Happy New Year! I remind you that your abstracts must contain your name and the title of the paper at a minimum. I have had to add these in by hand in a couple of recent cases. 7 new papers this time, from BrownR-Higgins, Jiang, Luo, Madsen-Weiss (the proof of the Mumford conjecture!), MauerOats, McClure-SmithJH, and Symonds. Mark Hovey New papers appearing on hopf between 12/01/02 and 01/08/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Higgins/cubabgp3 Title of Paper: Cubical abelian groups with connections\\ are equivalent to chain complexes Author(s): Ronald Brown and Philip J. Higgins AMS Classification numbers xxx LANL archive: math.AT/0212157 Addresses of Authors: Ronald Brown Mathematics Division \\ School of Informatics, \\ University of Wales, Bangor \\Gwynedd LL57 1UT, U.K. Philip J. Higgins, Department of Mathematical Sciences, \\ Science Laboratories, \\ South Rd., \\ Durham, DH1 3LE, U.K Email address of Authors r.brown---bangor.ac.uk p.j.higgins---durham.ac.uk Abstract: The theorem of the title is deduced from the equivalence between crossed complexes and cubical $\omega$-groupoids with connections proved by the authors in 1981. In fact we prove the equivalence of five categories defined internally to an additive category with kernels. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Jiang/realization Title of Paper: On the realization of the unstable modules Author: JIANG Dong Hua AMS Classification numbers: 55N99, 55S10 math.AT/0212054 Address of Author: LAGA, Institut Galilee, UMR 7539 University Paris Nord, Avenue Jean-Baptiste Clement 93430 VILLETANEUSE, FRANCE Email address of Author: donghua.jiang---m4x.org In this article, we give some restrictions about the structure of an unstable module, which should be verified providing this module is the reduced mod 2 cohomology of a space or a spectrum. We begin by studing the structure of the sub-modules of \Sigma^s H^\ast(B(Z/2)^{\oplus d}; Z/2)^{\oplus \alpha_d} (s \geq 0, \alpha_d > 0), i.e., the unstable modules whose nilpotent filtration has length 1. Next, we generelise this result for the unstable modules whose nilpotent filtration has a finite length, and who verified an additional condition. The result says that under some hypothesis, the reduced mod 2 cohomology of a space or a spectrum does not have arbitrary big gaps in its structure. This result is obtained by applying the famous Adams' theorem about the Hopf invariant and the classification of the injective unstable modules. For the unstable modules satisfing the condition of the theorem 3 (for example, any suspension of a sub-module of H^\ast(B(Z/2)^{\oplus d}; Z/2)^{\oplus \alpha_d}, the theorem 3 gives the upper bound of the length of the gaps in the modules, which means the module does not contain arbitrary big gaps. So when the module is reduced satisfing the condition of the theorem 4, its weight should be infinite. This gives us so many examples of the non-realizable unstable modules: F(n), any tensor product of F(n_i), etc. (These examples can also be proved by the theorem of Lionel Schwartz about the Kuhn conjecture, which was generalised by F-X. Dehon - G. Gaudens.) This article is written in french and the work is done under the direction of L. Schwartz. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Luo/pre Closed model categories for presheaves of simplicial groupoids and presheaves of 2-groupoids Zhi-ming Luo We prove that the category of presheaves of simplicial groupoids and the category of presheaves of 2-groupoids have Quillen closed model structures. We also show that the homotopy categories associated to the two categories are equivalent to the homotopy categories of simplicial presheaves and homotopy 2-types, respectively. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Madsen-Weiss/mumf The stable moduli space of Riemann surfaces: Mumford's conjecture Ib Madsen and Michael Weiss AMS classification numbers 57R50; 14H15, 32G15, 57R45, 57M99 Submitted to arXiv: math.AT/0212321 Institute for the Mathematical Sciences Aarhus University 8000 Aarhus C Denmark Department of Mathematics University of Aberdeen Aberdeen AB24 3UE United Kingdom imadsen---imf.au.dk m.weiss---maths.abdn.ac.uk The main result of this paper amounts to a complete evaluation of the integral cohomological structure of the stable mapping class group (i.e, the group of isotopy classes of automorphisms of a connected oriented surface of "large" genus). In particular it verifies the conjecture of D.Mumford about the rational cohomology of the stable mapping class group. It is part of a more recent development in the field which began with Ulrike Tillmann's result (Invent. Math., 1997) that the plus construction makes the classifying space of the stable mapping class group into an infinite loop space. This led to a stable homotopy theory version of Mumford's conjecture, stronger than the original (Madsen and Tillmann, Invent. Math., 2001). We prove the extended version of Mumford's conjecture by a mixture of techniques from singularity theory and from homotopy theory. The stability theorem of J.Harer (Annals of Math., 1985) and the "First Main theorem" of V.Vassiliev ("Complements of Discriminants of smooth maps: Topology and Applications", Trans. of Math. Monographs Vol.98, revised edition, Amer. Math. Soc. 1994) are prominent components of our proof. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/MauerOats/algebraic-calc Algebraic Goodwillie calculus and a cotriple model for the remainder Andrew Mauer-Oats We define an ``algebraic'' version of the Goodwillie tower, P_n^alg F(X), that depends only on the behavior of F on coproducts of X. When F is a functor to connected spaces or grouplike H-spaces, the functor P_n^alg F is the base of a fibration whose fiber is the simplicial space associated to a cotriple built from the (n+1) cross effect of the functor F. When the connectivity of X is large enough (for example, when F is the identity functor and X is connected), the algebraic Goodwillie tower agrees with the ordinary (topological) Goodwillie tower, so this theory gives a way of studying the Goodwillie approximation to a functor F in many interesting cases. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/McClure-SmithJH/McClureSmith2_1 Multivariable cochain operations and little $n$-cubes. James E. McClure and Jeffrey H. Smith 18D50, 55P48, 16E40 math.QA/0106024 Department of Mathematics Purdue University 150 N. University Street West Lafayette, IN 47907-2067 mcclure---math.purdue.edu jhs---math.purdue.edu This is a revision of a paper first posted June 4, 2001. It will appear in the Journal of the AMS. In this paper we construct a small $E_\infty$ chain operad $\S$ which acts naturally on the normalized cochains $S^*X$ of a topological space. We also construct, for each $n$, a suboperad $\S_n$ which is quasi-isomorphic to the normalized singular chains of the little $n$-cubes operad. The case $n=2$ leads to a substantial simplification of our earlier proof of Deligne's Hochschild cohomology conjecture. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Symonds/morava The Tate-Farrell cohomology of the Morava Stabilizer Group $S_{p-1}$ with coefficients in $E_{p-1}$ Peter Symonds We calculate the Tate-Farrell cohomology of the Morava stabilizer group $S_{p-1}$ with coefficients in the moduli space $E_{p-1}$ for odd primes $p$. ------------ ------------------------------ The dvipdf and dviselect programs don't seem to be working quite right on Hopf. Only a very few papers are affected, but if you have any trouble with pdf files, use the dvi file instead. 13 new papers this time, from BrownR, BrownR-Higgins, Chataur-Rodriguez-Scherer, Hovey, Hovey-Strickland (2 papers), Hung (4), Hung-Nam (2), and Marzantowicz-Prieto. Mark Hovey New papers appearing on hopf between 1/08/03 and 01/21/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/fields-artxx Title of Paper: Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems Author(s): Ronald Brown AMS Classification numbers: 01-01,16E05,18D05,18D35,55P15,55Q05 Already submitted to the xxx LANL archive, include the id. no., math.AT/0212271 Addresses of Authors: Ronald Brown Mathematics Division, School of Informatics, University of Wales, Bangor Gwynedd LL57 1UT, U.K. Email address of Authors r.brown---bangor.ac.uk 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. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Higgins/orbitgpdxx Title of Paper: The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action Author(s): Ronald Brown and Philip J. Higgins AMS Classification numbers: 0F34, 20L13, 20L15, 57S30 Already submitted to the xxx LANL archive, include the id. no., math.AT/0212271 Addresses of Authors: Ronald Brown Mathematics Division, School of Informatics, University of Wales, Bangor Gwynedd LL57 1UT, U.K. Philip J. Higgins Department of Mathematical Sciences, Science Laboratories, South Rd., Durham, DH1 3LE, U.K. Email address of Authors r.brown---bangor.ac.uk p.j.higgins---durham.ac.uk Text of Abstract (try for 20 lines or less) The main result is that the fundamental groupoid of the orbit space of a discontinuous action of a discrete group on a Hausdorff space which admits a universal cover is the orbit groupoid of the fundamental groupoid of the space. We also describe work of Higgins and of Taylor which makes this result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space. The main result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author's book on Topology (1988 edition). This is a somewhat edited, and in one point (on normal closures) corrected, version of those sections. Because of its provenance, this should be read as a graduate text rather than an article. The Exercises should be regarded as further propositions for which we leave the proofs to the reader. It is expected that this material will be part of a new edition of the book. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Chataur-Rodriguez-Scherer/operadplus Plus-construction of algebras over an operad, cyclic and Hochschild homologies up to homotopy David Chataur, Jose L. Rodriguez, and Jerome Scherer math.AT/0301130 CRM Barcelona, dchataur---crm.es Universidad de Almeria, jlrodri---ual.es Universidad Autonoma de Barcelona, jscherer---mat.uab.es The aim of this paper is to show how to apply the machinery of homotopical localization to the framework of differential graded algebras over an operad. By performing nullification with respect to a universal acyclic algebra one obtains a plus-construction, which doesn't affect Quillen homology and quotients out the maximal perfect ideal of $\pi_0$. For any associative algebra the general linear Lie (resp. Leibniz) algebra is a Lie (resp. Leibniz) algebra up to homotopy. The plus-construction yields then two new homology theories, closely related to cyclic and Hochschild homology (they coincide with the classical cyclic and Hochschild homology over the rational). We also compute the first homology groups of these theories, in analogy with the computation of the first $K$-theory groups of a ring. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/barcelona Chromatic phenomena in the algebra of BP_{*}BP-comodules Mark Hovey Wesleyan University mhovey---wesleyan.edu This paper begins with an exposition of the author's research on the category of BP_*BP-comodules, much of which is joint with Neil Strickland. We give an overview of the results obtained in the papers Hovey/comodule, Hovey-Strickland/torsion-comod, and Hovey-Strickland/derived-ln. The main result of that work is that the category of E(n)_*E(n)-comodules is equivalent to a localization of the category of BP_*BP-comodules (the localization is L_n, analogous to the topological L_n). The main new result in this paper is that, analogously, the stable homotopy category of E(n)_*E(n)-comodules is equivalent to a localization (the finite localization L_n^f this time, not L_n) of the stable homotopy category of BP_*BP-comodules. These stable homotopy categories were constructed in Hovey/comodule, and are supposed to model stable homotopy theory; it is like stable homotopy theory where there are no differentials in the Adams-Novikov spectral sequence. Our result embeds the Miller-Ravenel and Hovey-Sadofsky change of rings theorems as special cases of isomorphisms like [X,Y]=[L_n^f X, Y] for L_n^f-local objects Y. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Strickland/torsion-comod Comodules and Landweber exact homology theories Mark Hovey and Neil Strickland Wesleyan University University of Sheffield mhovey---wesleyan.edu N.P. Strickland---sheffield.ac.uk We show that, if E is a Landweber exact ring spectrum, then the category of E_*E-comodules is equivalent to the localization of the category of BP_*BP-comodules with respect to the hereditary torsion theory of v_n-torsion comodules, where n is the height of E. In particular, the category of E(n)_*E(n)-comodules is equivalent to the category of (v_n^{-1}BP)_*(v_n^{-1}BP)-comodules. We also prove structure theorems for E_*E-comodules; we show every E_*E-comodule has a primitive, we classify the invariant radical ideals, and we prove a version of the Landweber filtration theorem. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Strickland/derived-ln Local cohomology of BP_*BP-comodules Mark Hovey and Neil Strickland Wesleyan University University of Sheffield mhovey---wesleyan.edu N.P. Strickland---sheffield.ac.uk In the paper torsion-comod (announced above) on this archive, we showed that the category of E(n)_*E(n)-comodules is a localization of the category of BP_*BP-comodules. In this paper, we study the resulting localization functor L_n on the category of BP_*BP-comodules. It is an algebraic analogue of the usual topological localization L_n. It is left exact, so has right derived functors L_n^i. We show that these derived functors are closely related to the local cohomology groups of BP_*-modules studied by Greenlees and May; in fact, they coincide with Cech cohomology with respect to I_{n+1}. We also construct a spectral sequence of comodules analogous to the Greenlees-May spectral sequence (of modules) converging to BP_*(L_n X) whose E_2-term involves L_n^i(BP_*X). The proofs require getting a partial understanding of injective objects in the category of BP_*BP-comodules. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/2001 Title of Paper: On triviality of Dickson invariants in the homology of the Steenrod algebra Author: Nguy\^{e}n H. V. Hung 2000 Mathematics Subject Classification: Primary 55P47, 55Q45, 55S10, 55T15. Address of Author: Current Address: Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore MD 21218 - 2689 E-mail address: nhvhung---math.jhu.edu Permanent Address: Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung---vnu.edu.vn Abstract: Let ${\cal A}$ be the mod 2 Steenrod algebra and $D_k$ the Dickson algebra of $k$ variables. We study the Lannes-Zarati homomorphisms $$ \varphi_k: Ext_{\cal A}^{k,k+i}(F_2,F_2)\to (F_2\otimes_{\cal A} D_k)_i^*, $$ which correspond to an associated graded of the Hurewicz map $ H:\pi_*^s(S^0)\cong \pi_*(Q_0S^0)\to H_*(Q_0S^0)$. An algebraic version of the long-standing conjecture on spherical classes predicts that $\varphi_k=0$ in positive stems, for $k>2$. That the conjecture is no longer valid for $k=1$ and $2$ is respectively an exposition of the existence of Hopf invariant one classes and Kervaire invariant one classes. This conjecture has been proved for $k=3$ by Hung [Trans AMS 349 (1997), 3893-3910]. It has been shown that $\varphi_k$ vanishes on decomposable elements for $k>2$ [Hung and Peterson, Math. Proc. Camb. Phil. Soc. 124 (1998), 253-264] and on the image of Singer's algebraic transfer for $k>2$ [Hung, 1997; Hung and Nam, Trans AMS 353 (2001), 5029-5040]. In this paper, we establish the conjecture for $k=4$. To this end, our main tools include (1) an explicit chain-level representation of $\varphi_k$ and (2) a squaring operation $Sq^0$ on $(F_2\otimes_{\cal A} D_k)^*$, which commutes with the classical $Sq^0$ on $Ext_{\cal A}^k(F_2,F_2)$ through the Lannes-Zarati homomorphism. (To appear in Math. Proc. Camb. Phil. Soc. 134 (2003).) 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/2002h Title of Paper: The cohomology of the Steenrod algebra and representations of the general linear groups Author: Nguy\^{e}n H. V. Hung 2000 Mathematics Subject Classification: Primary 55P47, 55Q45, 55S10, 55T15. Address of Author: Current Address: Department of Mathematics, Wayne State University 656 W. Kirby Street, Detroit, MI 48202 (USA) E-mail address: nhvhung------math.wayne.edu Permanent Address: Department of Mathematics, Vietnam National University, Hanoi 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung------vnu.edu.vn Abstract: Let $Tr_k$ be the algebraic transfer that maps from the coinvariants of certain $GL_k$-representation to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer $tr_k: \pi_*^S((B\V _k)_+) \to \pi_*^S(S^0)$. It has been shown that the algebraic transfer is highly nontrivial, more precisely, that $Tr_k$ is an isomorphism for $k=1, 2, 3$ and that $Tr= \oplus_k Tr_k$ is a homomorphism of algebras. In this paper, we first recognize the phenomenon that if we start from any degree $d$, and apply $Sq^0$ repeatedly at most $(k-2)$ times, then we get into the region, in which all the iterated squaring operations are isomorphisms on the coinvariants of the $GL_k$-representation. As a consequence, every finite $Sq^0$-family in the coinvariants has at most $(k-2)$ non zero elements. Two applications are exploited. The first main theorem is that $Tr_k$ is not an isomorphism for $k\geq 5$. Furthermore, $Tr_k$ is not an isomorphism in infinitely many degrees for each $k > 5$. We also show that if $Tr_{\ell}$ detects a nonzero element in certain degrees of $\text{Ker}(Sq^0)$, then it is not a monomorphism and further, $Tr_k$ is not a monomorphism in infinitely many degrees for each $k>\ell$. The second main theorem is that the elements of any $Sq^0$-family in the cohomology of the Steenrod algebra, except at most its first $(k-2)$ elements, are either all detected or all not detected by $Tr_k$, for every $k$. Applications of this study to the cases $k=4$ and $5$ show that $Tr_4$ does not detect the three families $g$, $D_3$, $p'$ and $Tr_5$ does not detect the family $\{h_{n+1}g_n |\; n\geq 1\}$. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/HungTAMS01 Title of Paper: Spherical classes and the Lambda algebra Author: Nguy\^{e}n H. V. Hung 2000 Mathematics Subject Classification: Primary 55P47, 55Q45, 55S10, 55T15. Address: Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung---vnu.edu.vn Abstract: Let $\Gamma^{\wedge}= \oplus_k \Gamma_k^{\wedge}$ be Singer's invariant-theoretic model of the dual of the Lambda algebra with $H_k(\Gamma^{\wedge})\cong Tor_k^{\cal A}(F_2, F_2)$, where ${\cal A}$ denotes the mod 2 Steenrod algebra. We prove that the inclusion of the Dickson algebra, $D_k$, into $\Gamma_k^{\wedge}$ is a chain-level representation of the Lannes--Zarati dual homomorphism $$ \varphi_k^*: F_2\otimes_{\cal A} D_k \to Tor^{\cal A}_k(F_2,F_2) \cong H_k(\Gamma^{\wedge}). $$ The Lannes--Zarati homomorphisms themself, $\varphi_k$, correspond to an associated graded of the Hurewicz map $$ H:\pi_*^s(S^0)\cong \pi_*(Q_0S^0)\to H_*(Q_0S^0)\,. $$ Based on this result, we discuss some algebraic versions of the classical conjecture on spherical classes, which states that {\it Only Hopf invariant one and Kervaire invariant one classes are detected by the Hurewicz homomorphism.} One of these algebraic conjectures predicts that every Dickson element, i. e. element in $D_k$, of positive degree represents the homology class $0$ in $Tor^{\cal A}_k(F_2, F_2)$ for $k>2$. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/HungTAMS97 Title of Paper: Spherical classes and the algebraic transfer Author: Nguy\^{e}n H. V. Hung 1991 Mathematics Subject Classification: Primary 55P47, 55Q45, 55S10, 55T15. Address of Author: Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung---vnu.edu.vn Abstract: We study a weak form of the classical conjecture which predicts that there are no spherical classes in $Q_0S^0$ except the elements of Hopf invariant one and those of Kervaire invariant one. The weak conjecture is obtained by restricting the Hurewicz homomorphism to the homotopy classes which are detected by the algebraic transfer. We prove that the weak conjecture is equivalent to the following one: Every positive degree Dickson invariant of at least 3 variables belongs to the image of the Steenrod algebra acting on the corresponding polynomial algebra. This conjecture is proved for the case of 3 variables in two different ways. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung-Nam/HungNamJA01 Title of Paper: The hit problem for the modular invariants of linear groups Author: Nguy\^{e}n H. V. Hung and Tran Ngoc Nam 2000 Mathematics Subject Classification: Primary 55S10, Secondary 55Q45. Address of authors: Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung---vnu.edu.vn E-mail address: namtn---vnu.edu.vn Abstract: Let the mod 2 Steenrod algebra, ${\cal A}$, and the general linear group, $GL_k:=GL(k, F_2)$, act on $P_{k}:=F_2[x_{1},...,x_{k}]$ with $\deg(x_{i})=1$ in the usual manner. We prove that, for a family of some rather small subgroups $G$ of $GL_k$, every element of positive degree in the invariant algebra $P_{k}^G$ is hit by ${\cal A}$ in $P_{k}$. In other words, $(P_{k}^G)^+ \subset {\cal A}^+\cdot P_{k}$, where $(P_{k}^G)^+$ and ${\cal A}^+$ denote respectively the submodules of $P_{k}^G$ and ${\cal A}$ consisting of all elements of positive degree. This family contains most of the parabolic subgroups of $GL_k$. It should be noted that the smaller the group G is the harder the problem turns out to be. Remarkably, when $G$ is the smallest group of the family, the invariant algebra $P_{k}^G$ is a polynomial algebra in $k$ variables, whose degrees are $\leq 8$ and fixed while $k$ increases. It has been shown by Hung [Trans AMS 349 (1997), 3893-3910] that, for $G=GL_k$, the inclusion $(P_{k}^{GL_k})^+\subset {\cal A}^+\cdot P_{k}$ is equivalent to a week algebraic version of the long-standing conjecture stating that the only spherical classes in $Q_0S^0$ are the elements of Hopf invariant one and those of Kervaire invariant one. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung-Nam/HungNamTAMS01 Title of Paper: The hit problem for the Dickson algebra Author: Nguy\^{e}n H. V. Hung and Tran Ngoc Nam 2000 Mathematics Subject Classification: Primary 55S10, Secondary 55P47, 55Q45, 55T15. Address of authors: Department of Mathematics, Vietnam National University, Hanoi 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung---vnu.edu.vn E-mail address: namtn---vnu.edu.vn Abstract: Let the mod 2 Steenrod algebra, ${\cal A}$, and the general linear group, $GL(k, F_2)$, act on $P_{k}:= F_2[x_{1},...,x_{k}]$ with $|x_{i}|=1$ in the usual manner. We prove the conjecture of the first-named author in {\it Spherical classes and the algebraic transfer}, (Trans. AMS 349 (1997), 3893-3910) stating that every element of positive degree in the Dickson algebra $D_{k}:=(P_{k})^{GL(k,F_2)}$ is ${\cal A}$-decomposable in $P_{k}$ for arbitrary $k>2$. This conjecture was shown to be equivalent to a weak algebraic version of the classical conjecture on spherical classes, which states that the only spherical classes in $Q_0S^0$ are the elements of Hopf invariant one and those of Kervaire invariant one. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/Marzantowicz-Prieto/Marprieto The unstable equivariant fixed point index and the equivariant degree by Waclaw Marzantowicz and Carlos Prieto A correspondence between the equivariant degree introduced by Ize, Massab\'o, and Vignoli and an unstable version of the equivariant fixed point index defined by the second author and Ulrich is shown. With the help of conormal maps and properties of the unstable index, we prove a sum decomposition formula for the index and consequently also for the degree. As an application, we decompose equivariant homotopy groups as direct sums of smaller groups of fixed orbit types, and we give a geometric interpretation of each summand in terms of conormal maps. --------------- ------------------------------- 7 new papers this time, from Bokstedt-Ottosen, Chataur-Scherer, Hung, Jardine, Rosu (2), and Ruiz-Viruel. Mark Hovey New papers appearing on hopf between 1/21/03 and 03/01/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bokstedt-Ottosen/stringV4 Title: A spectral sequence for string cohomology Authors: Marcel Bokstedt and Iver Ottosen AMS Classification numbers: 55N91, 55P35, 18G50 Address of Authors: Institut for Matematiske Fag Aarhus Universitet Ny Munkegade DK-8000 Aarhus C Matematisk Afdeling Koebenhavns Universitet Universitetsparken 5 DK-2100 Koebenhavn OE Email address of Authors: marcel---imf.au.dk iver---math.ku.dk Abstract: Let $X$ be a 1-connected spaces with free loop space $\Lambda X$. We introduce two spectral sequences converging towards $H^*(\Lambda X;\ZZ /p)$ and $H^*((\Lambda X)_{hS^1};\ZZ /p)$. The $E_2$-terms are certain non Abelian derived functors applied to $H^*(X;\ZZ /p)$. When $H^*(X;\ZZ /p)$ is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If $X$ is a sphere it is a surprising fact that the spectral sequences collapse for $p=2$. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chataur-Scherer/fibrewise Fibrewise nullification and the cube theorem David Chataur and Jerome Scherer CRM Barcelona, dchataur---crm.es Universidad Autonoma de Barcelona, jscherer---mat.uab.es Our aim is to construct fibrewise localizations in model categories. For pointed spaces, the general idea is to decompose the total space of a fibration as a diagram over the category of simplices of the base and replace it by the localized diagram. This of course is not possible in an arbitrary category. We have thus to adapt another construction which heavily depends on Mather's cube theorem. Working with model categories in which the cube theorem holds, we characterize completely those who admit a fibrewise nullification. As an application we get fibrewise plus-construction and fibrewise Postnikov sections for algebras over an operad. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/HungMZ99 Title of Paper: The weak conjecture on spherical classes Author: Nguy\^{e}n H. V. Hung 1991 Mathematics Subject Classification: Primary 55P47, 55Q45, 55S10, 55T15. Address of Author: Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam E-mail address: nhvhung---vnu.edu.vn Abstract: Let $A$ be the mod 2 Steenrod algebra. We construct a chain-level representation of the dual of Singer's algebraic transfer, $$ Tr_k^*: Tor^A_k(F_2,F_2) \to F_2\otimes_A F_2[x_1,...,x_k], $$ which maps Singer's invariant-theoretic model of the lambda algebra to $F_2[x_1^{\pm},...,x_k^{\pm}]$ and is the inclusion of the Dickson algebra into the polynomial algebra $F_2[x_1,...,x_k]$. Based on this chain-level representation, we study some aspects of the weak conjecture on spherical classes and prove it in some special cases. (Address of Paper: Math. Zeit. 231 (1999), 727-743) 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/simpset3 Abstract: "Simplicial approximation", by J.F. Jardine This paper displays an approach to the construction of the homotopy theory of simplicial sets and the corresponding equivalence with the homotopy theory of topological spaces which is based on simplicial approximation techniques. The required simplicial approximation results for simplicial sets and their proofs are given in full. Subdivision behaves like a covering in the context of the techniques displayed here. Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada E-mail: jardine---uwo.ca URL: http://www.math.uwo.ca/~jardine/papers/ 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Rosu/ellc Title: Equivariant Elliptic Cohomology and Rigidity Author: Ioanid Rosu, AMS Classification numbers: 55N34; 55N91 xxx LANL archive ID number: AT/9912089 Addresses and emails of Authors: Ioanid Rosu, M.I.T., Cambridge, MA. ioanid---math.mit.edu Equivariant elliptic cohomology with complex coefficients was defined axiomatically by Ginzburg, Kapranov and Vasserot and constructed by Grojnowski. We give an invariant definition of S^1-equivariant elliptic cohomology, and use it to give an entirely cohomological proof of the rigidity theorem of Witten for the elliptic genus. We also state and prove a rigidity theorem for families of elliptic genera. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Rosu/kt Title: Equivariant K-theory and Equivariant Cohomology Author: Ioanid Rosu, with an appendix by Allen Knutson and Ioanid Rosu AMS Classification numbers: 55N91 xxx LANL archive ID number: AT/9912088 Addresses and emails of Authors: Ioanid Rosu, M.I.T., Cambridge, MA. ioanid---math.mit.edu Allen Knutson, University of California at Berkeley, CA allenk---math.berkeley.edu For T an abelian compact Lie group, we give a description of T-equivariant K-theory with complex coefficients in terms of equivariant cohomology. In the appendix we give applications of this by extending results of Chang-Skjelbred and Goresky-Kottwitz-MacPherson from equivariant cohomology to equivariant K-theory. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Ruiz-Viruel/rv TITLE: The classification of $p$-local finite groups over the extraspecial group of order $p^3$ and exponent $p$. AUTHORS: Albert Ruiz, LAGA Universit{\'e} Paris XIII 99av J.B.\ Cl{\'e}ment 93430 Villetaneuse France ruiz---math.univ-paris13.fr Antonio Viruel Dpto de {\'A}lgebra, Geometr{\'\i}a y Topolog{\'\i}a Universidad de M{\'a}laga Apdo correos 59 29080 M{\'a}laga Spain viruel---agt.cie.uma.es ABSTRACT: The concept of $p$-local finite group arise in the work of Broto-Levi-Oliver as a generalization of the classical concept of finite group. Therefore, the classification of $p$-local finite groups has interest, not only by itself but, as an opportunity to enlighten one of the highest mathematical achievements in the last decades: The Classification of Finite Simple Groups. In this work we classify all $p$-local finite group over the $p$-groups of type $p^{1+2}_+$. In this classification three new exotic $7$-local finite groups arise. ------------------------------------------------------ 10 new papers this time, from Anderson-Grodal-Moller-Viruel, Anjos-Granja, Bauer-McCarthy, Behrens-Pemmaraju, Budney-Conant-Scannell-Sinha, Donadze-Inassaridze-Porter, Dorabiala-Johnson, Kitchloo-Laures-Wilson, Salvatore, and Sinha. Mark Hovey New papers appearing on hopf between 3/01/03 and 4/09/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Anderson-Grodal-Moller-Viruel/classificationpodd Title: The classification of p-compact groups for p odd Authors: Kasper K. S. Andersen, Jesper Grodal, Jesper M. M{\o}ller, Antonio Viruel Subj-class: AT Algebraic Topology (GR Group theory; RT Representation Theory) MSC-class: 55R35 (Primary) 55P35, 57T10, 20G20 (Secondary) Comments: 87 pages \\ A p-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined p-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for p an odd prime, proving that there is a one-to-one correspondence between connected p-compact groups and finite reflection groups over the p-adic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as p-compact groups by their Weyl groups seen as finite reflection groups over the p-adic integers. Our method however leads to a largely self-contained proof of the entire classification theorem. \\ 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Anjos-Granja/homotopy.decomp.symplect Title: Homotopy decomposition of a group of symplectomorphisms of S^2\times S^2 Authors: Silvia Anjos and Gustavo Granja AMS Classification numbers: 57S05, 57R17, 55R35 Address of Authors: Departamento de Matematica Instituto Superior Tecnico Av. Rovisco Pais 1049-001 Lisboa Portugal Email address of Authors: sanjos---math.ist.utl.pt ggranja---math.ist.utl.pt Abstract: We continue the analysis started by Abreu, McDuff and Anjos of the topology of the group of symplectomorphisms of $S^2 \times S^2$ when the ratio of the areas of the two spheres lies in the interval (1,2]. We express the group, up to homotopy, as the amalgam of certain of its compact Lie subgroups. We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Bauer-McCarthy/mcbauer3 Kristine Bauer Department of Mathematics Johns Hopkins University 3400 N. Charles St. Baltimore, MD 21218 USA kbbauer---math.jhu.edu Randy McCarthy Department of Mathematics University of Illinois 1409 W. Green St. Urbana, IL 61801 USA randy---math.uiuc.edu On vanishing Tate cohomology and decompositions in Goodwillie calculus Mathematical Subject Classification: 55P65 (55P45, 13D03) Our main result is that if F is a functor from a pointed category C to spectra, the Goodwillie tower of F evaluated at X splits rationally when X is a co-H-object of C. We show that the layers of F(X) in this case are easy to identify. The splitting of the Goodwillie tower gives a decomposition of F(X) into a product of its layers. We use this to recover the rational decompositions of Hochschild and higher Hochschild homology by Pirashvili, Loday,and Gerstenhaber-Schack. Finally, we extend the main theorem to include dual calculus to recover the Poincar\'e-Birkhoff-Witt theorem, and improve the theorem in the special case in which the comultiplication map is cocommutative. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens-Pemmaraju/v2 On the existence of the self map v_2^9 on the Smith-Toda complex V(1) at the prime 3 Mark Behrens Department of Mathematics University of Chicago Chicago, IL 60637, U.S.A. mbehrens---math.uchicago.edu Satya Pemmaraju Fixed Income Derivatives UBS Warburg Stamford, CT 06901, U.S.A. Satya.Pemmaraju---ubsw.com AMS Classification: 55Q51; 55Q45, 55T15 math.AT/0303223 submitted to proceedings of the Northwestern University conference on algebraic topology, March 2002 Included EPS files: assE2.eps bss.eps eo_2V1.eps eo_2V1ASS.eps extP.eps splitting.eps Note: there is one chart created using the landscape package in LaTeX. On some dvi viewers, this chart does not display properly, but is viewable when converted to Postscript. Abstract Let V(1) be the Smith-Toda complex at the prime 3. We prove that there exists a map v_2^9: \Sigma^{144}V(1) \to V(1) that is a K(2) equivalence. This map is used to construct various v_2-periodic infinite families in the 3-primary stable homotopy groups of spheres. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Budney-Conant-Scannell-Sinha/selflink Title: New perspectives on self-linking Author: Ryan Budney, James Conant, Kevin P. Scannell, Dev P. Sinha AMS Class: 57M27; 55R80; 57R40; 57M25; 55P99 LANL ID: math.AT/0303034 Addresses: Departments of Mathematics, Rochester University, Cornell University, St. Louis University, University of Oregon Email: rybu---math.rochester.edu, jconant---polygon.math.cornell.edu, scannell---slu.edu, dps---math.uoregon.edu Abstract: We initiate the study of classical knots through the homotopy class of the n-th evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its n-th evaluation map realizes the space of knots as a subspace of what we call the n-th mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot. Note: The .dvi version is missing many (fun) figures - we strongly recommend downloading the .pdf file. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Donadze-Inassaridze-Porter/Hopfder N-fold Cech derived functors and generalized Hopf type formulas by Guram Donadze, Nick Inassaridze, and Timothy Porter, In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the n-fold Cech derived functors of the lower central series functors Z_k. The paper ends with an application to algebraic K-theory. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Dorabiala-Johnson/torsion The product theorem for parametrized topological Reidemeister torsion Wojtek Dorabiala Mark W. Johnson Primary: 19D10; Secondary: 18F25, 19Exx, 55R70 Department of Mathematics Penn State Altoona Altoona, PA 16601-3760 wud2---psu.edu mwj3---psu.edu The goal of this article is to prove the product formula for parametrized topological Reidemeister torsion. The theorem states that the product of the parametrized Euler characteristic of one fibration with the parametrized Reidemeister torsion class of another fibration yields the parametrized Reidemeister torsion class of the product fibration. In the process of establishing the theorem, several new products must be defined involving (derivative theories of) parametrized $\Aof$-theory and a detailed description of the coassembly map for parametrized $\Aof$-theory is included. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Laures-Wilson/klw2 Splittings of bicommutative Hopf algebras Nitu Kitchloo, Gerd Laures and W. Stephen Wilson Department of Mathematics Johns Hopkins University 3400 N. Charles Street Baltimore, MD 21218, USA Mathematisches Institut der Universit\"at Heidelberg, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany wsw---math.jhu.edu, nitu---math.jhu.edu, gerd---laures.de Abstract: We use the theory of Dieudonne modules to show that certain types of short exact sequences of Hopf algebras split. Several examples occur naturally with Morava K-theory. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Salvatore/config Title: Configuration spaces on the sphere and higher loop spaces Author: Paolo Salvatore AMS classification numbers: 55P48, 55R80, 55S12 xxx number: math.AT/0303290 Address: Dipartimento di Matematica, Universita` di Roma "Tor Vergata", Via della Ricerca Scientifica 1, 00133 Roma, Italy e-mail: salvator---mat.uniroma2.it Abstract: We show that the homology over a field of the space of free maps from the n-sphere to the n-fold suspension of X depends only on the cohomology algebra of X and compute it explicitly. We compute also the homology of the closely related labelled configuration space on the n-sphere with labels in X and of its completion, that depend only on the homology of X. In many but not all cases the homology of the configuration space coincides with the homology of the mapping space. In particular we obtain the homology of the unordered configuration spaces on a sphere. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Sinha/semifree Title: Bordism of semi-free $S^1$-actions. Author: Dev P. Sinha AMS Class: 57R85 (primary); 55R40 (secondary). LANL ID: math.AT/0303100 Addresses: Department of Mathematics, University of Oregon, Eugene OR Email: dps---math.uoregon.edu Abstract: We calculate the geometric and homotopical (or stable) bordism rings associated to semi-free $S^1$ actions on complex manifolds, giving explicit generators for the geometric theory. To calculate the geometric theory, we prove a case of the geometric realization conjecture, which in general would determine the geometric theory in terms of the homotopical. The determination of semi-free actions with isolated fixed points up to cobordism complements similar results from symplectic geometry. --------------------------------------------- 4 new papers this time, from BrownR, DavisD, Dwyer, and Gottlieb. Mark Hovey New papers appearing on hopf between 4/09/03 and 5/13/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/noncommut-at Title: Towards non commutative algebraic topology Author: Ronald Brown AMS Classification numbers: 55D15, 55U40, 18D35 Address of Author: Mathematics Division, School of Informatics, University of Wales, Bangor, Gwynedd LL57 1UT, UK. Email address of Author: r.brown---bangor.ac.uk Text of Abstract: These are the transparencies (slightly edited) for a seminar at University College, London, on May 7, 2003. They give a quick overview of some background and some directions taken for algebraic methods for higher dimensional, non commutative, local to global problems, including some algebraic models of homotopy types. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD/E7E8 Representation types and 2-primary homotopy groups of certain compact Lie groups Donald M. Davis 55Q52, 55T15, 57T20 Department of Mathematics Lehigh University Bethlehem, PA 18015 dmd1---lehigh.edu Abstract: Bousfield has shown how the 2-primary v1-periodic homotopy groups of certain compact Lie groups can be obtained from their representation ring with its decomposition into types and its exterior power operations. He has formulated a Technical Condition which must be satisfied in order that he can prove his description is valid. We prove that a simply-connected compact simple Lie group satisfies his Technical Condition if and only if it is not E6 or Spin(4k+2) with k not a 2-power. We then use his description to give an explicit determination of the 2-primary v1-periodic homotopy groups of E7 and E8. This completes a program, suggested to the author by Mimura in 1989, of computing the v1-periodic homotopy groups of all compact simple Lie groups at all primes. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer/local Localization W. G. Dwyer This is a largely expository paper, which describes the concept of localization, as it usually comes up in topology, and gives some examples of it. The examples include local homology and cohomology, homological localizations of spaces and spectra, and localization with respect to a map f. For appropriate choices of the map f, this last gives constructions related to the Goodwillie calculus and to motivic homotopy theory. There's also a proof that if a localization functor exists, the higher order categorical invariants associated to inverting the local equivalences are trivial. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Gottlieb/eigbndl EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE Daniel Henry Gottlieb Given a parameterized space of square matrices, the associated set of eigenvectors forms some kind of a structure over the parameter space. When is that structure a vector bundle? When is there a vector field of eigenvectors? We answer those questions in terms of three obstructions, using a Homotopy Theory approach. We illustrate our obstructions with five examples. One of those examples gives rise to a 4 by 4 matrix representation of the Complex Quaternions. This representation shows the relationship of the Biquaternions with low dimensional Lie groups and algebras, Electro-magnetism, and Relativity Theory. The eigenstructure of this representation is very interesting, and our choice of notation produces important mathematical expressions found in those fields and in Quantum Mechanics. In particular, we show that the Doppler shift factor is analogous to Berry's Phase. ----------------------------------------------------- These are mostly papers that just made it out of Clarence's e-mail. The new policy at Hopf is that e-mail submssions are strongly deprecated. Please use the web form if at all possible. There is a significant and unpredictable delay associated with e-mail submission and it is easier for papers to get misplaced. 11 new papers this time, from Chernov-Rudyak, Dugger (3), Dwyer-Wilkerson, Ibanez-Rudyak-Tralle, Ibanez-Rudyak-Tralle-Ugarte, Oliver, Oprea-Rudyak, Rudyak, and Wilkerson. Mark Hovey New papers appearing on hopf between 5/13/03 and 5/17/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Chernov-Rudyak/wavefronts Title: Affine Linking Numbers and Causality Relations for Wave Fronts Authors: Vladimir Chernov (Tshernov), Yuli Rudyak Addresses: V. Chernov, Department of Mathematics, 6199 Bradley Hall, Dartmouth College, Hanover NH 03755, USA Yuli Rudyak, Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105 Gainesville, FL 32611-8105 U.S.A e-mail: rudyak---math.ufl.edu e-mail: Vladimir.Chernov---dartmouth.edu 4 figures (eps files) Abstract: Let M be an oriented manifold. We study the causal relations between the wave fronts W and W' that originated at some points of M. We introduce a numerical topological invariant CRI(W, W') (the so-called causality relation invariant) that, in particular, gives the algebraic number of times the wave front W passed through the point that was the W' before the front W' originated. This invariant can be easily calculated from the current picture of wave fronts on M without the knowledge of the propagation law for the wave fronts. Moreover, in fact we even do not need to know the topology of M outside of a part V of M such that W and W' are null-homotopic in V. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/krspecDD Title: An Atiyah-Hirzebruch spectral sequence for KR-theory Author: Daniel Dugger Department of Mathematics, University of Oregon, Eugene, OR 97403 email: ddugger---math.uoregon.edu Abstract: We construct a spectral sequence for KR-theory which is analagous to the spectral sequence relating motivic cohomlogy to algebraic K-theory. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/mult1DD Title: Multiplicative structures on spectral sequences I Authors: Daniel Dugger Department of Mathematics, University of Oregon, Eugene, OR 97403 email: ddugger---math.uoregon.edu Abstract: This is mostly an expository paper, recording basic facts about towers of homotopy fiber sequences. We show that a pairing of towers induces an associated pairing of spectral sequences, for towers of spaces and towers of spectra. In the hope that this might eventually be a useful reference for people, feel free to send me suggestions for things that should be improved (with the understanding that it might be a while before I get around to implementing them). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/mult2DD Title: Multiplicative structures on spectral sequences II Authors: Daniel Dugger Department of Mathematics, University of Oregon, Eugene, OR 97403 email: ddugger---math.uoregon.edu Abstract: This paper summarizes the constructions of pairings for some of the standard spectral sequences in algebraic topology. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/NT Title: Normalizers of Tori Authors: W. G. Dwyer AND C. W. Wilkerson, Notre Dame and Purdue Suppose that G is a connected compact Lie group and that T G is a maximal torus, or in other words a maximal connected abelian subgroup. The normalizer NT of T lies in a short exact sequence (1.1) 1 -> T -> NT -> W -> 1 in which W is a finite group called the Weyl group of G. In this pa- per we reformulate some ideas of Tits [27 ] in order to describe exactly which groups appear as such an NT . This leads to an analogous deter- mination of which groups appear as the normalizer NT~ of a maximal 2-discrete torus in a connected 2-compact group (1.16). In the compact Lie group case, NT determines G up to isomorphism [3], and so in listing the possible NT 's we are giving an alternative approach to the classification of connected compact Lie groups them- selves. In contrast, it is not known that the normalizer of a maximal 2-discrete torus in a connected 2-compact group X determines X up to equivalence. However, this seems likely to be true [23 ] [19 ], and we hope that the results of this paper will eventually contribute to a classification of connected 2-compact groups. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Ibanez-Rudyak-Tralle/aspherical Title: On the fundamental groups of symplectically aspherical manifolds Authors: R. Ibanez, Yu. Rudyak, A. Tralle Adresses of Authors: R. Ibanez, Departamento de Matematicas, Facultad de Ciencias, Universidad del Pais Vasco, Apdo. 644, 48080 Bilbao, Spain Yu. Rudyak, Department of Mathematics, Universoty of Florida, 358 Little Hall, Gainesville, FL 32601, USA A. Tralle, Department of Mathematics, University of Warmia and Mazura, 10561 Olsztyn, Poland email: mtpibtor---lg.ehu.es rudyak---math.ufl.edu tralle---matman.uwm.edu.pl In this paper we are interested in the fundamental groups of closed symplectically aspherical manifolds; i.e., of symplectic manifolds whose symplectic form vanishes on 2-dimensional spherical homology classes. Motivated by some results of Gompf, we consider two classes of fundamental groups of symplectically aspherical manifolds: with trivial and-non-trivial second homotopy group. Relations between these classes are discussed. We show that several important classes of groups can be realized in both classes. Also, we notice that there are some dimensional phenomena in the realization problem. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Ibanez-Rudyak-Tralle-Ugarte/HomotopySymplecticKahler Title: On certain geometric and homotopy properties of closed symplectic manifolds Authors: R. Ibanez, Yu. Rudyak, A. Tralle, L. Ugarte Adresses of Authors: R. Ibanez, Departamento de Matematicas, Facultad de Ciencias, Universidad del Pais Vasco, Apdo. 644, 48080 Bilbao, Spain Yu. Rudyak, Department of Mathematics, Universoty of Florida, 358 Little Hall, Gainesville, FL 32601, USA A. Tralle, Department of Mathematics, University of Warmia and Mazura, 10561 Olsztyn, Poland L. Ugarte, Departamento de Matem\'aticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain email: mtpibtor---lg.ehu.es rudyak---math.ufl.edu tralle---matman.uwm.edu.pl ugarte\---posta.unizar.es The paper deals with relations between the Hard Lefschetz property, (non)vanishing of Massey products and the evenness of odd-degree Betti numbers of closed symplectic manifolds. It is known that closed symplectic manifolds can violate all these properties (in contrast with the case of Kaehler manifolds). However, the relations between such homotopy properties seem to be not analyzed. This analysis may shed a new light on topology of symplectic manifolds. In the paper, we summarize our knowledge in tables (different in the simply-connected and in symplectically aspherical cases). Also, we discuss the variation of symplectically harmonic Betti numbers on some 6-dimensional manifolds. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Oliver/limz Equivalences of classifying spaces completed at the prime two Bob Oliver We prove here the Martino-Priddy conjecture for the prime $2$: the $2$-completions of the classifying spaces of two groups $G$ and $G'$ are homotopy equivalent if and only if there is an isomorphism between their Sylow $2$-subgroups which preserves fusion. This is a consequence of a technical algebraic result, which says that for a finite group $G$, the second higher derived functor of the inverse limit vanishes for a certain functor $\calz_G$ on the $2$-subgroup orbit category of $G$. The proof of this result uses the classification theorem for finite simple groups. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Oprea-Rudyak/cat3man Title: Detecting Elements and Lusternik--Schnirelmann Category of 3-Manifolds Authors: John Oprea, Yuli Rudyak Addresses: John Oprea, Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115 U.S.A Yuli Rudyak, Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105 Gainesville, FL 32611-8105 U.S.A e-mail: rudyak---math.ufl.edu Abstract: In this paper, we give a new simplified calculation of the Lusternik-Schnirelmann category of closed 3-manifolds. We also describe when 3-manifolds have detecting elements and prove that 3-manifolds satisfy the equality of the Ganea conjecture. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Rudyak/PLstructures (This is an updated version of a paper already on the archive) Title: Piecewise linear structures on topological manifolds Author: Yuli Rudyak Address: Yuli Rudyak, Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105 Gainesville, FL 32611-8105, USA e-mail: rudyak---math.ufl.edu Abstract: This is a survey paper where we expose the Kirby--Siebenmann results on classification of PL structures on topological manifolds and, in particular, the homotopy equivalence TOP/PL=K(Z/2,3) and the Hauptvermutung for manifolds. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Wilkerson/e8-lab Lab Notes on the exceptional Lie group $E_8$ at the prime $2$ \author[C. W. Wilkerson]{Clarence W. Wilkerson, Jr.} \dedicatory{Dedicated to Morton L. Curtis (1921-1989).} \address{Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395} \thanks{Thanks to the National Science Foundation, Purdue University, Johns Hopkins University, and Fukuoka University for financial support during this research and the 2000 sabbatical of the author. Thanks to the Clay Foundation for travel support during this research.} \email{wilker---math.purdue.edu} This is an account of the author's use of computer algebra tools to explore the structure of the maximal elementary abelian $2$-subgroups of the exceptional Lie group $E_8$. The principal result obtained thus far by these methods is that any rank $8$ connected $2$-compact group $(BX,X)$ with Weyl group isomorphic to that of the exceptional Lie group $E_8$ has its normalizer of the maximal torus isomorphic to that of $E_8$ at the prime $2$. Similar results hold for the comparison of possible exotic forms of $G_2$, $DI(4)$, $F_4$, and $E_7/\Center(E_7)$ to the standard forms.\\ Corollaries of this result include that the Krull dimension of the mod $2$ cohomology of such $BX$ is $9$ and that the cohomology ring is not Cohen-Macaulay. \\ ----------------------------------------------- I had to do some hand editing of abstracts this time, so remember to include author's name and title of paper with the abstract, at the least. It is also suggested that you include author's e-mail and AMS subject classifications. 7 new papers this time, from Andersen-Bauer-Grodal-Pedersen, Baas-Dundas-Rognes, Granja, Grojnowski (this is his old paper about equivariant elliptic cohomology), Hornbostel, Kuhn, and Morava. Mark Hovey New papers appearing on hopf between 5/17/03 and 6/18/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Andersen-Bauer-Grodal-Pedersen/loopnotlie Title: A finite loop space not rationally equivalent to a compact Lie group Authors: Kasper K. S. Andersen, Tilman Bauer, Jesper Grodal, Erik K. Pedersen Subj-class: Algebraic Topology; Geometric Topology MSC-class: 55P35; 55P15, 55R35 Comments: 8 pages, arXiv : math.AT/0306234 We construct a connected finite loop space of rank $66$ and dimension $1254$ whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is minimal, i.e., that any finite loop space of rank less than $66$ is in fact rationally equivalent to a compact Lie group, extending the classical known bound of $5$. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Baas-Dundas-Rognes/segal60 Title of Paper: Two-vector bundles and forms of elliptic cohomology Authors: Nils A. Baas, Bjorn I. Dundas and John Rognes Addresses of Authors: Department of Mathematical Sciences The Norwegian University of Science and Technology NO-7491 Trondheim Norway Department of Mathematical Sciences The Norwegian University of Science and Technology NO-7491 Trondheim Norway Department of Mathematics University of Oslo NO-0316 Oslo Norway Email address of Authors: baas---math.ntnu.no, dundas---math.ntnu.no and rognes---math.uio.no In this paper we define 2-vector bundles as suitable bundles of 2-vector spaces over a base space, and compare the resulting 2-K-theory with the algebraic K-theory spectrum K(V) of the 2-category of 2-vector spaces, as well as the algebraic K-theory spectrum K(ku) of the connective topological K-theory spectrum ku. We explain how K(ku) detects v_2-periodic phenomena in stable homotopy theory, and as such is a form of elliptic cohomology. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Granja/notehpn Title: Self maps of HP^n via the unstable Adams spectral sequence Authors: Gustavo Granja AMS Classification numbers: 55S35,55S36,55S37 Address of Author: Departamento de Matematica Instituto Superior Tecnico Av. Rovisco Pais 1049-001 Lisboa Portugal Email address of Author: ggranja---math.ist.utl.pt Abstract: We use obstruction theory based on the unstable Adams spectral sequence to construct self maps of finite quaternionic projective spaces. As a result, a conjecture of Feder and Gitler regarding the classification of self maps up to homology is proved in two new cases. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Grojnowski/deloc Delocalized equivariant elliptic cohomology by Ian Grojnowski This is an old paper, which has been circulating quietly for almost a decade. It contains a definition of an equivariant elliptic cohomology theory for compact connected Lie groups and reasonable topological spaces. The theory is defined over Q, i.e. neglects torsion completely, and yet was still interesting. This is because of the well known heuristic identifying elliptic cohomology with something like the K-theory of the loop space. The functor of "loops into" is not local---there is no Mayer-Vietoris style patching. Yet elliptic cohomology has such a property. However the equivariant elliptic cohomology defined here does not satisfy such a naive locality propery. Instead, the elliptic cohomology of a space is a non-trivial bundle on the canonical abelian variety associated to the group. The crudest invariant of such a bundle is its first Chern class. This is a combinatorial shadow of the failure of locality. These same obstruction invariants occur in the study of semi-infinite D-modules on the infinitesimal neighbourhood of formal loops in the loop space of an algebraic variety; just as one would expect. -- Since this paper was written there have been several developments. Rosu and Ando used this theory to give a new proof of Witten rigidity, and Greenlees constructed a model for that part of rational equivariant S^1 homotopy that is seen by an elliptic cohomology theory. (There has also been the extraordinary work of Hopkins et al on tmf). 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hornbostel/Motchrom Chromatic motivic homotopy theory by Jens Hornbostel We construct a motivic version of the chromatic filtration and the chromatic spectral sequence. This should be used to study the stable ${\bf A}^1$-homotopy groups of the motivic sphere spectrum. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/Kuhn Title: Localization of Andre--Quillen--Goodwillie towers, and the periodic homology of infinite loopspaces Author: Nicholas J. Kuhn AMS classification numbers: 55P43, 55P47, 55N20, 18G55 Address: Department of Mathematics, University of Virginia, Charlottesville, VA 22903 email: njk4x---virginia.edu abstract: Let K(n) be the nth Morava K--theory at a prime p. This paper is a thorough study of questions like the following: to what extent does the K(n)--localization, or the K(n)--homology, of a spectrum X determine the K(n)--homology of its 0th space X_0? Our methods combine techniques from modern homotopical algebra with chromatic homotopy. In particular, we use the telescopic functors of Bousfield and the author (dependent on the Nilpotence Theorem of Devanitz, Hopkins, and Smith), as well as Topological Andre--Quillen Homology and Goodwillie calculus in nonconnective settings. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/SegalMS Heisenberg groups and algebraic topology by Jack Morava This paper overlaps considerably with earlier, sketchier papers about the Tate cohomology of circle actions and its connection to Heisenberg groups. It will appear in the Segal Festschrift: We study the Madsen-Tillmann spectrum $\C P^\infty_{-1}$ as a quotient of the Mahowald pro-object $\C P^{\infty}_{-\infty}$, which is closely related to the Tate cohomology of circle actions. That theory has an associated symplectic structure, whose symmetries define the Virasoro operations on the cohomology of moduli space constructed by Kontsevich and Witten. ------------------------------------------------ 6 new papers this time, from Baas-Dundas-Rognes (an update), Richter, Sinha, Strickland, and (Jim) Turner (2), Mark Hovey New papers appearing on hopf between 6/18/03 and 7/11/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Baas-Dundas-Rognes/segal60 Title of Paper: Two-vector bundles and forms of elliptic cohomology Authors: Nils A. Baas, Bjorn I. Dundas and John Rognes Email address of Authors: baas---math.ntnu.no, dundas---math.ntnu.no and rognes---math.uio.no In this paper we define 2-vector bundles as suitable bundles of 2-vector spaces over a base space, and compare the resulting 2-K-theory with the algebraic K-theory spectrum K(V) of the 2-category of 2-vector spaces, as well as the algebraic K-theory spectrum K(ku) of the connective topological K-theory spectrum ku. We explain how K(ku) detects v_2-periodic phenomena in stable homotopy theory, and as such is a form of elliptic cohomology. (This is an new version of a paper previously on Hopf). 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Richter/Richter-Lambda-EHP Title: Lambda algebra unstable composition products and the Lambda EHP sequence Author: William Richter AMS Classification numbers: 55T15, 55Q40, 55Q25 Address: Math Department, Northwestern University, Evanston IL 6020 Email: richter---math.nwu.edu Abstract: Simple combinatorial proofs are given of Lambda algebra results, mostly due to Priddy & the 6 authors, but also the ``Adams filtration better'' unstable Lambda products of Wang, Mahowald and Singer: Lambda^{s,t}(n) --- Lambda(n+t ) ---> Lambda(n) which imply the folklore Lambda EHP sequence Lambda(n) >-E--> Lambda(n+1) -H-->> Lambda(2n+1) The 6 authors proved Lambda(n) is a chain complex, but not that H is a chain map. A careful reader could deduce a proof from the papers of Wang, Mahowald and Singer, but Singer, who best stated the formulas, gave no proofs. New results: combinatorial proofs of the Lambda admissible monomial basis; the differential d is well-defined. The paper should be accessible to geometers interested in forthcoming applications with Mahowald on 3-cell Poincare complexes. Perhaps the Lambda algebra is undergoing a Renaissance, as 2 young people, Mark Behrens and Mizuho Hikida are doing interesting new work in it. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Sinha/compactify Title: Manifold theoretic compactifications of configuration spaces. Author: Dev P. Sinha AMS Class: 55R80; 32J05 LANL ID: math.GT/0306385 Addresses: Departments of Mathematics, University of Oregoni, Eugene, OR 97403 Email: dps---math.uoregon.edu Abstract: We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton-MacPherson and Axelrod-Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification. Our constructions are elementary and give simple global coordinates for the compactified configuration space of a general manifold embedded in Euclidean space. We stratify the canonical compactification, identifying the diffeomorphism types of the strata in terms of spaces of configurations in the tangent bundle, and give completely explicit local coordinates around the strata as needed to define a manifold with corners. We analyze the quotient map from the canonical to the simplicial compactification, showing it is a homotopy equivalence. We define projection maps and diagonal maps, which for the simplicial variant satisfy cosimplicial identities. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Strickland/axsurv Axiomatic stable homotopy--a survey by N. P. Strickland We survey various approaches to axiomatic stable homotopy theory, with examples including derived categories, categories of (possibly equivariant or localized) spectra, and stable categories of modular representations of finite groups. We focus mainly on representability theorems, localisation, Bousfield classes, and nilpotence. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Turner/finite On simplicial commutative algebras with finite Andre-Quillen homology by James M. Turner L. Avramov, following D. Quillen, posed a conjecture to the effect that if $R \to A$ is a homomorphism of Noetherian rings then the Andr\'e-Quillen homology on the category of A-modules satisfies: $D_{s}(A|R;-) = 0$ for $s\gg 0$ implies $D_{s}(A|R;-) = 0$ for s>2. In an earlier paper, the author posed an extended version of this conjecture which considered A to be a simplicial commutative R-algebra with Noetherian homotopy such that the characteristic of $\pi_{0}A$ is non-zero. In addition, a homotopy characterization of such algebras was described. The main goal of this paper is to develop a strategy for establishing this extended conjecture and provide a complete proof when R is Cohen-Macaulay of characteristic 2. Note: this paper replaces "Nilpotency in the homotopy of simplicial commutative algebras". 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Turner/Gorenstein Characterizing Simplicial Commutative Algebras with Vanishing Andr'e-Quillen Homology by James M. Turner The use of homological and homotopical devices, such as Tor and Andr\'e-Quillen homology, have found substantial use in characterizing commutative algebras. The primary category setting has been differentially graded algebras and modules, but recently simplicial categories have also proved to be useful settings. In this paper, we take this point of view up a notch by extending some recent uses of homological algebra in characterizing Noetherian commutative algebras to characterizing simplicial commutative algebras having finite Noetherian homotopy through the use of simplicial homotopy theory. These characterizations involve extending the notions of locally complete intersections and locally Gorenstein algebras to the simplicial homotopy setting. --------------------------------------------------------- 4 new papers this time, from Bartels-Reich, Goodwillie (Calc III!), Gorbounov-Malikov, and Kuhn. Mark Hovey New papers appearing on hopf between 7/11/03 and 8/21/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bartels-Reich/isoIIhopf Title: On the Farrell-Jones Conjecture for higher algebraic K-theory Authors: Arthur Bartels, Holger Reich e-mail adresses: bartelsa---math.uni-muenster.de, reichh---math.uni-muenster.de arxiv: math.AT/0308030 Abstract: We prove the Farrell-Jones Isomorphism Conjecture about the algebraic K-theory of a group ring RG in the case where the group G is the fundamental group of a closed Riemannian manifold with strictly negative sectional curvature. The coefficient ring R is an arbitrary associative ring with unit and the result applies to all dimensions. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Goodwillie/calculus3 Title: Calculus III, Taylor series Author: Thomas G. Goodwillie Author's e-mail address: tomg---math.brown.edu Abstract: We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal n-excisive approximation, which may be thought of as its n-excisive part. Homogeneous functors, meaning n-excisive functors with trivial (n-1)-excisive part, can be classified: they correspond to symmetric functors of n variables that are 1-excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen's algebraic K-theory. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Gorbounov-Malikov/LG-CY-try Vertex algebras and the Landau-Ginzburg/Calabi-Yau correspondence Vassily Gorbounov and Fyodor Malikov We construct a spectral sequence that converges to the cohomology of the chiral de Rham complex over a Calabi-Yau hypersurface and whose first term is a vertex algebra closely related to the Landau-Ginburg orbifold. As an application, we prove an explicit orbifold formula for the elliptic genus of Calabi-Yau hypersurfaces. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/Tate Title: Tate cohomology and periodic localization of polynomial functors Author: Nicholas J. Kuhn AMS classification numbers: Primary 55P65; Secondary 55N22, 55P60, 55P91 Address: Department of Mathematics, University of Virginia, Charlottesville, VA 22903 email: njk4x---virginia.edu abstract: In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a v_n self map of a finite S--module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T(n) is independent of choices. Goodwillie's general theory says that to any homotopy functor F from S--modules to S--modules, there is an associated tower under F, {P_dF}, such that F --> P_dF is the universal arrow to a d--excisive functor. Our first theorem says that P_dF --> P_{d-1}F always admits a homotopy section after localization with respect to T(n) (and so also after localization with respect to Morava K--theory K(n)). Thus, after periodic localization, polynomial functors split as the product of their homogeneous factors. This theorem follows from our second theorem which is equivalent to the following: for any finite group G, the Tate spectrum t_G(T(n)) is weakly contractible. This strengthens and extends previous theorems of Greenlees--Sadofsky, Hovey--Sadofsky, and Mahowald--Shick. The Periodicity Theorem is used in an essential way in our proof. The connection between the two theorems is via a reformulation of a result of McCarthy on dual calculus. ------------------------------------------------ 12 new papers this time, from Bendersky-DavisD-Mahowald, Dugger-Isaksen, Jessup-Lupton, KrauseH, Lupton, Lupton-SmithSB (2 papers), Nofech, Pengelley-Williams, Pitsch-Scherer, Toen-Vezzosi, and ZhouXueguang. Mark Hovey New papers appearing on hopf between 8/21/03 and 9/26/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bendersky-DavisD-Mahowald/sgd2 Stable geometric dimension of vector bundles over even-dimensional real projective spaces Martin Bendersky, Donald M. Davis, and Mark Mahowald mbenders---shiva.hunter.cuny.edu dmd1---lehigh.edu mark---math.northwestern.edu Abstract In 1981, Davis, Gitler, and Mahowald determined the geometric dimension of stable vector bundles of order 2^e over RP^{2n} if e > 74 and n is sufficiently large. In this paper, we use the Bendersky-Davis computation of v1-periodic homotopy groups of SO(m) to determine this geometric dimension for all values of e (still provided that n is sufficiently large). The same formula that worked for e>74 works for e>5, but for e \le 5 the geometric dimension is often different due to anomalies in the v1-periodic homotopy groups of SO(m) when m<11. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Isaksen/hopfDI The Hopf condition for bilinear forms over arbitrary fields Daniel Dugger (ddugger---math.uoregon.edu) Daniel C. Isaksen (isaksen---math.wayne.edu) We settle an old question about the existence of certain "sums-of-squares" formulas over a field F. A classical result, due originally to Hopf and proven via topological methods, says that if such a formula exists over a field of characteristic 0 then certain binomial coefficients must be even. We use motivic methods to prove that the result also holds for fields of characteristic p. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Jessup-Lupton/JessLup Title: Free Torus Actions and Two-Stage Spaces Author(s): Barry Jessup, Gregory Lupton Author's e-mail address: Bjessup---sciences.uottawa.ca, G.Lupton---csuohio.edu AMS classification number: 55P62, 57S99 Other useful information: math.AT/0309434. To appear, Math. Proc. Camb, Philos. Soc. Abstract: We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/KrauseH/quotient Title: Cohomological quotients and smashing localizations Author: Henning Krause Email: henning---maths.leeds.ac.uk Abstract: The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier's construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S, a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic K-theory and demonstrates the relevance of the telescope conjecture for derived categories. Another application leads to a derived analogue of an almost module category in the sense of Gabber-Ramero. It is shown that the derived category of an almost ring is of this form. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton/Catell Title: The Rational Toomer Invariant and Certain Elliptic Spaces Author(s): Gregory Lupton Author's e-mail address: G.Lupton---csuohio.edu AMS classification number: Primary 55P62, 55M30; Secondary 55T10 Other useful information: math.AT/0309392. Contemporary Mathematics, Vol. 316 (2002), 135--146 Abstract: We give an explicit formula for the rational category of an elliptic space whose minimal model has a homogeneous-length differential. We also show that for such a space, there are no gaps in the sequence of integers realized as the rational Toomer invariant of some cohomology class. With an additional hypothesis, we show a result from which we deduce the relation dim(H^*(X;Q)) >= 2 cat_0(X). 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-SmithSB/Cyclic Title: Cyclic Maps in Rational Homotopy Theory Author(s): Gregory Lupton, Samuel Bruce Smith Author's e-mail address: G.Lupton---csuohio.edu, smith---sju.edu AMS classification number: 55P62, 55Q05 Other useful information: math.AT/0309423 Abstract: The notion of a cyclic map g: A -> X is a natural generalization of a Gottlieb element in pi_n(X). We investigate cyclic maps from a rational homotopy theory point of view. We show a number of results for rationalized cyclic maps which generalize well-known results on the rationalized Gottlieb groups. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-SmithSB/Gseq Title: Rationalized Evaluation Subgroups of a Map and the Rationalized G-Sequence Author(s): Gregory Lupton, Samuel Bruce Smith Author's e-mail address: G.Lupton---csuohio.edu, smith---sju.edu AMS classification number: 55P62, 55D23 Other useful information: math.AT/0309432 Abstract: Let f: X -> Y be a based map of simply connected spaces. The corresponding evaluation map w: map(X,Y;f) -> Y induces a homomorphism of homotopy groups whose image in pi_n(Y) is called the nth evaluation subgroup of f. The nth Gottlieb group of X occurs as the special case in which Y = X and f = 1_X. We identify the homomorphism induced on rational homotopy groups by this evaluation map, in terms of a map of complexes of derivations constructed using Sullivan minimal models. Our identification allows for the characterization of the rationalization of the nth evaluation subgroup of f. It also allows for the identification of several long exact sequences of rational homotopy groups, including the long exact sequence induced on rational homotopy groups by the evaluation fibration. As a consequence, we obtain an identification of the rationalization of the so-called G-sequence of the map f. This is a sequence---in general not exact---of groups and homomorphisms that includes the Gottlieb groups of X and the evaluation subgroups of f. We use these results to study the G-sequence in the context of rational homotopy theory. We give new examples of non-exact G-sequences and uncover a relationship between the homology of the rational G-sequence and negative derivations of rational cohomology. We also relate the splitting of the rational G-sequence of a fibre inclusion to a well-known conjecture in rational homotopy theory. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Nofech/e2 An $E^2$-type closed model category for bisimplicial groups Alexander Nofech anofech---shaw.ca A closed model category structure is defined on the category of bisimplicial groups in which the weak equivalences are isomorphisms on bigraded homotopy groups $\pi_{k,l}$ and at the same time isomorphisms on the $E^2$ term of the Quillen spectral sequence. There is an analogue of the spiral exact sequence of Dwyer-Kan-Stover. One of the reasons for looking specifically at groups rather than at a general construction of a $E^2$-type model category is that it is easier to find the abelianization of a cofibrant group. This structure is considered as a convenient setting for a study of the relation between bigraded homotopy and hyperhomology. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/subsmalg Global Structure of the mod 2 Symmetric Algebra over the Steenrod algebra. David J. Pengelley (davidp---nmsu.edu) Frank Williams (frank---nmsu.edu) The algebra S of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod algebra A and is isomorphic to the mod two cohomology of BO, the classifying space for vector bundles. We provide a minimal presentation for S in the category of unstable A-algebras, i.e., a minimal set of generators and a minimal set of relations. From this we produce minimal presentations for various unstable A-algebras associated with the cohomology of related spaces, such as the BO(2^n - 1) that classify finite dimensional vector bundles, and the connected covers of BO. The presentations then show that certain of these unstable A-algebras coalesce to produce the mod 2 Dickson algebras, and we speculate about possible related topological realizability. Our methods also produce a related simple A-module presentation of the cohomology of infinite-dimensional real projective space, with a filtration having well-known filtered quotients. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Pitsch-Scherer/completion Title: Homology fibrations and group completion revisited Authors : Jerome SCHERER and Wolfgang PITSCH e-mail : jscherer---mat.uab.es and Wolfgang.Pitsch---math.unige.ch AMS classification : Primary 55U10; Secondary 19D06 arXiv : math.AT/0307339 Abstract : We give a proof of the Jardine-Tillmann generalized group completion theorem. It is much in the spirit of the original homology fibration approach by McDuff and Segal, but follows a modern treatment of homotopy colimits, using as little simplicial technology as possible. We compare simplicial and topological definitions of homology fibrations. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Toen-Vezzosi/bravenew Title: ``Brave New'' Algebraic Geometry and global derived moduli spaces of ring spectra Authors: Bertrand Toen, Gabriele Vezzosi Author's e-mail address: toen---picard.ups-tlse.fr ; vezzosi---dm.unibo.it Other useful information: arXive submission numbermath.AT\0309145 Abstract: We develop homotopical algebraic geometry (see math.AG/0207028) in the special context where the base symmetric monoidal model category is that of spectra S, i.e. what might be called, after Waldhausen, "brave new algebraic geometry". We discuss various model topologies on the model category of commutative algebras in S, the associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry), and finally show how to define global moduli spaces of associative ring spectra structures as geometric S-stacks. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/ZhouXueguang/zhouxin title of the paper: A reply author: Zhou Xueguang AMS classification numbers: Q55 Address of author:Department of Mathematics, Nankai University, Tianjin 300071, People's Republic of China Email address of author: zhengqb---eyou.com Abstract: In this paper, we answer the question why V(n) exists for all non-negative integers $n$. ----------------------------------------------------------------------- 10 new papers this time, from Dugger-Isaksen, Flores, Gaudens, Kitchloo-Wilson, Klein, LinJP, Luo, Nam, Sauvageot, and Schwede. Mark Hovey New papers appearing on hopf between 9/26/03 and 10/24/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Isaksen/motcell Title: Motivic cell structures Authors: Daniel Dugger and Daniel C. Isaksen Authors' e-mail address: ddugger---math.uoregon.edu and isaksen---math.wayne.edu Abstract: An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We explain why the algebraic K-theory and algebraic cobordism spectra are both cellular, and prove some Kunneth theorems for cellular objects. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Flores/draft1 NULLIFICATION AND CELLULARIZATION OF CLASSIFYING SPACES OF FINITE GROUPS by RAM'ON J. FLORES Departamento de Matem'aticas, Universidad Aut'onoma de Barcelona, E-08193 Bellaterra, Spain E-mail address: ramonj---mat.uab.es Mathematical subject classification: 55P20, 55P80. Abstract. In this note we discuss the effect of the BZ/p-nullification and the BZ/p-cellularization functors over classifying spaces of finite groups, and we compare them with the corresponding ones with regard to Moore spaces, that have been intensively studied in the last years. We describe the BZ/p- nullification of BG by means of a Postnikov fibration, and we classify all finite groups G for which BG is BZ/p-cellular. In particular, we relate the effect these (co)localizations have over the fundamental group with the analogous functors in the category of groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Gaudens/bocksteinnul Title: A remark on N. Kuhn's unbounded strong realization conjecture Author(s): Gerald Gaudens Author's e-mail address: gaudens---math.univ-nantes.fr AMS classification number: 55S10; 57S35 Abstract: N. Kuhn has given several conjectures on the special features satisfied by the singular cohomology of topological spaces with coefficients in a finite prime field, as modules over the Steenrod algebra. The so-called Realization conjecture was solved in special cases By N. Kuhn and in complete generality by L. Schwartz. The more general Strong realization conjecture has been settled at the prime 2, as a consequence of the work of L. Schwartz, and the subsequent work of F.-X. Dehon and the author. In this note, we are interested in the even more general Unbounded strong realization conjecture. We shall prove that it holds at the prime $2$ for the class of spaces whose cohomology has a trivial Bockstein action in high degrees. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Wilson/kitchloo-wilson Title: On fibrations related to real spectra Authors: Nitu Kitchloo and W. Stephen Wilson E-mail addresses: nitu---math.jhu.edu, wsw---math.jhu.edu Address: Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 Abstract: We consider real spectra, collections of Z/(2)-spaces indexed over Z direct sum Z_\alpha with compatibility conditions. We produce fibrations connecting the homotopy fixed points and the spaces in these spectra. We also evaluate the map which is the analogue of the forgetful functor from complex to reals composed with complexification. Our first fibration is used to connect the real 2^{n+2}(2^n-1)-periodic Johnson-Wilson spectrum ER(n) to the usual 2(2^n-1)-periodic Johnson-Wilson spectrum, E(n). Our main result is the fibration \Sigma^{\lambda(n)} ER(n) -> ER(n) -> E(n), where \lambda(n) = 2^{2n+1}-2^{n+2}+1. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Klein/embclass Title: On embeddings in the sphere Author: John R. Klein Author's e-mail address: klein---math.wayne.edu Abstract: We consider embeddings of a finite complex in a sphere. We give a homotopy theoretic classification such embeddings in a wide range. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/LinJP/Lin=HspaceAnalog1 (This abstract was sent in dvi form; the program we use to convert is not perfect). H-spaces analogous to E8 mod 3 Dedicated to the memory of Masahiro Sugawara James P. Lin Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112, U.S.A. email:jimlin---euclid.ucsd.edu Abstract: Let p be an odd prime. Let X0 be a finite, p-local, simply connected homotopy associative H-space. Suppose H* (X0; Zp) contains the subalgebra Zp [x0,_z0]_xp p(r0, P1 r0, Pp P1 r0, y0) 0, z0 satisfying z0 = Pp x0 = Q0Pp P1 r0, Pp P1 r0 = P1 y0 for r0 2 H3 (X0; Zp). The only known examples occur for p = 3 and involve the Lie group E8. In this note we prove that if X0 exists, then p must be 3. Thus there are no homotopy associative H-space analogues of E8mod 3 for primes bigger than 3. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Luo/pre (This is an updated version) Closed model categories for presheaves of simplicial groupoids and presheaves of 2-groupoids Zhi-ming Luo We prove that the category of presheaves of simplicial groupoids and the category of presheaves of 2-groupoids have Quillen closed model structures. We also show that the homotopy categories associated to the two categories are equivalent to the homotopy categories of simplicial presheaves and homotopy 2-types, respectively. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Nam/transfer Title : Transfert alg'ebrique et repr'esentation modulaire du groupe lin'eaire Author : Tran Ngoc Nam Author's e-mail address : trngnam---hotmail.com Author's mailing address : LAGA, Universit'e Paris 13, 93430 Villetaneuse, France Abstract : On se propose de d'eterminer la dimension d'une repr'esentation du groupe lin'eaire d'efinie par un sous-espace vectoriel de l'alg`ebre `a puissances divis'ees, d'expliciter l'image du transfert alg'ebrique en degr'e g'en'erique et celle du transfert alg'ebrique quadruple, d'identifier les ind'ecomposables de degr'e pair de l'alg`ebre polynomiale `a 4 variables, vue comme module sur l'alg`ebre de Steenrod mod 2. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Sauvageot/thesis STABILISATION DES COMPLEXES CROISES Orin R. Sauvageot orin.sauvageot---epfl.ch Ecole Polytechnique Federale de Lausanne Institute of Mathematics This is my PhD thesis in FRENCH, 158 pages. The graphic files C-tensor-I.eps, pi-delta-4.eps and pi-xc.eps are included in the zip archives thesis-print.dvi.zip and thesis-screen.dvi.zip. (Note from Mark; you should get these eps files individually if you get the dvi file. The file thesis.dvi is thesis-print.dvi; thesis-screen.dvi is in case you have trouble viewing the diagrams in thesis.dvi on your screen. The files thesis.ps and thesis.pdf already have the eps files embedded) Abstract In this doctoral thesis we present a stabilization of the category of crossed complexes. Our work is motivated by the difficulty one has in performing algebraic calculations in Boardman's stable homotopy category, since products and actions are defined only up to homotopy in the underlying category of spectra, as defined by Bousfield and Friedlander. To correct this lack of precision, a number of new models of the stable homotopy category have been developed in which algebraic constructions are exactly defined. One such model is the category of symmetric spectra on simplicial sets, the manipulation of which is still not easy, however. The idea behind this thesis is to stabilize the category of crossed complexes, as it is an interesting approximation to the category of simplicial sets, reflecting certain, though not all, nonabelian homotopical information concerning simplicial sets. We have stabilized it according to the procedure codified in Hovey's "Spectra and symmetric spectra in general model categories". Stabilization requires that the category of crossed complexes satisfies certain properties. We have succeeded in proving these properties, in each case establishing a previously unknown result. For example, we have shown that it is cofibrantly generated and that it is a symmetric monoidal model category. Furthermore we have verified that it is a proper, cellular category. In proving the properness we have answered an open question posed by Brown and Golasinski. In the course of establishing these properties we have established a nonabelian version of the 5-Lemma. A crossed complex is a generalization of a chain complex of abelian groups. We have shown, however, that the stabilization of crossed complexes is homotopy equivalent to that of the category of chain complexes. On the other hand, the situation of unpointed crossed complexes is different, and it is very likely that their stabilization is not that of chain complexes. In order to argue so, we have constructed an innovative simplicial model of the Hopf map. It remains then to give a topological meaning to an unpointed stabilization. An attempt of answer is sketched. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Schwede/Morita Title: Morita theory in abelian, derived and stable model categories Author: Stefan Schwede e-mail address: sschwede---math.uni-muenster.de This is a survey paper, based on lectures given at the Workshop on "Structured ring spectra and their applications" which took place January 21-25, 2002, at the University of Glasgow. The term `Morita theory' is usually used for results concerning equivalences of various kinds of module categories. We focus on the covariant form of Morita theory, so the basic question is: When do two `rings' have `equivalent' module categories ? We discuss this question in different contexts and illustrate it by examples: (Classical) When are the module categories of two rings equivalent as categories ? (Derived) When are the derived categories of two rings equivalent as triangulated categories ? (Homotopical) When are the module categories of two ring spectra Quillen equivalent as model categories ? There is always a related question, which is in a sense more general: What characterizes the category of modules over a `ring' ? The answer is, mutatis mutandis, always the same: modules over a `ring' are characterized by the existence of a `small generator', which plays the role of the free module of rank one. The precise meaning of `small generator' depends on the context, be it an abelian category, a derived category or a stable model category. --------------------------------------- 12 new papers this time, from Aouina-Klein, Chalupnik (3), Fausk-Oliver, Grandis (2), Knudson-Walker, Notbohm-Ray, Sauvageot, Troesch, and ZhouXueguang. Mark Hovey New papers appearing on hopf between 10/24/03 and 11/25/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Aouina-Klein/config_stable Title: On the homotopy invariance of configuration spaces Author(s): Mokhtar Aouina and John R. Klein Author's e-mail address: aouina---math.wayne.edu, klein---math.wayne.edu AMS classification number: Primary 55R80; Secondary 57Q35, 55R70. Abstract: For a closed PL manifold M, we consider the configuration space F(M,k) of ordered k-tuples of distinct points in M. We show that a suitable iterated suspension of F(M,k) is a homotopy invariant of M. The number of suspensions we require depends on three parameters: the number of points k, the dimension of M and the connectivity of M. Our proof uses a mixture of embedding theory and fiberwise algebraic topology. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/cohsdr Title of Paper: Schur_De-Rham complex and its cohomology Author: Marcin Chalupnik Email: mchal---mimuw.edu.pl Abstract: We associate to a Young diagram a complex of strict polynomial functors which we call the Schur-De-Rham complex. Its cohomology turns out to reflect deep combinatorial properties of a diagram. We show that if a ground field is of characteristic p, the Schur-De-Rham complex is acyclic when the p-core of a diagram is nontrivial. We also compute its cohomology for a diagram with a trivial p-core and p-quotient consisting of a single diagram. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/extpol Title of Paper: Extensions of strict polynomial functors Author: Marcin Chalupnik Email: mchal---mimuw.edu.pl Abstract: We compute Ext-groups between various strict polynomial functors important in representation theory (eg. between twisted Weyl and Schur functors). Our method utilizes: computation of the Ext-groups between twisted divided and symmetric powers due to Franjou-Friedlander-Scorichenko-Suslin, resolutions of functors by divided and symmetric powers, interplay between functors and representations of the symmetric group. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/extws Title of Paper: Extensions of Weyl and Schur functors Author: Marcin Chalupnik Abstract: We use here the Schur-De-Rham complex to extend calculations of the Ext-groups between twisted Weyl and Schur functors initiated in the paper ``Extensions of strict polynomial functors''. The main result is a full calculation of those groups in the case of a pair of diagrams which can be obtained from diagrams of the same weights by the operation F described in ``Schur-De-Rham complex and its cohomology''. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk-Oliver/piperfect Title: Continuity of p-perfection for compact Lie groups Authors: Halvard Fausk and Bob Oliver Author's e-mail address: fausk---math.uio.no and bob---math.univ-paris13.fr AMS classification number: 55P91 Abstract: Let G be a compact Lie group, and let pi be any prime or set of primes. We construct a ``pi-perfection map'': a continuous function from the space of conjugacy classes of all closed subgroups of G to the space of conjugacy classes of pi-perfect subgroups with finite index in their normalizer. We use this to show that the idempotent elements of the Burnside ring of G localized at pi are in bijective correspondence with the open and closed subsets of the space of conjugacy classes of pi-perfect subgroups of G with finite index in their normalizer. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Grandis/Grandis.Bsy2 (Note: this paper is only available in pdf format) Normed combinatorial homology and noncommutative tori Marco Grandis Keywords: Cubical sets, noncommutative C*-algebras, combinatorial homology, normed abelian groups. Dipartimento di Matematica Universita` di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis---dima.unige.it http://www.dima.unige.it/~grandis/ Notes: Dip. Mat. Univ. Genova, Preprint 484 (2003), 14 p. Abstract. Cubical sets have a directed homology, studied in a previous paper and consisting of preordered abelian groups, with a positive cone generated by the structural cubes. By this additional information, cubical sets can provide a sort of "noncommutative topology", agreeing with some results of noncommutative geometry but lacking the metric aspects of C*-algebras. Here, we make such similarity stricter by introducing normed cubical sets and their normed directed homology, formed of normed preordered abelian groups. The normed cubical sets associated with "irrational" rotations have thus the same classification up to isomorphism as the well-known irrational rotation C*-algebras. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Grandis/Grandis.Dht1 (Note: this paper is only available in pdf format) Directed homotopy theory, I. The fundamental category Marco Grandis Key words: homotopy theory, homotopical algebra, directed homotopy, fundamental category. Dipartimento di Matematica Universita di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis---dima.unige.it http://www.dima.unige.it/~grandis/ Notes: to appear in: Cahiers Topologie Geometrie Differentielle Categoriques Preprint: Dip. Mat. Univ. Genova, Preprint 443 (2001), 26 p. Revised version: 5 Nov 2001. Abstract. Directed Algebraic Topology is beginning to emerge from various applications. The basic structure we shall use for the foundations of such a theory, a d-space, is a topological space equipped with a family of directed paths, closed under some operations. This allows for directed homotopies, generally non reversible, represented by a cylinder and cocylinder functors. The existence of 'pastings' (colimits) yields a geometric realisation of cubical sets as d-spaces, together with homotopy constructs which will be developed in a sequel. Here, the fundamental category of a d-space is introduced and a 'Seifert-van Kampen' theorem proved; its homotopy invariance rests on directed homotopy of categories. In the process, new shapes appear, for d-spaces but also for small categories, their elementary algebraic model. Applications of such tools are briefly considered or suggested, for objects which model a directed image, or a portion of space-time, or a concurrent process. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Knudson-Walker/hom11-19 Title: Homology of linear groups via cycles in BG x X Author1: Kevin P. Knudson Author2: Mark E. Walker email1: knudson---math.msstate.edu email2: mwalker---math.unl.edu Abstract: Let G be an algebraic group and let X be a smooth integral scheme over a field k. In this paper we construct homology-type groups H_i(X,G) by considering cycles in the simplicial scheme BG x X (an idea suggested by Andrei Suslin). We discuss the basic properties of these groups and construct a spectral sequence, beginning with the groups H_i(\Delta^j,G), which converges to the etale cohomology of the simplicial group BG. These groups are therefore connected with the study of Friedlander's generalized isomorphism conjecture. We also compute some examples, focusing in particular on the case X=Spec(k). In the case where k is the real numbers, there is a connection between the groups H_i and the Z/2-equivariant cohomology of the classifying space of the discrete group G(R). 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Notbohm-Ray/djthrational On Davis-Januszkiewicz Homotopy Types I; Formality and Rationalisation by Dietrich Notbohm} and Nigel Ray For an arbitrary simplicial complex $K$, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of $K$. Subsequently, Buchstaber and Panov gave an alternative construction, which they showed to be homotopy equivalent to Davis and Januszkiewicz's examples. It is therefore natural to investigate the extent to which the homotopy type of a space $X$ is determined by having such a cohomology ring. We begin this study here, in the context of model category theory. In particular, we extend work of Franz by showing that the singular cochain algebra of $X$ is formal as a differential graded noncommutative algebra. We then specialise to the rationals, by proving the corresponding property for Sullivan's {\it commutative\/} cochain algebra; this confirms that the rationalisation of $X$ is unique. In a sequel, we will consider the uniqueness of $X$ at each prime separately, and apply Sullivan's arithmetic square to produce global results in special families of cases. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Sauvageot/simpl-hopf-model A simplicial model for the Hopf map Orin R. Sauvageot Ecole Polytechnique Federale de Lausanne orin.sauvageot---epfl.ch We give an explicit simplicial model for the Hopf map S^3 -> S^2. For this purpose, we construct a model of S^3 as a principal twisted cartesian product K x_{eta} S^2, where K is a simplicial model for S^1 acting by left multiplication on itself, S^2 is given the simplest simplicial model and the twisting map is eta:(S^2)_n -> (K)_{n-1}. We construct a Kan complex for the simplicial model K of S^1. The simplicial model for the Hopf map is then the projection K x_{eta} S^2 -> S^2. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Troesch/troesch_Resolution_of_symmetric_powers Title: A propos d'une question de Friedlander et Suslin I -- Une r'esolution injective des puissances sym'etriques twist'ees (in French) Author: Alain Troesch Address of Author: Institut de Mathematiques de Jussieu, Case 82 4 place Jussieu, F-75252 PARIS CEDEX 05 e-mail address: troesch---math.jussieu.fr Abstract. Some years ago, Friedlander and Suslin constructed an explicit injective resolution of twisted symmetric powers in the category of strict polynomial functors over a ground field of characteristic 2. The factors in this resolutions are given by direct sums of tensor products of (non twisted) symmetric powers. The case of a symmetric power twisted only once is a well-known result: it is some kind of Koszul complex. In characteritic p>2, nothing similar was known up to now, even for a single twist. In this paper, we construct such injective resolutions. The resolutions we construct are in fact "p-resolutions", that is, the differential does not vanish when composed twice, but only when composed p times. This result should unable us to constuct an injective resolution of any twisted functor if we know an injective resolution of the corresponding non twisted functor. This will be the subject of another paper. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/ZhouXueguang/zhou2 title of the paper: The answer to an email of Mr. Douglas C. Ravenel author: Zhou Xueguang Address of author:Department of Mathematics, Nankai University, Tianjin 300071, People's Republic of China Email address of author: zhengqb---eyou.com Abstract: In this paper, we answer the question why V(n) exists for all non-negative integers n. -----------------