-------------------------------------------- Sorry once again for the long delay. 12 new papers this time, from BrownR, BrownR-Sivera, Glover-Henn, Harper, Henn-Karamanov-Mahowald, LinJP, SmithL (2 papers), SmithL-Stong (4 papers). Mark Hovey New papers appearing on hopf between 9/6/08 and 11/20/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/fields-art Author: Ronald Brown Title: Crossed complexes and higher homotopy groupoids as non commutative tools for higher dimensional local-to-global problems Abstract: We outline the main features of the definitions and applications of crossed complexes and cubical $\omega$-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain some non commutative results and compute some homotopy types in non simply connected situations. Web page: www.bangor.ac.uk/r.brown Futher information: This is a revised version (2008) of a paper published in Fields Institute Communications 43 (2004) 101-130, which was an extended account of a lecture given at the meeting on `Categorical Structures for Descent, Galois Theory, Hopf algebras and semiabelian categories', Fields Institute, September 23-28, 2002. The author is grateful for support from the Fields Institute and a Leverhulme Emeritus Research Fellowship, 2002-2004, and to M. Hazewinkel for helpful comments on a draft. This paper is to appear in Michiel Hazewinkel (ed.), Handbook of Algebra, volume 6, Elsevier, 2008/2009. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Sivera/fibcat Title: Algebraic colimit calculations in homotopy theory using fibred and cofibred categories Author(s): Ronald Brown and Rafael Sivera Abstract: Higher Homotopy van Kampen Theorems allow the computation as colimits of certain homotopical invariants of glued spaces. One corollary is to describe homotopical excision in critical dimensions in terms of induced modules and crossed modules over groupoids. This paper shows how fibred and cofibred categories give an overall context for discussing and computing such constructions, allowing one result to cover many cases. A useful general result is that the inclusion of a fibre of a fibred category preserves connected colimits. The main homotopical application are to pairs of spaces with several base points, but we also describe briefly the situation for triads. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Glover-Henn/oct28-2008 Title: On the mod-p cohomology of Out(F_{2(p-1)}) Authors: Henry Glover Hans-Werner Henn Abstract: We study the mod-p cohomology of the group Out(F_n) of outer automorphisms of the free group F_n in the case n=2(p-1) which is the smallest n for which the p-rank of this group is 2. For p=3 we give a complete computation, at least above the virtual cohomological dimension of Out(F_4) (which is 5). More precisley, we calculate the equivariant cohomology of the p-singular part of outer space for p=3. For a general prime p>3 we give a recursive description in terms of the mod-p cohomology of Aut(F_k) for k less or equal to p-1. In this case we use the Out(F_{2(p-1)})-equivariant cohomology of the poset of elementary abelian p-subgroups of Out(F_n). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Harper/ModulesSpectra14 Title: Homotopy theory of modules over operads in symmetric spectra Author: John E. Harper Author's mailing address: Institute of geometry, algebra and topology, EPFL, CH-1015 Lausanne, Switzerland Comments: 33 pages, uses xy-pic. Significant revision. Abstract: This paper establishes model category structures on modules and algebras over operads in symmetric spectra, and studies when a morphism of operads induces a Quillen equivalence between corresponding categories of modules (resp. algebras) over operads. *** Please note: this is not a new submission to the Hopf arxiv, but a revision of an earlier manuscript with the same title. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Henn-Karamanov-Mahowald/hkm Title: The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited Authors: Hans-Werner Henn, Nasko Karamanov and Mark Mahowald Abstract: In this paper we use the approach introduced in an earlier paper by Goerss, Henn, Mahowald and Rezk in order to analyze the homotopy groups of L_{K(2)}V(0), the mod-3 Moore spectrum V(0) localized with respect to Morava K-theory K(2). These homotopy groups have already been calculated by Shimomura. The results are very complicated so that an independent verification via an alternative approach is of interest. In fact, we end up with a result which is more precise and also differs in some of its details from that of Shimomura. An additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the K(2)-localization of the spectrum TMF of topological modular forms and related spectra. Even more, the Adams-Novikov differentials for L_{K(2)}V(0) can be read off from those for TMF. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/LinJP/Lin08 Homology Rings of Homotopy Associative $H$--spaces James P. Lin Let $X$ be a homotopy associative mod $p$ $H$--space for $p$ an odd prime. The homology $H_*(X; \mathbb{F}_p)$ is an associative ring, but not necessarily commutative. We study conditions when $[\overline{x}, \overline{y}] \neq 0$ for $\overline{x}, \overline{y}$ elements of $H_*(X; \mathbb{F}_p)$. Under certain conditions $[\overline{x}, \overline{y}] \neq 0$ imply $ad^l (\overline{x},\overline{y}) \neq 0$ for $l=p-2$ or $p-1$. These methods can be used to prove results about homology commutators that were previously obtained using the adjoint action (Hamanaka et. al., 1996), (Kono et. al., 1993), (Kono et. al., 2003). We also generalize results of (Kane, 2006) to nonfinite mod $p$ homotopy associative $H$--spaces. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL/stable Title: Stable Invariants of Finite General Linear Groups and Symmetric Groups in Odd Characteristic Author: Larry Smith (AG-Invariantentheorie) We show that the stable invariants of the finite general linear group $\GL(n, \F_q)$ over a Galois field $\F_q$ with an odd characteristic coincide with the Hilbert ideal. The same argument applies to the tautological representation of the symmetric group in odd characteristic. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL/steintri Title: On R. Steinberg's Theorem on Algebras of Coinvariants Author Larry Smith (AG-Invariantentheorie} Steinberg's Theorem on the coinvariant algebra $\C[V]_G$ of a complex representation $\rho : G \hra \GL(n, \C)$ of a finite group $G$ says that $\C[V]_G$ is a Poincar\'e duality algebra if and only if the invariant algebra $\C[V]^G$ is a polynomial algebra. The extension of this to the nonmodular case has been achieved in stages, the final result being obtained by W.G. Dwyer and C.W. Wilkerson. We show that the main module theoretic tool they use extends to the following characteristic free result: If $\F[V]_G$ is a Poincar\'e duality algebra of formal dimension $d$\/, then $\F[V]^G$ is a polynomial algebra if and only if $\Hom_{\F[V]^G} (\F[V], \F[V])$ contains a nonzero element of degree $-d$\/. In the nonmodular case an easy transfer argument then recovers their extension of Steinberg's Theorem by means of some representation theory. Combined with some new results concerning the $\Delta$ operators of Demazure, our characteristic free result yields the following for reflection groups: A reflection group $G$ for which $\F[V]_G$ is a Poincar\'e duality algebra in which the trivial $G$-representation $1_G$ occurs only once as a subrepresentation has a polynomial algebra for its invariant algebra $\F[V]^G$\/. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL-Stong/binary Title: Invariants of Binary Forms Modulo Two Authors: Larry Smith (AG-Invariantentheorie) and R.E. Stong (University of Virginia) We examine the invariant theory of binary bilinear forms over the field $\F_2$ of two elements that arises in the classification of (standardly graded) Poincar\'e duality algebras with two algebra generators over the field $\F_2$ of two elements. We compute the corresponding ring of invariants and find seperating invariants for the orbit space. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL-Stong/pbi Title : Projective Bundle Ideals : Construction of Maximal Primary Irreducible Ideals in Polynomial Algebras Authors: Larry Smith and R. E. Stong Summary: We formalize the algebra of the Projective Bundle Theorem and use it to construct and study maximal primary irreducible ideals in polynomial algebras. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL-Stong/pda_quos Title: Poincar\'e Duality Algebras Modulo Two and Macaulay's Inverse Systems Authors: Larry Smith (AG-Invariantentheorie) and R.E. Stong (University of Virginia) If $H$ is a Poincar\'e duality algebra generated by its homogeneous component of degree $1$ it is called {\bf standardly graded} and the dimension of its homogeneous component $H_1$ of degree one is called its {\bf rank}\/. Standardly graded Poincar\'e duality algebras occur as quotient algebras of a (standardly graded) polynomial algebra by a maximal primary irreducible ideal Such ideals were studied in the work of F. S. Macaulay at the start of the last century who developed an elegant means of constructing them. The fact that these quotients are Poincar\'e duality algebras is a special case of a result of W. Gr\"obner. In this note we study the classification of Poincar\'e duality algebras over the field $\F_2$ of two elements. We obtain a complete classification of surfaces, i.e., Poincar\'e duality algebras of formal dimension two. To do so we determine the Grothendieck group of standardly graded surface algebras over an arbitrary field under the operation of connected sum. This group turns out to be $\Z$\/, hence finitely generated, and mirrors faithfully the topological classification of closed surfaces. By contrast, for Poincar\'e duality algebras (standardly graded or not) of formal dimension strictly greater than two the Grothendieck group fails to be finitely generated. We make a systematic study of standardly graded threefolds, i.e., Poincar\'e duality algebras of formal dimension three that are generated by their elements of degree one. The isomorphism classes of threefolds of rank at most three are in bijective correspondence with the orbits of the action of $\GL(3, \F_2)$ on a $10$-dimensional vector space, the space of catalecticant matrices. To determine the number of isomorphism classes we count the number of orbits using invariant theory. As a byproduct we obtain a classification of arbitrary bilinear forms in up to three variables. We determine explicitly all the standardly graded threefolds of rank at most three. There are 21 isomorphism classes. Twelve of these admit an unstable Steenrod algebra action, so could in theory be realized as the mod $2$ cohomology of a closed manifold. We exhibit for each such example a corresponding manifold; most of these are obvious, but there is one example of a slightly exotic $3$-manifold that is a torus bundle over a circle to which we devote some space. For threefolds of higher rank we explain one of several ways to construct such algebras that are not connected sums using Macaulay's theory of inverse systems. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL-Stong/rank_two Title: On Maximal Primary Irreducible Ideals in $\F[x, y]$ Authors: Larry Smith (AG-Invariantentheorie) and R.E. Stong (University of Virginia) At the beginning of the last century F.~S.~Macaulay developed an elegant theory describing homogeneous ideals in polynomial rings. This theory makes the maximal-primary irriducible ideals $I \subset \F[z_1\commadots z_n]$ correspond to a single homogeneous inverse polynomial $\theta_I \in \F[z_1^{-1} \commadots z_n^{-1}]$\/. Macaulay's theory has recently attracted attention in connection with problems arising in invariant theory and algebraic topology. In this note we show how given an inverse binary form $\theta \in \F[x^{-1}, y^{-1}]$ one may explicitly write down generators of the corresponding maximal-primary irreducible ideal $I(\theta) \subset \F[x, y]$\/. As a bonus we obtain an elementary proof of a theorem of Vasconcelos that such an ideal is always generated by a regular sequence. -----------------