Subject: new Hopf listings From: Mark Hovey Date: 07 Jan 1999 04:45:01 -0500 A blizzard :) of papers this time, 15 in all. One of them is the errata to my book on model categories--if you see any other errors in it, please let me know. Don't miss the orthogonal spectra of Mandell and May--they combine the best features of S-modules and symmetric spectra, it seems to me. Mark Hovey New papers uploaded to hopf between 12/16/98 and 1/7/99: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Browder-Pakianthan/pakianat Title: Cohomology of Uniformly Powerful p-groups. Authors' names: William Browder and Jonathan Pakianathan AMS Classifications: Primary 20J06, 17B50 Secondary 17B56 institutions: Princeton University and University of Wisconsin, Madison. email: browder@math.princeton.edu and pakianat@math.wisc.edu Abstract: In this paper, the cohomology of p-central, powerful, p-groups with a certain extension property are studied. Such groups naturally correspond to Lie algebras and the paper exploits this relation to calculate their Fp-cohomology as a module over the Steenrod algebra. For example, a formula for the Bockstein based on the structure constants of the Lie algebra is obtained. Then the first few terms of the Bockstein spectral sequence are calculated and expressed in terms of the corresponding Lie algebra cohomologies. This is then used to study the integral cohomology of these p-groups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Butowiez-TurnerP/umc Title: Unstable Multiplicative Cohomology Operations Authors: Jean-Yves Butowiez and Paul Turner Email: jbutowiez@lemel.fr and pt@maths.abdn.ac.uk Abstract: We investigate the relationship between multiplicative unstable cohomolgy operations G^0(-) to E^0(-) and formal group laws for a certain important class of theories. As an application we study additive multiplicative idempotents. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/DavisD/e8 From Representation Thoery to Homotopy Groups Donald M. Davis Lehigh University dmd1@lehigh.edu http://www.lehigh.edu/~dmd1/dmd1.html Bousfield recently gave a formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations. In this paper, we apply Bousfield's theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of all compact simple Lie groups, a project suggested by Mimura in 1989. The method is completely different than that used by the author in previous papers. There is no homotopy theoretic input, and no spectral sequence calculation. The input is the second exterior power operation in the representation ring of E8, which we determine using specialized software. This can be interpreted as giving the Adams operation psi^2 in K(E8). Eigenvectors of psi^2 must also be eigenvectors of psi^k for any k. The matrix of these eigenvectors is the key to the analysis. Its determinant is closely related to the homotopy decomposition of E8 localized at each prime. By taking careful combinations of eigenvectors, we obtain a set of generators of K(E8) on which we have a nice formula for all Adams operations. Bousfield's theorem (and considerable Maple computation) allows us to obtain from this the v1-periodic homotopy groups. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/model-err Errata for the book Model Categories AMS Mathematical Surveys and Monographs vol. 63 by Mark Hovey hovey@member.ams.org The title is self-explanatory. No wrong theorems so far, but one wrong lemma, an incorrect proof, and a couple of incorrect definitions. Please send further errors to me at the above address. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell-May/mmnov14 Orthogonal spectra and S-modules M.A. Mandell MIT mandell@math.mit.edu J.P. May University of Chicago may@math.uchicago.edu There are two general approaches to the construction of symmetric monoidal categories of spectra, one based on an encoding of operadic structure in the definition of the smash product and the other based on the categorical observation that categories of diagrams with symmetric monoidal domain are symmetric monoidal. The first was worked out by Elmendorf, Kriz, and the authors in the theory of ``S-modules''. The second was worked out in the case of symmetric spectra by Hovey, Shipley, and Smith and, in a general topological setting, by Schwede, Shipley, and the authors. A comparison between symmetric spectra and S-modules was given by Schwede. Orthogonal spectra are intermediate between symmetric spectra and S-modules: they are defined in the same diagrammatic fashion as symmetric spectra, but, as with S-modules, their stable weak equivalences are just the maps that induce isomorphisms on homotopy groups. We prove that the categories of orthogonal spectra and S-modules are Quillen equivalent and that this equivalence induces Quillen equivalences between the respective categories of ring spectra, of modules over a ring spectrum, and of commutative ring spectra. The equivalence is given by a functor that is closely related to an older and more intuitive functor from orthogonal spectra to S-modules, and a comparison between the two leads to a precise understanding in terms of a category of Thom spaces of the relationship between the definitions of orthogonal spectra and of S-modules. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell-May-Schwede-Shipley/mm ss1nov14 Diagram spaces, diagram spectra, and FSP's M.A. Mandell MIT mandell@math.mit.edu J.P. May University of Chicago may@math.uchicago.edu S. Schwede Universitet Bielefeld, Germany schwede@mathematik.uni-bielefeld.de B. Shipley Purdue University and University of Chicago bshipley@math.purdue.edu Working in the category $T$ of based spaces, we give the basic theory of diagram spaces, diagram spectra, and functors with smash product. For a small topological category $D$, a $D$-space is just a continuous functor $D >--> T$. There is an external smash product that takes a pair of $D$-spaces to a $(D x D)$-space. If $D$ is symmetric monoidal, there is an internalization of this smash product that makes the category $DT$ of $D$-spaces a symmetric monoidal category. This allows the definition of monoids R in $DT$, modules over monoids R, and, when R is commutative, monoids in the category of R-modules. These structures are defined in terms of the internal smash product, but they all have more elementary descriptions in terms of the external smash product. A monoid R is a symmetric monoidal functor $D >--> T$, and the external version of an R-module is a $D$-spectrum over R. We show that there is a new category $D_R$ such that a $D_R$-space has the same structure as a $D$-spectrum over R. When R is commutative, the external version of a monoid in the category of R-modules is a $D$-FSP (functor with smash product) over R. We are especially interested in functors relating categories such as these as $D$ varies. With R taken as a canonical sphere diagram space, examples include Symmetric spectra, as defined by Jeff Smith. Orthogonal spectra, a coordinate free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category. Gamma-spaces, as defined by Graeme Segal. $W$-spaces, an analogue of Gamma-spaces with finite sets replaced by finite CW complexes in the domain category. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell-May-Schwede-Shipley/mm ss2nov14bis Model categories of diagram spectra M.A. Mandell MIT mandell@math.mit.edu J.P. May University of Chicago may@math.uchicago.edu S. Schwede Universitet Bielefeld, Germany schwede@mathematik.uni-bielefeld.de B. Shipley Purdue University and University of Chicago bshipley@math.purdue.edu In this sequel to our paper ``Diagram spaces, diagram spectra, and FSP's", we construct and compare model structures on the categories of prespectra, symmetric spectra, orthogonal spectra, Gamma-spaces, and $W$-spaces defined there. With the caveat that Gamma-spaces are always connective, the homotopy categories associated to all of these model categories are equivalent to the classical stable homotopy category. In all cases, there is a levelwise model structure, in which the weak equivalences and fibrations are defined levelwise. Actually, it is often convenient or necessary to modify this by considering some but not all levels. There is then a stable model structure in which the cofibrations are the cofibrations in the level model structure and the weak equivalences are the stable weak equivalences. In the cases of prespectra, orthogonal spectra, Gamma-spaces, and W-spaces, stable weak equivalences are just maps whose associated maps of prespectra induce isomorphisms of homotopy groups. In the case of symmetric spectra, a stable weak equivalence f: X >--> Y is a map such that f^*:[Y,E] >--> [X,E] is an isomorphism for all symmetric Omega-spectra E, where the brackets refer to the levelwise homotopy category. Modulo the caveat about Gamma-spaces, the model categories of prespectra, symmetric spectra, orthogonal spectra, Gamma-spaces, and $W$-spaces are Quillen equivalent and thus have equivalent homotopy categories. In favorable cases, the subcategories of ring spectra, module spectra over a ring spectrum, and commutative ring spectra are also model categories. Prespectra do not form a symmetric monoidal category, this being the main reason for interest in the other categories. In all other cases, the respective categories of ring spectra are model categories and, with the caveat about Gamma-spaces, they are all Quillen equivalent and thus have equivalent homotopy categories. A similar statement holds for module spectra over ring spectra. The categories of commutative symmetric ring spectra and commutative orthogonal ring spectra are model categories and are Quillen equivalent. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/May/boardman The hare and the tortoise J.P. May The University of Chicago may@math.uchicago.edu This is a little tribute to Mike Boardman, for inclusion in the proceedings honoring his 60th birthday. It describes some of the history behind his definition of the stable homotopy category, sketches how his original construction worked, and describes its relationship to the recent ``all frills attached'' construction due to Elmendorf, Kriz, Mandell, and myself. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/May/history STABLE ALGEBRAIC TOPOLOGY, 1945-1966 J. P. MAY Contents 1. Setting up the foundations 3 2. The Eilenberg-Steenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6 5. Steenrod operations, K(pi; n)'s, and characteristic classes 8 6. The introduction of cobordism 10 7. The route from cobordism towards K-theory 12 8. Bott periodicity and K-theory 14 9. The Adams spectral sequence and Hopf invariant one 15 10. S-duality and the introduction of spectra 18 11. Oriented cobordism and complex cobordism 21 12. K-theory, cohomology, and characteristic classes 23 13. Generalized homology and cohomology theories 25 14. Vector fields on spheres and J(X) 28 15. Further applications and refinements of K-theory 31 16. Bordism and cobordism theories 34 17. Further work on cobordism and its relation to K-theory 37 18. High dimensional geometric topology 40 19. Iterated loop space theory 42 20. Algebraic K-theory and homotopical algebra 43 21. The stable homotopy category 45 References 50 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/May/Melpaper Equivariant orientations and Thom isomorphisms J.P. May University of Chicago may@math.uchicago.edu Despite a great deal of work, notably by Costenoble and Waner, there is still not a fully satisfactory theory of orientations and Thom isomorphisms of equivariant bundles. Working in a given RO(G)-graded cohomology theory, one wants somehow to grade Thom classes on fiber representations. A G-space B is G-connected if each of its fixed point subspaces B^H is non-empty and connected. Over such base spaces, one can fix the same fiber representation for all fibers, and work of Lewis and myself gives a satisfactory theory. In any approach to more general base spaces, one must parametrize changes of fiber representation as one moves around B on the equivariant fundamental groupoid pi(B), which depends on all components of all fixed point spaces and all paths connecting them. Costenoble and Waner package the complexity in a generalization of RO(G)-graded cohomology. I propose an alternative. Giving up the idea that an orientation should be a single cohomology class, I propose that orientations should be compatible collections of cohomology classes in the cohomologies of the Thom H-spaces of the pullbacks of the given bundle to the ``H-connected covers'' of B. This allows one to quote rather than generalize the theory of Lewis and myself. The H-connected covers introduced for this purpose should have other uses. As in the case of orientations, they provide a substitute in the equivariant world for the standard first step in so many nonequivariant arguments, namely: ``We may assume without loss of generality that X is connected''. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Neusel/symplectic TITLE: The Invariants of the Symplectic Groups AUTHOR: Mara D. Neusel AG Invariantentheorie, Germany EMAIL: mdn@sunrise.uni-math.gwdg.de maramara@steenrod.mast.queensu.ca mneusel@cfgauss.uni-math.gwdg.de neusel@math.umn.edu These are lecture notes of the talks "Invarianten klassischer Gruppen III/IV", held at the Oberseminar Algebraische Topologie und Invariantentheorie, University of G\"ottingen, Germany, winter semester 1998/9. In this notes we study the invariant rings of the symplectic groups in odd characteristic in their tautological representation, and try to make the original paper by Carlisle and Kropholler more readable and understandable, i.e., the only new thing is the expository, in particular that/how and where the Steenrod algebra is used is my contribution. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rognes/whdiff Two-primary algebraic K-theory of pointed spaces John Rognes Department of Mathematics University of Oslo P.O. Box 1053, Blindern Norway rognes@math.uio.no We compute the mod 2 cohomology of Waldhausen's algebraic K-theory spectrum A(*) of the category of finite pointed spaces, as a module over the Steenrod algebra. This also computes the mod 2 cohomology of the smooth Whitehead spectrum of a point, denoted Wh^{DIFF}(*). Using an Adams spectral sequence we compute the 2-primary homotopy groups of these spectra in dimensions * <= 18, and up to extensions in dimensions 19 <= * <= 21. As applications we show that the linearization map L : A(*) -> K(Z) induces the zero homomorphism in mod 2 spectrum cohomology in positive dimensions, the space level Hatcher-Waldhausen map hw : G/O -> Omega Wh^{DIFF}(*) does not admit a four-fold delooping, and there is a 2-complete spectrum map M : Wh^{DIFF}(*) \to Sigma g/o_{oplus} which is precisely 9-connected. Here g/o_{oplus} is a spectrum whose underlying space has the 2-complete homotopy type of G/O. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Stanley/dona Title:``The sectional category of spherical fibrations'' Author: Don Stanley AMS-classification number: 55P50 Don Stanley Freie Universitaet Berlin Institut fur Mathematik II Arnimallee 3 14195 Berlin Germany email: stanley@math.fu-berlin.de Abstract: We give homological conditions which determine sectional category, secat, for rational spherical fibrations. In the odd dimensional case the secat is the least power of the Euler class which is trivial. In the even dimensional case secat is one when a ceratin homology class in twice the dimension of the sphere is -1 times a square. Otherwise secat is two. We apply out results to construct a fibration $p$ such that $secat(p)=2$ and $genus(p)=\infty$. We also observe that secat, unlike cat, can decrease in a field extension of $\mathbb Q$. 14. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Stanley/ls4 Title:``On the Lusternik-Schnirelmann category of maps''(revised version) Author: Don Stanley AMS-classification number: 55P50 Address: Don Stanley Freie Universitaet Berlin Institut fur Mathematik II Arnimallee 3 14195 Berlin Germany email: stanley@math.fu-berlin.de Abstract: We give conditions when $cat(f \times g) before you type get . To put a paper of yours on the archive, 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/pub/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.