Subject: new Hopf listings Date: 21 Jan 2003 09:54:34 -0500 From: Mark Hovey Reply-To: mhovey@wesleyan.edu To: dmd1@lehigh.edu 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. ---------------------Instructions----------------------------- To subscribe or unsubscribe to this list, send a message to Don Davis at dmd1@lehigh.edu with your e-mail address and name. Please make sure he is using the correct e-mail address for you. To see past issues of this mailing list, point your WWW browser to http://math.wesleyan.edu/~mhovey/archive/ If this doesn't work or is missing a few issues, try http://www.lehigh.edu/~dmd1/algtop.html which also has the other messages sent to Don's list. To get the papers listed above, point your Web browser to the URL listed. The general Hopf archive URL is http://hopf.math.purdue.edu There is a web form for submitting papers to Hopf on this site as well. You can also use ftp, explained below. The largest archive of math preprints is at http://xxx.lanl.gov There is an algebraic topology section in this archive. The most useful way to browse it or submit papers to it is via the front end developed by Greg Kuperberg: http://front.math.ucdavis.edu To get the announcements of new papers in the algebraic topology section at xxx, send e-mail to math@xxx.lanl.gov with subject line "subscribe" (without quotes), and with the body of the message "add AT" (without quotes). You can also access Hopf through ftp. Ftp to hopf.math.purdue.edu, and login as ftp. Then cd to pub. Files are organized by author name, so papers by me are in pub/Hovey. If you want to download a file using ftp, you must type binary before you type get . To put a paper of yours on the archive, go to http://hopf.math.purdue.edu and use the web form. You can also use anonymous ftp as above. First cd to /pub/incoming. Transfer the dvi file using binary, by first typing binary then put You should also transfer an abstract as well. Clarence has explicit instructions for the form of this abstract: see http://hopf.math.purdue.edu/new-html/submissions.html In particular, your abstract is meant to be read by humans, so should be as readable as possible. I reserve the right to edit unreadable abstracts. You should then e-mail Clarence at wilker@math.purdue.edu telling him what you have uploaded. I am solely responsible for these messages---don't send complaints about them to Clarence. Thanks to Clarence for creating and maintaining the archive.