There are 9 new papers on Hopf this time, 7 with Jeffrey Strom as one of the authors, one by Bill Dwyer and Clarence Wilkerson, and one by G. Meigneiz. Mark Hovey New papers appearing on hopf between 10/17/01 and 11/13/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Oshima-Strom/H-Inverses The Inverses of an H-Space Martin Arkowitz, Hideaki Oshima and Jeffrey Strom MSC: 55P45, 55P62 Department of Mathematics Dartmouth College Hanover, NH 03755 USA Jeffrey.Strom@Dartmouth.edu Martin.Arkowitz@Dartmouth.edu Ibaraki University Mito, Ibaraki 310-8512 JAPAN ooshima@mito.ipc.ibaraki.ac.jp ABSTRACT A multiplication on an H-space X has a left inverse \lambda and a right inverse \rho. They are mutual inverses and \lambda = \rho if and only if \lambda^2 = id. In this paper we investigate the order |\lambda| of \lambda. We give an example of a multiplication with |\lambda|=6, and prove that for any finite H-complex X there are finitely many left inverses of finite order. Conditions are given for there to be infinitely many multiplications on X with the same left inverse. We then give conditions for a left inverse to have infinite order. We apply these results to specific Lie groups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Oshima-Strom/NonComm Non commutativity of the group of self homotopy classes of classical simple Lie groups Martin Arkowitz, Hideaki Oshima and Jeffrey Strom MSC: 55Q05 Department of Mathematics Dartmouth College Hanover, NH 03755 USA Ibaraki University Mito Ibaraki 310-8512 Japan Martin.Arkowitz@Dartmouth.edu ooshima@mito.ipc.ibaraki.ac.jp Jeffrey.Strom@Dartmouth.edu ABSTRACT For a large class of simple Lie groups G we prove that [G,G] is nonabelian. For certain special Lie groups we show that \nil [G,G] > 2. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Strom/NearlyTrivial Nearly Trivial Homotopy Classes Between Finite Complexes Martin Arkowitz and Jeffrey Strom 2000 MSC: Primary 55P99; Secondary 55M30, 55P60 Department of Mathematics Dartmouth College Hanover, NH 03755 USA Martin.Arkowitz@Dartmouth.edu Jeffrey.Strom@Dartmouth.edu ABSTRACT We construct examples of essential maps of finite complexes f : X --> Y which are trivial of order at least n. This latter condition implies that for any space K with cone length at most n, the induced map f_* = 0:[K,X] --> [K,Y]. The main result establishes a connection between the skeleta of the infinite dimensional domains of essential phantom maps and the finite dimensional domains of maps which are trivial of order at least n. In particular, there are essential maps f: \Sigma^2i ( CP^t / S^2 ) --> M( Z/p^s, 2l+3) which are trivial of order at least n. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/normalizers/tits-final Title: Cartan Involutions and the normalizer of the maximal torus Authors: William G. Dwyer and Clarence W. Wilkerson Email: dwyer.1@nd.edu cwilkers@purdue.edu Classification codes: 22E15 (55R35 55S40) One consequence of Tits' well known work \cite{rTits} on the structure of the normalizer of the maximal torus in a connected compact Lie group is that twice the $k$-invariant classifying the extension $$\{e\} \to T_G \to N_G(T_G) \to W(G) \to \{e\}$$ is zero. In this note we observe that this conclusion follows directly from the existence of an unstable Adams map of type $\Psi^{-1}$ on the classifying space $BG$. Work from the 1970's using etale methods or more recent diagramatic methods produce a $\Psi^{\alpha}$ self-map of $BG$ whenever $\alpha$ is relatively prime to the order of $W(G)$, so the $k$-invariant bound follows. However, the Lie algebra version of ${\Psi^{-1}}$ (the Cartan involution) is classical. This note discusses the Cartan involution, and shows how for a connected compact Lie group it gives rise to a self map of type $\Psi^{-1}$.\\ Analogues of $\{\Psi^{-1}\}$ are not known for the general $2$-compact group context of Dwyer-Wilkerson \cite{rDW1}. While this could be a possible divergence point for $2$-compact group theory from classical Lie theory, the authors speculate that it is not. { This was written for the Grand Lake, CO Bastille Day 2001 conference in honor of Brooke Shipley and Kevin Corlette. It has been submitted to Publ. Res. Inst. Math. Sci., RIMS, Kyoto. } 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Fernandez-Suarez-Gomez-Tato-Strom-Tanre/Sp(3) The Lusternik-Schnirelmann Category of Sp(3) Lucia Fernandez-Suarez, Antonio Gomez-Tato, Jeffrey Strom and Daniel Tanre MSC: 55M30, 22E20 Departamento de Matematica (CMAT) Universidade do Minho (Gualtar) 4710 Braga, Portugal lfernandez@math.uminho.pt Departamento de Xeometria e Topoloxia Universidade de Santiago de Compostela 15706 Santiago de Compostela Espana agtato@zmat.usc.es Department of Mathematics Dartmouth College Hanover, NH 03755 U.S.A. Jeffrey.A.Strom@Dartmouth.edu Departement de Mathematiques UMR 8524 Universite de Lille 1 59655 Villeneuve d'Ascq Cedex, France Daniel.Tanre@agat.univ-lille1.fr ABSTRACT We show that the Lusternik-Schnirelmann category of the symplectic group Sp(3) is 5. This L-S category coincides with the cone length and the stable weak category. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Meigniez/sfb Title: Submersions, fibrations & bundles. Author: G. Meigniez Abstract --- When does a submersion have the homotopy lifting property ? When is it a locally trivial fibre bundle ? We establish characterizations in terms of consistency in the topology of the neighbouring fibres. -- Universite de Bretagne Sud, Centre de Recherche, Campus de Tohannic, B.P. 573, F-56017 Vannes, France. Phone: (33)6.87.49.79.45. Fax: (33)2.97.68.42.12. http://www.univ-ubs.fr/lmam/meigniez/ 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Strom/Diagonal Decomposition of the Diagonal Map Jeffrey Strom 2000 MSC: Primary: 55M30 Secondary: 55Q25 Department of Mathematics Dartmouth College Hanover, NH 03755 USA Jeffrey.Strom@Dartmouth.edu ABSTRACT This paper presents a new method for using cup product information to draw conclusions about the Lusternik-Schnirelmann category of a space. The key idea is that of the Hopf set in X of a map f : S^{n-1} --> L; if K = L \cup_f D^n is a subcomplex of X, then cat_X (K) = cat_X (L) if and only if * is in the Hopf set in X of f. The main result explicitly constructs elements of the Hopf set in X of f in terms of members of the Hopf set in X of the attaching maps of lower dimensional cells. Applications include: a calculation of the category of Sp(2) without higher order cohomology operations; new, easily used upper bounds for Lusternik-Schnirelmann category that apply to any space; and new information about the category of the CW skeleta of loop spaces and free loop spaces on even-dimensional spheres. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Strom/MillerSpaces Miller Spaces and Spherical Resolvability of Finite Complexes Jeffrey Strom MSC: 55Q05, 55P50 Department of Mathematics Dartmouth College Hanover, NH 03755 USA Jeffrey.Strom@Dartmouth.edu ABSTRACT We show that if K is a nilpotent finite complex, then the loop space of K can be built from spheres using fibrations and homotopy (inverse) limits. This is applied to show that if map_*(X,S^n) is weakly contractible for all n, then map_*(\s X,K) is weakly contractible for any nilpotent finite complex K. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Strom/S1xS1 The Lusternik-Schnirelmann Category of S^1_\QQ\cross S^1 and S^1_\QQ\cross S^1_\QQ Jeffrey Strom MSC: 55M30 Department of Mathematics Dartmouth College Hanover, NH 03755 USA Jeffrey.Strom@Dartmouth.edu (From Mark: My guess is that the subscript \QQ indicates the rationalization). ABSTRACT We answer a question of Rudyak by showing that cat(S^1_\QQ\cross S^1) = cat(S^1_\QQ\cross S^1_\QQ) = 3. The second formula shows that X= S^1_\QQ is an example of a space for which \cat(X\cross X) < 2 \cat(X). These calculations are derived from a general formula for the category weight of elements of H^*(BG;\pi) that is of independent interest.