----------------------------------- 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 $n$. --------------