There are 7 new papers this time. Mark Hovey New papers appearing on hopf between 6/1/01 and 6/21/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Strom/Equivalences The group of homotopy equivalences of products of spheres and of Lie groups Martin Arkowitz and Jeffrey Strom AMS Classifications 55P10, 55P60, 55S37 Dartmouth College, Hanover, NH 03755 Martin.Arkowitz@Dartmouth.edu Jeffrey.Strom@Dartmouth.edu Abstract We investigate the group E_#(X) of self homotopy equivalences of a space X which induce the identity homomorphism on all homotopy groups. We obtain results on the structure of E_#(X) provided the p-localization X_(p) of X has the homotopy type of a p-local product of odd-dimensional spheres. In particular, we show that E_#(X)_(p) is a semidirect product of certain homotopy groups pi_n(X_(p)). We also show that E_#(X)_(p) has a central series whose successive quotients are pi_n(X_(p)), which are direct sums of homotopy groups of p-local spheres. This leads to a determination of the order of the p-torsion subgroup of E_#(X) and an upper bound for its p-exponent. These results apply to any Lie group G at a regular prime p. We derive some general properties of E_\#(G) and give numerous explicit calculations using MAPLE. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Kitchloo/BrKiCorregit This is a corrected version (the diagrams are better) of the paper announced last time, so I will just give the title: Classifying spaces of Kac-Moody groups by Carles Broto and Nitu Kitchloo 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Costenoble-May-Waner/CMWFinal Equivariant orientation theory by S.R. Costenoble, J.P. May, and S. Waner subjclass: Primary 55P91; Secondary 18B40, 20L15, 55N25, 55N91, 55P20, 55R91, 57Q91, 57R91 Hofstra University, University of Chicago, and Hofstra University Steven.R.Costenoble@Hofstra.edu, may@uchicago.edu, matszw@hofstra.edu We give a long overdue theory of orientations of G-vector bundles, topological G-bundles, and spherical G-fibrations, where G is a compact Lie group. The notion of equivariant orientability is clear and unambiguous, but it is surprisingly difficult to obtain a satisfactory notion of an equivariant orientation such that every orientable G-vector bundle admits an orientation. Our focus here is on the geometric and homotopical aspects, rather than the cohomological aspects, of orientation theory. Orientations are described in terms of functors defined on equivariant fundamental groupoids of base G-spaces, and the essence of the theory is to construct an appropriate universal target category of G-vector bundles over orbit spaces G/H. The theory requires new categorical concepts and constructions that should be of interest in other subjects where analogous structures arise. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/IsaksenD/ethtpy Etale realization on the A^1-homotopy theory of schemes Daniel C. Isaksen 14F42 (primary), 14F35 (secondary) Department of Mathematics University of Notre Dame Notre Dame, IN 46556 isaksen.1@nd.edu We compare Friedlander's definition of etale homotopy for simplicial schemes to another definition involving homotopy colimits of pro-simplicial sets. This can be expressed as a notion of hypercover descent for etale homotopy. We use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the etale site of schemes over S to the category of pro-spaces. After completing away from the characteristics of the residue fields of S, we get a functor from the Morel-Voevodsky A^1-homotopy category of schemes to the homotopy category of pro-spaces. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/IsaksenD/prospace (This is a revised version of a paper announced 6/99) A model structure on the category of pro-simplicial sets Daniel C. Isaksen 18E25, 55Pxx, 55U35 Department of Mathematics University of Notre Dame Notre Dame, IN 46556 Abstract: We study the category pro-SSet of pro-simplicial sets, which arises in etale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on pro-SSet so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for pro-spaces we use local systems on pro-spaces. We also give several alternative descriptions of weak equivalences, including a cohomological characterization. We outline dual constructions for ind-spaces. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/May/AddJan01 The additivity of traces in triangulated categories J.P. May University of Chicago may@math.uchicago.edu This paper is a much expanded version of the Appendix of the previously posted paper entitled "Picard groups, Grothendieck rings, and Burnside rings of categories. In it, we explain a fundamental additivity theorem for Euler characteristics and generalized trace maps in triangulated categories. The proof depends on a refined axiomatization of symmetric monoidal categories with a compatible triangulation. The refinement consists of several new axioms relating products and distinguished triangles. The axioms hold in the examples and shed light on generalized homology and cohomology theories. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/McClure-SmithJH/McClureSmith2 Multivariable cochain operations and little $n$-cubes. James E. McClure and Jeffrey H. Smith 18D50, 55P48, 16E40 math.QA/0106024 Department of Mathematics, Purdue University, West Lafayette, IN 47907--1395 mcclure@math.purdue.edu jhs@math.purdue.edu In this paper we construct a small $E_\infty$ chain operad $\S$ which acts naturally on the normalized cochains $S^*X$ of a topological space. We also construct, for each $n$, a suboperad $\S_n$ which is quasi-isomorphic to the normalized singular chains of the little $n$-cubes operad. The case $n=2$ leads to a substantial simplification of our earlier proof of Deligne's Hochschild cohomology conjecture.