Subject: new Hopf listings--January 2001 From: Mark Hovey Date: 05 Feb 2001 09:52:19 -0500 These are the January papers, of which there are 13. Mark Hovey New papers appearing on hopf between 1/1/01 and 2/3/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Baker/regquotients On the homology of regular quotients Andrew Baker Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland. a.baker@maths.gla.ac.uk We construct a free resolution of $R/I^s$ over $R$ where $I\ideal R$ is generated by a (finite or infinite) regular sequence. This generalizes the Koszul complex for the case $s=1$. We easily deduce that for $s>1$, the algebra structure of $\Tor^R_*(R/I,R/I^s)$ is trivial and the reduction $R/I^s\lra R/I^{s-1}$ induces the trivial map of algebras. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Baker-Lazarev/Rmod-ASS On the Adams Spectral Sequence for $R$-modules Andrew Baker \& Andrej Lazarev Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland. a.baker@maths.gla.ac.uk Department of Mathematics, University of Bristol, Bristol BS8 1TW, England. A.Lazarev@bris.ac.uk We consider the Adams Spectral Sequence for $R$-modules based on commutative localized regular quotient ring spectra of a commutative $S$-algebra $R$ in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral sequence is similar to the classical case, and we reduce to algebra involving the cohomology of certain `brave new Hopf algebroids' $E^R_*E$. In order to work out the details we resurrect Adams' original approach to Universal Coefficient Spectral Sequences for modules over an $R$ ring spectrum. We show that the Adams Spectral Sequence for $S_R$ based on $E=R/I[X^{-1}]$ converges to the homotopy of the $E$-nilpotent completion which has homotopy \[ \pi_*\hat{\mathrm{L}}^R_ES_R=R_*[X^{-1}]\sphat_{I_*}. \] We also show that $\hat{\mathrm{L}}^R_ES_R$ is equivalent to $\L^R_ES_R$, the Bousfield localization of $S_R$ with respect to $E$-theory. This seems surprising since the spectral sequence collapses at $\E_2$, but $\E_r$ does not have a vanishing line because of the presence of polynomial generators of positive cohomological degree, thus only one of Bousfield's two standard convergence criteria applies here even though we have this equivalence. The details involve a construction of the internal $I$-adic tower \[ R/I\la R/I^2\la\cdots\la R/I^s\la R/I^{s+1}\la\cdots \] whose homotopy limit is $\hat{\mathrm{L}}^R_ES_R$. Finally, we describe some examples for the case $R=MU$. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bruner-Greenlees/kubg The Connective K-theory of Finite Groups Robert Bruner and John Greenlees MSC2000: Primary 19L41, 19L47, 19L64, 55N15. Secondary 20J06, 55N22, 55N91, 55T15, 55U20, 55U25, 55U30. Department of Mathematics, School of Mathematics and Statistics, Wayne State University, Hicks Building, Detroit MI 48202-3489, Sheffield S3 7RH, USA. UK. rrb@math.wayne.edu, j.greenlees@sheffield.ac.uk Included graphics files: AdamsA4.eps AdamsBip.eps AdamsC2.eps AdamsC4.eps AdamsC5.eps AdamsD8.eps AdamsQ8.eps AdamsSl23.eps AdamsV2.eps AdamsX.eps ExtIE.eps Extku.eps Extl.eps L.eps Qrank4.eps Qrank4lc.eps T3rank6.eps T3rank6lc.eps Xku.eps rank8.eps string.eps tku2.eps Abstract: This paper is devoted to the connective K homology and cohomology of finite groups G. We attempt to give a systematic account from several points of view. In Chapter 1, following Quillen, we use the methods of algebraic geometry to study the ring ku^*(BG) where ku denotes connective complex K-theory. We describe the variety in terms of the category of abelian p-subgroups of G for primes p dividing the group order. The variety is obtained by splicing that of periodic complex K-theory and that of integral ordinary homology, the interest lying in the way these parts fit together. The main technical obstacle is that the Kunneth spectral sequence does not collapse, so we have to show that it collapses up to isomorphism of varieties. In Chapter 2 we give several families of new complete and explicit calculations of the ring ku^*(BG). In Chapter 3 we consider the associated homology ku_*(BG), as a module over ku^*(BG) by using the local cohomology spectral sequence. This gives new specific calculations, but also illuminating structural information, including remarkable duality properties. Finally, in Chapter 4 we make a particular study of elementary abelian groups V. Despite the group-theoretic simplicity of V, the detailed calculation of ku^*(BV) and ku_*(BV) exposes a very intricate structure, and gives a striking illustration of our methods. Unlike earlier work, our description is natural for the action of GL(V). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/JohnsonM/shfloop Loop Spaces as Sheaves: A Sheaf-Theoretic View of Loop Spaces Mark W. Johnson \address {Department of Mathematics\\ University of Notre Dame\\ Notre Dame, IN 46556} \email{johnson.295@nd.edu} The context of enriched sheaf theory introduced in \cite{thesis} provides a convenient viewpoint for models of the stable homotopy category as well as categories of finite loop spaces. Also, the languages of algebraic geometry and algebraic topology have been interacting quite heavily in recent years, primarily due to the work of Voevodsky and that of Hopkins. Thus, the language of Grothendieck topologies is becoming a necessary tool for the algebraic topologist. The current document is intended to give a somewhat relaxed introduction to this language of sheaves in a topological context, using familiar examples such as $n$-fold loop spaces and pointed $G$-spaces. This language also includes the diagram categories of spectra from \cite{MMSS} as well as spectra in the sense of \cite{Lewis}, which will be discussed in some detail. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Larusson/excision Title: Excision for simplicial sheaves on the Stein site and Gromov's Oka Principle Author: Finnur Larusson AMS classification numbers: Primary: 32Q28; secondary: 18F10, 18F20, 18G30, 18G55, 32E10, 32H02, 55U35 arXiv:math.CV/0101103 Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada larusson@uwo.ca ABSTRACT: A complex manifold $X$ satisfies the Oka-Grauert property if the inclusion $\Cal O(S,X) \hookrightarrow \Cal C(S,X)$ is a weak equivalence for every Stein manifold $S$, where the spaces of holomorphic and continuous maps from $S$ to $X$ are given the compact-open topology. Gromov's Oka principle states that if $X$ has a spray, then it has the Oka-Grauert property. The purpose of this paper is to investigate the Oka-Grauert property using homotopical algebra. We embed the category of complex manifolds into the model category of simplicial sheaves on the site of Stein manifolds. Our main result is that the Oka-Grauert property is equivalent to $X$ representing a finite homotopy sheaf on the Stein site. This expresses the Oka-Grauert property in purely holomorphic terms, without reference to continuous maps. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McAuley/revised-hilbert This is another revised version of the proof of the Hilbert-Smith conjecture. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Looptan Title: The equivariant tangent bundle of a free smooth loopspace Author: Jack Morava AMS classification: 58Dxx; 53C29, 55P91 Address: The Johns Hopkins Uniperversity e-mail: jack@math.jhu.edu ABSTRACT: The space of free loops on a manifold X inherits an action of the circle group \T. A Riemannian metric on X defines an equivariant isomorphism of the complexified tangent bundle of the loopspace with \bT X \otimes (\oplus \C(n)), where \C(n) is the standard one-dimensional representation of \T, and \bT X \otimes \C is an equivariant bundle on the loopspace, nonequivariantly isomorphic to the pullback of the complexified tangent bundle of X along evaluation at the basepoint. On a flat manifold, this analogue of Fourier analysis is quite familiar. [Perhaps this is all nonsense; if so, please let me know.] 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/PGGravity5 Title: Pretty Good Gravity Author: Jack Morava AMS Classification: 19Dxx, 57Rxx, 83Cxx (not yet on xxx, but will be soon) Address: Dept. of Mathematics, the Johns Hopkins Uniperversity e-mail address: jack@math.jhu.edu Abstract: A theory of topological gravity is a homotopy-theoretic representation of the Segal-Tillmann topologification of a two-category with cobordisms as morphisms. This note describes a relatively accessible example of such a thing, suggested by the wall-crossing formulas of Donaldson theory. [This is a writeup of a talk at the RIMS Symposium on algebraic geometry and integrable systems related to string theory, June 12-16, 2000.] 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Tate2MU Title: Duality of Tate cohomology of framed circle actions Author: Jack Morava AMS classification: 19Dxx, 57Rxx, 83Cxx Address: The Johns Hopkins University Baltimore 21218 Maryland e-mail: Abstract: The complex Mahowald pro-spectrum \CP^{\infty}_{-\infty} is not, as might seem at first sight, Spanier-Whitehead self-dual; rather, its S-dual is its own double suspension. This assertion makes better sense as a claim about the Tate cohomology spectrum t_{\T}S^0 defined by circle actions on framed manifolds. A subtle twist in some duality properties of infinite-dimensional projective space results, which has consequences [via work of Madsen and Tillmann] for the Virasoro symmetries [discovered by Witten and Kontsevich] of the stable cohomology of the Riemann moduli space. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Moreno/moreno Author: Guillermo Moreno Title: Alternative elements in the Cayley--Dickson algebras We describe the alternative elements in the Cayley-Dickson algebras for n>3. Also we ``measure'' the failure of these algebras of being a normed algebra in terms of the alternative elements. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rezk-Schwede-Shipley/simplicia l Title: Simplicial structures on model categories and functors Authors: Charles Rezk, Stefan Schwede, Brooke Shipley To appear in American Journal of Mathematics Institute for Advanced Study School of Mathematics Olden Lane Princeton, NJ 08540, USA rezk@ias.edu Fakultat fur Mathematik Universitat Bielefeld 33615 Bielefeld, Germany schwede@mathematik.uni-bielefeld.de Department of Mathematics Purdue University West Lafayette, IN 47907, USA bshipley@math.purdue.edu We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or `continuous', functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Shimomura/ks-hgr The homotopy groups $\pi_*(L_nT(m)\wedge V(n-2))$ Katsumi Shimomura Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520 Japan katsumi@math.kochi-u.ac.jp Let $V_{T(m)}(n)$ denote the spectrum such that $BP_*(V_{T(m)}(n))=BP_*/I_{n+1}[t_1,\dots, t_m]$ for the ideal $I_{n+1}=(p,v_1,\dots, v_{n})$. In the title, we write $T(m)\wedge V(n-2)$ as $V_{T(m)}(n-2)$. Ravenel determined the structure of the Adams-Novikov $E_2$-term for the homotopy groups $\pi_*(L_nV_{T(m)}(n-1))$ for $n\le m+2$ and $n3$. ---------------------Instructions----------------------------- To subscribe or unsubscribe to this list, send a message to Don Davis at dmd1@lehigh.edu with your e-mail address and name. Please make sure he is using the correct e-mail address for you. To see past issues of this mailing list, point your WWW browser to http://www.math.wesleyan.edu/~mhovey/archive/ If this doesn't work or is missing a few issues, try http://www.lehigh.edu/~dmd1/algtop.html which also has the other messages sent to Don's list. To get the papers listed above, point your WWW client (Netscape< Internet Explorer) to the URL listed. The general Hopf archive URL is http://hopf.math.purdue.edu There are links to Purdue seminars, and other math related things on this page as well. 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