give some examples of such generalizations. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Gonzalez-Landweber/symmotion Title: Symmetric topological complexity of projective and lens spaces Authors: Jesus Gonzalez and Peter Landweber Adresses: Departamento de Matematicas, CINVESTAV-IPN, Mexico City 07000, MEXICO Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA Abstract: For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps, and (c) the topological complexity are known to be three facets of the same problem. This paper describes the corresponding relationship between the symmetrized versions of (b) and (c) to the Euclidean embedding dimension of projective spaces. Extensions to the case of lens spaces and complex projective spaces are discussed. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/bpbp1 The homotopy of MString and MU<6> at large primes Mark Hovey We use Hopf rings to compute the homotopy rings $\pi_{*}\MO{8}$ and $\pi_{*}\MU{6}$ at primes $>3$. In this case, the additive structure is well-known, but the ring structure is not polynomial. Instead, these rings are quotients of polynomial rings by infinite regular sequences. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/intersection-homology Intersection homological algebra Mark Hovey We investigate the abelian category which is the target of intersection homology. Recall that, given a stratified space $X$, we get intersection homology groups $I^{\perversity{p}}H_{n}X$ depending on the choice of an $n$-perversity $\perversity{p}$. The $n$-perversities form a lattice, and we can think of $IH_{n}X$ as a functor from this lattice to abelian groups, or more generally $R$-modules. Such perverse $R$-modules form a closed symmetric monoidal abelian category. We study this category and its associated homological algebra. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge/ssrs Semisimple ring spectra Mark Hovey and Keir Lockridge Abstract. We define global dimension and weak dimension for the structured ring spectra that arise in algebraic topology. We provide a partial classification of ring spectra of global dimension 0, the semisimple ring spectra of the title. These ring spectra are closely related to classical rings whose projective modules admit the structure of a triangulated category. Applications to two analogues of the generating hypothesis in algebraic topology are given. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Kashiwabara/hqs2008 The Hopf ring for Bockstein-nil homology of QSn Takuji Kashiwabara Institut Fourier UMR au CNRS 5582 BP 74 38402, St-Martin-d'H`eres CEDEX FRANCE In this paper we give a generator-relation description of mod $p$ Bockstein-nil homology of $QS^n$ for odd prime $p$. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/loopSS Title: Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology Author: Nicholas J. Kuhn Address: University of Virginia, Charlottesville, VA 22904 AMS classification number: 55S10 arXiv:0806.3281 Abstract: We prove a strengthened version of a theorem of Lionel Schwartz that says that certain modules over the Steenrod algebra cannot be the mod 2 cohomology of a space. What is most interesting is our method, which replaces his iterated use of the Eilenberg--Moore spectral sequence by a single use of the spectral sequence converging to the mod 2 cohomology of Omega^nX obtained from the Goodwillie tower for the suspension spectrum of Omega^nX. Much of the paper develops basic properties of this spectral sequence. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Monico-Neusel/counting TITLE: Counting Special Monomials AUTHORS: Chris Monico and Mara D. Neusel Department of Mathematics and Statistics, MS 1042, Texas Tech University, Lubbock, Texas 79409 ABSTRACT: In this paper we study the number of orbits of special monomials of G acting by permutations on the polynomials in n variables. We give formulae for several crucial families of groups, for direct sums of representations, as well as for vector invariants. In addition we give two algorithms for arbitrary permutation groups, one relying on the geometry of G acting on the underlying vector space, the other relying on the representation theory of the symmetric groups. ----------------- BABYL OPTIONS: -*- rmail -*- Version: 5 Labels: Note: This is the header of an rmail file. Note: If you are seeing it in rmail, Note: it means the file has no messages in it.  1, edited,, Mail-from: From dmd1@lehigh.edu Sat Jan 17 18:19:15 1998 Received: from nss4.cc.Lehigh.EDU (root@nss4.CC.Lehigh.EDU [128.180.1.13]) by mail.wesleyan.edu (8.8.6/8.7.3) with ESMTP id SAA10234 for ; Sat, 17 Jan 1998 18:20:52 -0500 (EST) Received: from ns4-1.CC.Lehigh.EDU (root@ns4-1.CC.Lehigh.EDU [128.180.1.42]) by nss4.cc.Lehigh.EDU (8.8.8/8.8.5) with ESMTP id SAA119110; Sat, 17 Jan 1998 18:23:02 -0500 Received: (from dmd1@localhost) by ns4-1.CC.Lehigh.EDU (8.8.5/8.8.5) id SAA39528; Sat, 17 Jan 1998 18:19:17 -0500 Message-Id: <199801172319.SAA39528@ns4-1.CC.Lehigh.EDU> Date: Sat, 17 Jan 1998 18:19:15 EST From: dmd1@lehigh.edu (DONALD M. DAVIS) X-Mailer: SENDM [Version 2.0.17] Subject: new Hopf listings To: Distribution.List@lehigh.edu (toplist) Content-Type: text X-UIDL: aacd4710beaa4a6483935a131ded8f1b Lines: 256 Xref: picard.math.wesleyan.edu davis:291 X-Gnus-Newsgroup: davis:291 Sun Jan 18 06:29:52 1998 *** EOOH *** Date: Sat, 17 Jan 1998 18:19:15 EST From: dmd1@lehigh.edu (DONALD M. DAVIS) Subject: new Hopf listings To: Distribution.List@lehigh.edu (toplist) Content-Type: text Xref: picard.math.wesleyan.edu davis:291 *** EOOH *** Date: Sat, 17 Jan 1998 18:19:15 EST From: dmd1@lehigh.edu (DONALD M. DAVIS) Subject: new Hopf listings To: Distribution.List@lehigh.edu (toplist) X-UIDL: aacd4710beaa4a6483935a131ded8f1b Xref: picard.math.wesleyan.edu davis:291 X-Gnus-Newsgroup: davis:291 Sun Jan 18 06:29:52 1998 Hmm, I seem to be losing momentum on these Hopf announcements. 9 new papers this time, from Behrens-Davis, Gillespie-Hovey, Gonzalez-Landweber, Hovey (2), Hovey-Lockridge, Kashiwabara, Kuhn, and Monico-Neusel. Mark Hovey New papers appearing on hopf between 5/15/08 and 9/6/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens-Davis/funcspec15 Title of Paper: The homotopy fixed point spectra of profinite Galois extensions Authors: Mark Behrens, Daniel G. Davis AMS Classification numbers: 55N20, 55P43 ArXiv ID: math.AT/0808.1092 Abstract: Let E be a k-local profinite G-Galois extension of an E_\infty-ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes's Galois correspondence extends to the profinite setting. We show the function spectrum F_A((E^hH)_k, (E^hK)_k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]])^hK)_k where H and K are closed subgroups of G. Applications to Morava E-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived functor of fixed points. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Gillespie-Hovey/gorenstein Gorenstein model structures and generalized derived categories James Gillespie and Mark Hovey In a previous paper, the second author introduced the Gorenstein projective and Gorenstein injective model structures on $R$-Mod, the category of $R$-modules, where $R$ is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce; the Gorenstein flat model structure. The homotopy category with respect to each of these is called the stable homotopy category of $R$. Here we show that if such a ring $R$ has finite global dimension, the graded ring $R[x]/(x^2)$ is Gorenstein and the three associated Gorenstein model structures on $R[x]/(x^2)$-Mod, the category of graded $R[x]/(x^2)$-modules, are nothing more than the usual projective, injective and flat model structures on Ch($R$), the category of chain complexes of $R$-modules. Although these correspondences only recover these model structures on Ch($R$) when $R$ has finite global dimension, we can set $R = \Z$ and use general techniques from model category theory to lift the projective model structure from Ch($\Z$) to Ch($R$) for an arbitrary ring $R$. This shows that homological algebra is a special case of Gorenstein homological algebra. Moreover, this method of constructing and lifting model structures carries through when $\Z[x]/(x^2)$ is replaced by many other graded Gorenstein rings (or Hopf algebras, which lead to monoidal model structures). This gives us a natural way to generalize both chain complexes over a ring $R$ as well as the derived category of $R$ and we give some examples of such generalizations. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Gonzalez-Landweber/symmotion Title: Symmetric topological complexity of projective and lens spaces Authors: Jesus Gonzalez and Peter Landweber Adresses: Departamento de Matematicas, CINVESTAV-IPN, Mexico City 07000, MEXICO Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA Abstract: For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps, and (c) the topological complexity are known to be three facets of the same problem. This paper describes the corresponding relationship between the symmetrized versions of (b) and (c) to the Euclidean embedding dimension of projective spaces. Extensions to the case of lens spaces and complex projective spaces are discussed. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/bpbp1 The homotopy of MString and MU<6> at large primes Mark Hovey We use Hopf rings to compute the homotopy rings $\pi_{*}\MO{8}$ and $\pi_{*}\MU{6}$ at primes $>3$. In this case, the additive structure is well-known, but the ring structure is not polynomial. Instead, these rings are quotients of polynomial rings by infinite regular sequences. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/intersection-homology Intersection homological algebra Mark Hovey We investigate the abelian category which is the target of intersection homology. Recall that, given a stratified space $X$, we get intersection homology groups $I^{\perversity{p}}H_{n}X$ depending on the choice of an $n$-perversity $\perversity{p}$. The $n$-perversities form a lattice, and we can think of $IH_{n}X$ as a functor from this lattice to abelian groups, or more generally $R$-modules. Such perverse $R$-modules form a closed symmetric monoidal abelian category. We study this category and its associated homological algebra. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge/ssrs Semisimple ring spectra Mark Hovey and Keir Lockridge Abstract. We define global dimension and weak dimension for the structured ring spectra that arise in algebraic topology. We provide a partial classification of ring spectra of global dimension 0, the semisimple ring spectra of the title. These ring spectra are closely related to classical rings whose projective modules admit the structure of a triangulated category. Applications to two analogues of the generating hypothesis in algebraic topology are given. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Kashiwabara/hqs2008 The Hopf ring for Bockstein-nil homology of QSn Takuji Kashiwabara Institut Fourier UMR au CNRS 5582 BP 74 38402, St-Martin-d'H`eres CEDEX FRANCE In this paper we give a generator-relation description of mod $p$ Bockstein-nil homology of $QS^n$ for odd prime $p$. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/loopSS Title: Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology Author: Nicholas J. Kuhn Address: University of Virginia, Charlottesville, VA 22904 AMS classification number: 55S10 arXiv:0806.3281 Abstract: We prove a strengthened version of a theorem of Lionel Schwartz that says that certain modules over the Steenrod algebra cannot be the mod 2 cohomology of a space. What is most interesting is our method, which replaces his iterated use of the Eilenberg--Moore spectral sequence by a single use of the spectral sequence converging to the mod 2 cohomology of Omega^nX obtained from the Goodwillie tower for the suspension spectrum of Omega^nX. Much of the paper develops basic properties of this spectral sequence. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Monico-Neusel/counting TITLE: Counting Special Monomials AUTHORS: Chris Monico and Mara D. Neusel Department of Mathematics and Statistics, MS 1042, Texas Tech University, Lubbock, Texas 79409 ABSTRACT: In this paper we study the number of orbits of special monomials of G acting by permutations on the polynomials in n variables. We give formulae for several crucial families of groups, for direct sums of representations, as well as for vector invariants. In addition we give two algorithms for arbitrary permutation groups, one relying on the geometry of G acting on the underlying vector space, the other relying on the representation theory of the symmetric groups. ---------------------Instructions----------------------------- To subscribe or unsubscribe to this listserv, go to https://lists.lehigh.edu/mailman/listinfo/algtop-l. To see past issues of new submissions to Hopf, go to http://math.wesleyan.edu/~mhovey/archive/ To get the papers listed above, go to the URL listed. The general Hopf archive URL is http://hopf.math.purdue.edu There is a web form for submitting papers to Hopf on this site as well. You should submit an abstract as well. Clarence has explicit instructions for the form of this abstract: see http://hopf.math.purdue.edu/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 at math.purdue.edu telling him what you have uploaded. The largest archive of math preprints is at http://arxiv.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 the arXiv, send e-mail to math@arxiv.org with subject line "subscribe" (without quotes), and with the body of the message "add AT" (without quotes). I am solely responsible for these messages---don't send complaints about them to Clarence. Thanks to Clarence for creating and maintaining the archive.  -------------------------------------------- Sorry once again for the long delay. 12 new papers this time, from BrownR, BrownR-Sivera, Glover-Henn, Harper, Henn-Karamanov-Mahowald, LinJP, SmithL (2 papers), SmithL-Stong (4 papers). Mark Hovey New papers appearing on hopf between 9/6/08 and 11/20/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/fields-art Author: Ronald Brown Title: Crossed complexes and higher homotopy groupoids as non commutative tools for higher dimensional local-to-global problems Abstract: We outline the main features of the definitions and applications of crossed complexes and cubical $\omega$-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain some non commutative results and compute some homotopy types in non simply connected situations. Web page: www.bangor.ac.uk/r.brown Futher information: This is a revised version (2008) of a paper published in Fields Institute Communications 43 (2004) 101-130, which was an extended account of a lecture given at the meeting on `Categorical Structures for Descent, Galois Theory, Hopf algebras and semiabelian categories', Fields Institute, September 23-28, 2002. The author is grateful for support from the Fields Institute and a Leverhulme Emeritus Research Fellowship, 2002-2004, and to M. Hazewinkel for helpful comments on a draft. This paper is to appear in Michiel Hazewinkel (ed.), Handbook of Algebra, volume 6, Elsevier, 2008/2009. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Sivera/fibcat Title: Algebraic colimit calculations in homotopy theory using fibred and cofibred categories Author(s): Ronald Brown and Rafael Sivera Abstract: Higher Homotopy van Kampen Theorems allow the computation as colimits of certain homotopical invariants of glued spaces. One corollary is to describe homotopical excision in critical dimensions in terms of induced modules and crossed modules over groupoids. This paper shows how fibred and cofibred categories give an overall context for discussing and computing such constructions, allowing one result to cover many cases. A useful general result is that the inclusion of a fibre of a fibred category preserves connected colimits. The main homotopical application are to pairs of spaces with several base points, but we also describe briefly the situation for triads. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Glover-Henn/oct28-2008 Title: On the mod-p cohomology of Out(F_{2(p-1)}) Authors: Henry Glover Hans-Werner Henn Abstract: We study the mod-p cohomology of the group Out(F_n) of outer automorphisms of the free group F_n in the case n=2(p-1) which is the smallest n for which the p-rank of this group is 2. For p=3 we give a complete computation, at least above the virtual cohomological dimension of Out(F_4) (which is 5). More precisley, we calculate the equivariant cohomology of the p-singular part of outer space for p=3. For a general prime p>3 we give a recursive description in terms of the mod-p cohomology of Aut(F_k) for k less or equal to p-1. In this case we use the Out(F_{2(p-1)})-equivariant cohomology of the poset of elementary abelian p-subgroups of Out(F_n). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Harper/ModulesSpectra14 Title: Homotopy theory of modules over operads in symmetric spectra Author: John E. Harper Author's mailing address: Institute of geometry, algebra and topology, EPFL, CH-1015 Lausanne, Switzerland Comments: 33 pages, uses xy-pic. Significant revision. Abstract: This paper establishes model category structures on modules and algebras over operads in symmetric spectra, and studies when a morphism of operads induces a Quillen equivalence between corresponding categories of modules (resp. algebras) over operads. *** Please note: this is not a new submission to the Hopf arxiv, but a revision of an earlier manuscript with the same title. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Henn-Karamanov-Mahowald/hkm Title: The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited Authors: Hans-Werner Henn, Nasko Karamanov and Mark Mahowald Abstract: In this paper we use the approach introduced in an earlier paper by Goerss, Henn, Mahowald and Rezk in order to analyze the homotopy groups of L_{K(2)}V(0), the mod-3 Moore spectrum V(0) localized with respect to Morava K-theory K(2). These homotopy groups have already been calculated by Shimomura. The results are very complicated so that an independent verification via an alternative approach is of interest. In fact, we end up with a result which is more precise and also differs in some of its details from that of Shimomura. An additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the K(2)-localization of the spectrum TMF of topological modular forms and related spectra. Even more, the Adams-Novikov differentials for L_{K(2)}V(0) can be read off from those for TMF. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/LinJP/Lin08 Homology Rings of Homotopy Associative $H$--spaces James P. Lin Let $X$ be a homotopy associative mod $p$ $H$--space for $p$ an odd prime. The homology $H_*(X; \mathbb{F}_p)$ is an associative ring, but not necessarily commutative. We study conditions when $[\overline{x}, \overline{y}] \neq 0$ for $\overline{x}, \overline{y}$ elements of $H_*(X; \mathbb{F}_p)$. Under certain conditions $[\overline{x}, \overline{y}] \neq 0$ imply $ad^l (\overline{x},\overline{y}) \neq 0$ for $l=p-2$ or $p-1$. These methods can be used to prove results about homology commutators that were previously obtained using the adjoint action (Hamanaka et. al., 1996), (Kono et. al., 1993), (Kono et. al., 2003). We also generalize results of (Kane, 2006) to nonfinite mod $p$ homotopy associative $H$--spaces. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL/stable Title: Stable Invariants of Finite General Linear Groups and Symmetric Groups in Odd Characteristic Author: Larry Smith (AG-Invariantentheorie) We show that the stable invariants of the finite general linear group $\GL(n, \F_q)$ over a Galois field $\F_q$ with an odd characteristic coincide with the Hilbert ideal. The same argument applies to the tautological representation of the symmetric group in odd characteristic. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL/steintri Title: On R. Steinberg's Theorem on Algebras of Coinvariants Author Larry Smith (AG-Invariantentheorie} Steinberg's Theorem on the coinvariant algebra $\C[V]_G$ of a complex representation $\rho : G \hra \GL(n, \C)$ of a finite group $G$ says that $\C[V]_G$ is a Poincar\'e duality algebra if and only if the invariant algebra $\C[V]^G$ is a polynomial algebra. The extension of this to the nonmodular case has been achieved in stages, the final result being obtained by W.G. Dwyer and C.W. Wilkerson. We show that the main module theoretic tool they use extends to the following characteristic free result: If $\F[V]_G$ is a Poincar\'e duality algebra of formal dimension $d$\/, then $\F[V]^G$ is a polynomial algebra if and only if $\Hom_{\F[V]^G} (\F[V], \F[V])$ contains a nonzero element of degree $-d$\/. In the nonmodular case an easy transfer argument then recovers their extension of Steinberg's Theorem by means of some representation theory. Combined with some new results concerning the $\Delta$ operators of Demazure, our characteristic free result yields the following for reflection groups: A reflection group $G$ for which $\F[V]_G$ is a Poincar\'e duality algebra in which the trivial $G$-representation $1_G$ occurs only once as a subrepresentation has a polynomial algebra for its invariant algebra $\F[V]^G$\/. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL-Stong/binary Title: Invariants of Binary Forms Modulo Two Authors: Larry Smith (AG-Invariantentheorie) and R.E. Stong (University of Virginia) We examine the invariant theory of binary bilinear forms over the field $\F_2$ of two elements that arises in the classification of (standardly graded) Poincar\'e duality algebras with two algebra generators over the field $\F_2$ of two elements. We compute the corresponding ring of invariants and find seperating invariants for the orbit space. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL-Stong/pbi Title : Projective Bundle Ideals : Construction of Maximal Primary Irreducible Ideals in Polynomial Algebras Authors: Larry Smith and R. E. Stong Summary: We formalize the algebra of the Projective Bundle Theorem and use it to construct and study maximal primary irreducible ideals in polynomial algebras. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL-Stong/pda_quos Title: Poincar\'e Duality Algebras Modulo Two and Macaulay's Inverse Systems Authors: Larry Smith (AG-Invariantentheorie) and R.E. Stong (University of Virginia) If $H$ is a Poincar\'e duality algebra generated by its homogeneous component of degree $1$ it is called {\bf standardly graded} and the dimension of its homogeneous component $H_1$ of degree one is called its {\bf rank}\/. Standardly graded Poincar\'e duality algebras occur as quotient algebras of a (standardly graded) polynomial algebra by a maximal primary irreducible ideal Such ideals were studied in the work of F. S. Macaulay at the start of the last century who developed an elegant means of constructing them. The fact that these quotients are Poincar\'e duality algebras is a special case of a result of W. Gr\"obner. In this note we study the classification of Poincar\'e duality algebras over the field $\F_2$ of two elements. We obtain a complete classification of surfaces, i.e., Poincar\'e duality algebras of formal dimension two. To do so we determine the Grothendieck group of standardly graded surface algebras over an arbitrary field under the operation of connected sum. This group turns out to be $\Z$\/, hence finitely generated, and mirrors faithfully the topological classification of closed surfaces. By contrast, for Poincar\'e duality algebras (standardly graded or not) of formal dimension strictly greater than two the Grothendieck group fails to be finitely generated. We make a systematic study of standardly graded threefolds, i.e., Poincar\'e duality algebras of formal dimension three that are generated by their elements of degree one. The isomorphism classes of threefolds of rank at most three are in bijective correspondence with the orbits of the action of $\GL(3, \F_2)$ on a $10$-dimensional vector space, the space of catalecticant matrices. To determine the number of isomorphism classes we count the number of orbits using invariant theory. As a byproduct we obtain a classification of arbitrary bilinear forms in up to three variables. We determine explicitly all the standardly graded threefolds of rank at most three. There are 21 isomorphism classes. Twelve of these admit an unstable Steenrod algebra action, so could in theory be realized as the mod $2$ cohomology of a closed manifold. We exhibit for each such example a corresponding manifold; most of these are obvious, but there is one example of a slightly exotic $3$-manifold that is a torus bundle over a circle to which we devote some space. For threefolds of higher rank we explain one of several ways to construct such algebras that are not connected sums using Macaulay's theory of inverse systems. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL-Stong/rank_two Title: On Maximal Primary Irreducible Ideals in $\F[x, y]$ Authors: Larry Smith (AG-Invariantentheorie) and R.E. Stong (University of Virginia) At the beginning of the last century F.~S.~Macaulay developed an elegant theory describing homogeneous ideals in polynomial rings. This theory makes the maximal-primary irriducible ideals $I \subset \F[z_1\commadots z_n]$ correspond to a single homogeneous inverse polynomial $\theta_I \in \F[z_1^{-1} \commadots z_n^{-1}]$\/. Macaulay's theory has recently attracted attention in connection with problems arising in invariant theory and algebraic topology. In this note we show how given an inverse binary form $\theta \in \F[x^{-1}, y^{-1}]$ one may explicitly write down generators of the corresponding maximal-primary irreducible ideal $I(\theta) \subset \F[x, y]$\/. As a bonus we obtain an elementary proof of a theorem of Vasconcelos that such an ideal is always generated by a regular sequence. -----------------