--------------------------------- 13 new papers this month, by Angeltveit-Rognes, Arkowitz-Oshima-Strom, Arkowitz-Stanley-Strom, Barker-Snaith, Broto-Castellana-Grodal-Levi-Oliver, Broto-Levi-Oliver, Iwase-Stanley-Strom, Jardine, Levi-Oliver, Lupton-SmithSB, Nendorf-Scoville-Strom, and Rognes (2). Mark Hovey New papers appearing on hopf between 2/5/05 and 3/5/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Angeltveit-Rognes/vigleik Title: Hopf algebra structure on topological Hochschild homology Author(s): Vigleik Angeltveit and John Rognes Author's e-mail address: Abstract: The topological Hochschild homology THH(R) of a commutative S-algebra (E-infinity ring spectrum) R naturally has the structure of a Hopf algebra over R, in the homotopy category. We show that under a flatness assumption this makes the Bokstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to study some interesting examples, including the commutative S-algebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of THH(ku) and THH(ko) after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic K-theory of S-algebras, using topological cyclic homology. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Oshima-Strom/Equiv Homotopy classes of self-maps and induced homomorphisms of homotopy groups Martin Arkowitz Hideaki Oshima Jeffrey Strom For a based space X, we consider the group of all self homotopy classes of $X$ such that which induce the identity on homotopy groups in dimensions 1 through n, and the group of all homotopy classes which loop to the identity. Analogously, we study the semigroups defined by replacing `identity' by `0' above. There is a chain of containments of these groups and semigroups, and we discuss examples for which the containment is proper. We then obtain various conditions on X which ensure that these groups are equal, or when the semigroups are equal. When X is a group-like space, we derive lower bounds for the order of these groups and their localizations. In the last section we make specific calculations for certain low dimensional Lie groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Stanley-Strom/Length1 The A-category and A-cone length of a map Martin Arkowitz Donald Stanley Jeffrey Strom For any collection A of spaces we define two numerical invariants of maps: A-category of f and the A-cone length of f. These invariants are defined axiomatically, and our first results give equivalent constructive definitions in terms of mapping cone decompositions. We show that if A is the collection of all spaces, then the A-category of f is the category of f as defined by Fadell and Husseini and the A-category of f is the cone length of f as defined by Marcum. By specializing to the unique maps from and to a one-point space, we obtain four invariants of spaces. Each of these four invariants has its own axiomatic and constructive definitions. We compare them similar invariants defined by Scheerer and Tanr\'e. We conclude by giving lower bounds for these invariants in terms of cohomology. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Barker-Snaith/psi3triangle3 \psi^3 as an upper triangular matrix Jonathan Barker and Victor Snaith 55S25 (Primary) 55P42 (Secondary) math.AT/0502472 Jonathan Barker Building 54 (School of Mathematics) University of Southampton Highfield Southampton SO17 1BJ UK Victor Snaith Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK In the 2-local stable homotopy category the group of left-bu-module automorphisms of bu\wedge bo which induce the identity on mod 2 homology is isomorphic to the group of infinite upper triangular matrices with entries in the 2-adic integers. We identify the conjugacy class of the matrix corresponding to 1\wedge\psi^3, where \psi^3 is the Adams operation. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Castellana-Grodal-Levi-Oliver/bcglo2 EXTENSIONS OF p-LOCAL FINITE GROUPS C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver A $p$-local finite group consists of a finite $p$-group $S$, together with a pair of categories which encode ``conjugacy'' relations among subgroups of $S$, and which are modelled on the fusion in a Sylow $p$-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as $p$-completed classifying spaces of finite groups. In this paper, we study and classify extensions of $p$-local finite groups, and also compute the fundamental group of the classifying space of a $p$-local finite group. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Levi-Oliver/blo4 A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS Carles Broto, Ran Levi, and Bob Oliver A saturated fusion system consists of a finite $p$-group $S$, together with a category which encodes ``conjugacy'' relations among subgroups of $S$, and which satisfies certain axioms which are motivated by properties of the fusion in a Sylow $p$-subgroup of a finite group. We describe here new ways of constructing abstract saturated fusion systems, first as fusion systems of spaces with certain properties, and then via certain graphs. Subject class: Primary 55R35. Secondary 55R40, 20D20 Keywords: classifying space, $p$-completion, finite groups, fusion. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Iwase-Stanley-Strom/GaneaCond Implications of the Ganea Condition Norio Iwase Donald Stanley Jeffrey Strom Suppose the spaces X and X x A have the same Lusternik-Schnirelmann category: cat(X x A) = cat(X). Then there is a strict inequality cat(X x (A \halfsmash B)) < cat (X) + cat(A \halfsmash B) for every space B, provided the connectivity of A is large enough (depending only on X). This is applied to give a partial verification of a conjecture of Iwase on the category of products of spaces with spheres. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/diagrams Author: J.F. Jardine Author's mailing address: Department of Mathematics University of Western Ontario London, ON N6A 5B7 Canada Suppose that A is a small presheaf of categories enriched in simplicial sets on a small Grothendieck site. It is shown that the homotopy theory of enriched A-diagrams taking values in simplicial sets can be identified with the homotopy theory of simplicial presheaves fibred over the diagonalized nerve dBA of A. One can also identify the set [*,dBA] of morphisms in the simplicial presheaf homotopy category with path components of the category of A-torsors, suitably defined. These statements are special cases of localized results which hold when the corresponding localized model structures are proper. Examples of the latter include the motivic homotopy category of Morel and Voevodsky, and so these results lead to a theory of motivic A-torsors which is classifiable up to equivalence by a family of morphisms in the motivic homotopy category. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Levi-Oliver/sol-corr Correction to: CONSTRUCTION OF 2-LOCAL FINITE GROUPS OF A TYPE STUDIED BY SOLOMON AND BENSON by Ran Levi and Bob Oliver A $p$-local finite group is an algebraic structure with a classifying space which has many of the properties of $p$-completed classifying spaces of finite groups. In our paper \cite{Sol}, we constructed a family of 2-local finite groups which are ``exotic'' in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of $\Spin_7(q)$ ($q$ an odd prime power) shown by Solomon not to occur as the 2-fusion in any actual finite group. As predicted by Benson, the classifying spaces of these 2-local finite groups are very closely related to the Dwyer-Wilkerson space $BDI(4)$. An error in our paper \cite{Sol} was pointed out to us by Andy Chermak, and we correct that error here. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-SmithSB/pi_one(map) Title: Rank of the fundamental group of a component of a function space Authors: Gregory Lupton and Samuel Bruce Smith ArXive: math.AT/0502311 MSC-class: 55Q52; 55P15 We compute the rank of the fundamental group of an arbitrary connected component of the space map(X, Y) for X and Y nilpotent CW complexes with X finite. For the general component corresponding to a homotopy class f : X --> Y, we give a formula directly computable from the Sullivan model for f. For the component of the constant map, our formula expresses the rank in terms of classical invariants of X and Y. Among other applications and calculations, we obtain the following: Let G be a compact simple Lie group with maximal torus T^n. Then the fundamental group of map(S^2, G/T^n; f) is a finite group if and only if f: S^2 --> G/T^n is essential. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Nendorf-Scoville-Strom/Seq1 Categorical Sequences Rob Nendorf Nick Scoville Jeffrey Strom We define and study the categorical sequence of a space, which is a new formalism that streamlines the computation of the Lusternik-Schnirelmann category of a space X by induction on its CW skelta. The k-th term in the categorical sequence of a CW complex X, is the least integer n for which the n-skeleton of X has L-S category at least k. We show that the categorical sequence of X is a well-defined homotopy invariant. We prove that the sequence is `superadditive' which is one of three keys to the power of categorical sequences. In addition to this formula, we provide formulas relating the categorical sequences of spaces and some of their algebraic invariants, including their cohomology algebras and their rational models; we also find relations between the categorical sequences of the spaces in a fibration sequence and give a preliminary result on the categorical sequence of a product of two spaces in the rational case. We completely characterize the sequences which can arise as categorical sequences of formal rational spaces. The most important of the many examples that we offer is a simple proof of a theorem of Ghienne: if X is a member of the Mislin genus of the Lie group Sp(3), then cat(X) = cat(Sp(3)) (which is known to be 5). 12. http://hopf.math.purdue.edu/cgi-bin/generate?/Rognes/dualizable Title: Stably dualizable groups Author: John Rognes Abstract: We extend the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by (Dwyer and) J.R. Klein and the p-complete study for p-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the E-local stable homotopy category, for any spectrum E. The principal new examples occur in the K(n)-local category, where the Eilenberg-Mac Lane spaces G = K(Z/p, q) are stably dualizable for all 0 <= q <= n. We show how to associate to each E-locally stably dualizable group G a stably defined representation sphere S^{adG}, called the dualizing spectrum, which is dualizable and invertible in the E-local category. Each stably dualizable group is Atiyah-Poincare self-dual in the E-local category, up to a shift by S^{adG}. There are dimension-shifting norm- and transfer maps for spectra with G-action, again with a shift given by S^{adG}. The stably dualizable group G also admits a kind of framed bordism class [G] in the homotopy of L_E S, in degree dim_E(G) = [S^{adG}] of the Pic_E-graded homotopy groups of the E-localized sphere spectrum. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/Rognes/galois Title: Galois extensions of structured ring spectra Author: John Rognes Abstract: We introduce the notion of a Galois extension of commutative S-algebras (E-infinity ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological K-theory, Lubin-Tate spectra and cochain S-algebras. We establish the main theorem of Galois theory in this generality. Its proof involves the notions of separable (and etale) extensions of commutative S-algebras, and the Goerss-Hopkins-Miller theory for E-infinity mapping spaces. We show that the global sphere spectrum~S is separably closed (using Minkowski's discriminant theorem), and we estimate the separable closure of its localization with respect to each of the Morava K-theories. We also define Hopf-Galois extensions of commutative S-algebras, and study the complex cobordism spectrum MU as a common integral model for all of the local Lubin-Tate Galois extensions. ------------