-------------------------------------------------- Back to my old inefficiency. There are 7 new papers this time, from BrownR, Dwyer-Wilkerson, Kuhn, Levi-Seeliger, and Pengelley-Williams (3). Mark Hovey New papers appearing on hopf between 1/7/10 and 6/1/10 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/freeloopspace Title: Crossed modules and the homotopy 2-type of the free loop space Author: Ronald Brown Author's home page: www.bangor.ac.uk/r.brown AMS classification number: 18D15,55Q05,55Q52 arXive submission number: arXiv:1003.5617 Key words: free loop space, crossed module, crossed complex, closed category, classifying space, higher homotopies Abstract: The question was asked by Niranjan Ramachandran: how to describe the fundamental groupoid of LX, the free loop space of a space X? We show how this depends on the homotopy 2-type of X by assuming X to be the classifying space of a crossed module over a group, and then describe completely a crossed module over a groupoid determining the homotopy 2-type of LX; that is we describe crossed modules representing the 2-type of each component of LX. The method requires detailed information on the monoidal closed structure on the category of crossed complexes. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/GorensteinCoinvariants POINCAR'E DUALITY AND STEINBERG'S THEOREM ON RINGS OF COINVARIANTS W. G. DWYER AND C. W. WILKERSON In this note we use elementary methods to prove Steinberg's result for fields of characteristic 0 or of characteristic prime to the order of W . This gives a new proof even in the characteristic zero case. 1.1. Theorem. Let k be a field, V an r-dimensional k-vector space, and W a finite subgroup of Aut k(V ). Let S = S[V #] be the symmetric algebra on V # the k-dual of V, and R = S^W the ring of invariants of under the natural action of W on S. Define P* to be the quotient algebra S i\tensor_R k. If the characteristic of k is zero or prime to the order of W and P* satisfies Poincar'e duality, then R is isomorphic to a polynomial algebra on r generators. Steinberg [9] has shown that R is polynomial if k is the field of complex numbers and the quotient algebra P* = S\tensor_R k satisfies Poincar'e duality (1.3). Steinberg's result was extended by Kane [3, 4] to other fields of characteristic zero, and by T.-C. Lin [5] to the case in which k is a finite field of characteristic prime to the order of W . The current proof is independent of previous methods. (Revised Jan. 25, 2010 to correct typos and to incorporate some remarks by R. J. Shank .) 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/kuhnnilpotence Title: Nilpotence in Group Cohomology Author: Nicholas J. Kuhn AMS classification number: 20J06, 55R40 arXiv:1002.4662 Subject class: math.GR; math.AT Abstract: We study bounds on nilpotence in H*(BG), the mod p cohomology of the classifying space of a compact Lie group G. Part of this is a report of our previous work on this problem, updated to reflect the consequences of Peter Symonds recent verification of Dave Benson's Regularity Conjecture. New results are given for finite p--groups, leading to good bounds on nilpotence in H*(BP) determined by the subgroup structure of the p--group P. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Levi-Seeliger/LoopSol TITLE: Loop space homology associated to the mod 2 Dickson invariants} AUTHORS: Ran Levi, Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, U.K.} Nora Seeliger LAGA, Institut Galilee, Av. J-B Clement, 93430 Villetaneuse, France AMS CLASSIFICATION: Primary 55R35. Secondary 55R40, 20D20 arXiv:1001.3581v1 ABSTRACT The spaces BG_2 and BDI(4) have the property that their mod 2 cohomology is given by the rank 3 and 4 Dickson invariants respectively. Associated with these spaces one has for q odd the classifying spaces of the finite groups BG_2(q)and the exotic family of classifying spaces of 2-local finite groups BSol(q). In this article compute the mod 2 loop space homology of the 2-completed classifying space of G_2(q) and of BSol(q) for all odd primes q, as algebras over the Steenrod algebra, and the associated Bockstein spectral sequences. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/beyond Beyond the hit problem: Minimal presentations of odd-primary Steenrod modules, with application to CP(infinity) and BU. David J. Pengelley New Mexico State University Las Cruces, NM 88003 Frank Williams New Mexico State University Las Cruces, NM 88003 Primary 55R40; Secondary 55R45, 55S05, 55S10 We describe a minimal unstable module presentation over the Steenrod algebra for the odd-primary cohomology of infinite-dimensional complex projective space and apply it to obtain a minimal algebra presentation for the cohomology of the classifying space of the infinite unitary group. We also show that there is a unique Steenrod module structure on any unstable cyclic module that has dimension one in each complex degree (half the topological degree) with a fixed alpha-number (sum of `digits') and is zero in other degrees. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/FiniteRealProj46 Unstable module presentations for the cohomology of real projective spaces David J. Pengelley New Mexico State University Las Cruces, NM 88003 Frank Williams New Mexico State University Las Cruces, NM 88003 Primary 55R40; Secondary 55R45, 55S05, 55S10 There is much we still do not know about projective spaces. We describe here how the mod two cohomology of each real projective space is built as an unstable module over the Steenrod algebra A, or equivalently, over K, the algebra of inherently unstable mod two "lower operations" originally introduced by Steenrod. In particular, to produce the cohomology of projective space of each dimension we consider the well-known minimal set of unstable module generators and construct a minimal set of unstable relations. Three new perspectives we blend for this purpose are: 1. to focus solely on the two-power Steenrod squares that generate A to understand the A-action in a process we call "shoveling ones"; 2. to describe every element in a canonical way from a particular unstable generator by composing operations from the algebra K; 3. to shift attention when studying an unstable A-module to consid- ering and analyzing it directly as an equivalent K-module. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/symmetric-hit-problem-hopf The hit problem for symmetric algebras over the Steenrod algebra David Pengelley New Mexico State University Las Cruces, NM 88003 Frank Williams New Mexico State University Las Cruces, NM 88003 The hit problem for a cohomology module over the Steenrod algebra A asks for a minimal set of A-generators for the module. In this paper we consider the symmetric algebra over the field with p elements, for p an arbitrary prime, and treat the equivalent problem of determing the set of A-primitive elements in its dual. We produce a method for generating new A-primitives from known ones via a new action of the Kudo-Araki-May algebra, K, and consider the K-module structure of the A-primitives, which form a sub K-algebra of the dual of the symmetric algebra over the Steenrod algebra. ----------------