Four papers published by AGT and five papers posted in preview by GT

Discussion in 'Math Research' started by Geometry and Topology, Oct 22, 2008.

  1. Four papers have been published by Algebraic & Geometric Topology in
    Volume 8; papers (1) and (2) complete issue 3 and papers (3) and (4)
    open issue 4:

    (1) Algebraic & Geometric Topology 8 (2008) 1811-1832
    The homology of the stable nonorientable mapping class group
    by Oscar Randal-Williams
    URL: http://www.msp.warwick.ac.uk/agt/2008/08-03/p066.xhtml
    DOI: 10.2140/agt.2008.8.1811

    (2) Algebraic & Geometric Topology 8 (2008) 1833-1853
    Commensurability classes of (-2,3,n) pretzel knot complements
    by Melissa L Macasieb and Thomas W Mattman
    URL: http://www.msp.warwick.ac.uk/agt/2008/08-03/p067.xhtml
    DOI: 10.2140/agt.2008.8.1833

    (3) Algebraic & Geometric Topology 8 (2008) 1855-1959
    Model structures on the category of small double categories
    by Thomas M Fiore, Simona Paoli and Dorette Pronk
    URL: http://www.msp.warwick.ac.uk/agt/2008/08-04/p068.xhtml
    DOI: 10.2140/agt.2008.8.1855

    (4) Algebraic & Geometric Topology 8 (2008) 1961-1987
    The R(S^1)-graded equivariant homotopy of THH(F_p)
    by Teena Meredith Gerhardt
    URL: http://www.msp.warwick.ac.uk/agt/2008/08-04/p069.xhtml
    DOI: 10.2140/agt.2008.8.1961

    Five papers have been posted on the preview page for Geometry &
    Topology Volume 13 (2009):

    (5) Geometry & Topology 13 (2009) 1-48
    Gromov-Witten invariants of blow-ups along submanifolds with
    convex normal bundles
    by Hsin-Hong Lai
    URL: http://www.msp.warwick.ac.uk/gt/2009/13-01/p001.xhtml
    DOI: 10.2140/gt.2009.13.1

    (6) Geometry & Topology 13 (2009) 49-86
    K-duality for stratified pseudomanifolds
    by Claire Debord and Jean-Marie Lescure
    URL: http://www.msp.warwick.ac.uk/gt/2009/13-01/p002.xhtml
    DOI: 10.2140/gt.2009.13.49

    (7) Geometry & Topology 13 (2009) 87-98
    Global fixed points for centralizers and Morita's Theorem
    by John Franks and Michael Handel
    URL: http://www.msp.warwick.ac.uk/gt/2009/13-01/p003.xhtml
    DOI: 10.2140/gt.2009.13.87

    (8) Geometry & Topology 13 (2009) 99-139
    On the homology of the space of knots
    by Ryan Budney and Fred Cohen
    URL: http://www.msp.warwick.ac.uk/gt/2009/13-01/p004.xhtml
    DOI: 10.2140/gt.2009.13.99

    (9) Geometry & Topology 13 (2009) 141-187
    Snowflake groups, Perron-Frobenius eigenvalues and isoperimetric spectra
    by Noel Brady, Martin R Bridson, Max Forester and Krishnan Shankar
    URL: http://www.msp.warwick.ac.uk/gt/2009/13-01/p005.xhtml
    DOI: 10.2140/gt.2009.13.141

    Abstracts follow

    (1) The homology of the stable nonorientable mapping class group
    by Oscar Randal-Williams

    Combining results of Wahl, Galatius--Madsen--Tillmann--Weiss and
    Korkmaz one can identify the homotopy type of the classifying space of
    the stable nonorientable mapping class group N_infty (after
    plus-construction). At odd primes p, the F_p-homology coincides with
    that of Q_0({HP}^infty_+), but at the prime 2 the result is less
    clear. We identify the F_2-homology as a Hopf algebra in terms of the
    homology of well-known spaces. As an application we tabulate the
    integral stable homology of N_infty in degrees up to six.

    As in the oriented case, not all of these cohomology classes have a
    geometric interpretation. We determine a polynomial subalgebra of
    H^*(N_infty;F_2) consisting of geometrically-defined characteristic
    classes.


    (2) Commensurability classes of (-2,3,n) pretzel knot complements
    by Melissa L Macasieb and Thomas W Mattman

    Let K be a hyperbolic (-2,3,n) pretzel knot and M its complement in
    the 3-sphere. For these knots, we verify a conjecture of Reid and
    Walsh: there are at most three knot complements in the
    commensurability class of M. Indeed, if n is not 7, we show that M is
    the unique knot complement in its class. We include examples to
    illustrate how our methods apply to a broad class of Montesinos knots.


    (3) Model structures on the category of small double categories
    by Thomas M Fiore, Simona Paoli and Dorette Pronk

    In this paper we obtain several model structures on DblCat, the
    category of small double categories. Our model structures have three
    sources. We first transfer across a categorification-nerve
    adjunction. Secondly, we view double categories as internal categories
    in Cat and take as our weak equivalences various internal equivalences
    defined via Grothendieck topologies. Thirdly, DblCat inherits a model
    structure as a category of algebras over a 2-monad. Some of these
    model structures coincide and the different points of view give us
    further results about cofibrant replacements and cofibrant objects. As
    part of this program we give explicit descriptions for and discuss
    properties of free double categories, quotient double categories,
    colimits of double categories, horizontal nerve and horizontal
    categorification.


    (4) The R(S^1)-graded equivariant homotopy of THH(F_p)
    by Teena Meredith Gerhardt

    The main result of this paper is the computation of TR^n_alpha(F_p;p)
    for alpha in R(S^1). These R(S^1)-graded TR-groups are the
    equivariant homotopy groups naturally associated to the S^1-spectrum
    THH (F_p), the topological Hochschild S^1-spectrum. This computation,
    which extends a partial result of Hesselholt and Madsen, provides the
    first example of the R(S^1)-graded TR-groups of a ring. These groups
    arise in algebraic K-theory computations and are particularly
    important to the understanding of the algebraic K-theory of
    non-regular schemes.


    (5) Gromov-Witten invariants of blow-ups along submanifolds with convex normal bundles
    by Hsin-Hong Lai

    When the normal bundle N_{Z/X} is convex with a minor assumption, we
    prove that genus-0 GW-invariants of the blow-up Bl_Z X of X along a
    submanifold Z, with cohomology insertions from X, are identical to
    GW-invariants of X. Under the same hypothesis, a vanishing theorem is
    also proved. An example to which these two theorems apply is when
    N_{Z/X} is generated by its global sections. These two main theorems
    do not hold for arbitrary blow-ups, and counterexamples are included.


    (6) K-duality for stratified pseudomanifolds
    by Claire Debord and Jean-Marie Lescure

    This paper continues our project started in [J. Funct. Anal. 219
    (2005) 109-133] where Poincare duality in K-theory was studied for
    singular manifolds with isolated conical singularities. Here, we
    extend the study and the results to general stratified
    pseudomanifolds. We review the axiomatic definition of a smooth
    stratification S of a topological space $X$ and we define a groupoid
    T^{S}X, called the S-tangent space. This groupoid is made of
    different pieces encoding the tangent spaces of strata, and these
    pieces are glued into the smooth noncommutative groupoid T^{S}X using
    the familiar procedure introduced by Connes for the tangent groupoid
    of a manifold. The main result is that C^*(T^{S}X) is Poincare dual
    to C(X), in other words, the S-tangent space plays the role in
    K-theory of a tangent space for X.


    (7) Global fixed points for centralizers and Morita's Theorem
    by John Franks and Michael Handel

    We prove a global fixed point theorem for the centralizer of a
    homeomorphism of the two-dimensional disk D that has
    attractor-repeller dynamics on the boundary with at least two
    attractors and two repellers. As one application we give an
    elementary proof of Morita's Theorem, that the mapping class group of
    a closed surface S of genus g does not lift to the group of C^2
    diffeomorphisms of S and we improve the lower bound for g from 5 to 3.


    (8) On the homology of the space of knots
    by Ryan Budney and Fred Cohen

    Consider the space of 'long knots' in R^n, K_{n,1}. This is the space
    of knots as studied by V Vassiliev. Based on previous work [Budney:
    Topology 46 (2007) 1-27], [Cohen, Lada and May: Springer Lecture Notes
    533 (1976)], it follows that the rational homology of K_{3,1} is free
    Gerstenhaber-Poisson algebra. A partial description of a basis is
    given here. In addition, the mod-p homology of this space is a 'free,
    restricted Gerstenhaber-Poisson algebra'. Recursive application of
    this theorem allows us to deduce that there is p-torsion of all orders
    in the integral homology of K_{3,1}.

    This leads to some natural questions about the homotopy type of the
    space of long knots in R^n for n>3, as well as consequences
    for the space of smooth embeddings of S^1 in S^3 and embeddings of
    S^1 in R^3.


    (9) Snowflake groups, Perron-Frobenius eigenvalues and isoperimetric spectra
    by Noel Brady, Martin R Bridson, Max Forester and Krishnan Shankar

    The k-dimensional Dehn (or isoperimetric) function of a group bounds
    the volume of efficient ball-fillings of k-spheres mapped into
    k-connected spaces on which the group acts properly and cocompactly;
    the bound is given as a function of the volume of the sphere. We
    advance significantly the observed range of behavior for such
    functions. First, to each nonnegative integer matrix P and positive
    rational number r, we associate a finite, aspherical 2-complex X_r,P
    and determine the Dehn function of its fundamental group G_r,P in
    terms of r and the Perron-Frobenius eigenvalue of P. The range of
    functions obtained includes elta(x)= x^s, where s is arbitrary
    rational number at least 2. Next, special features of the groups G_r,P
    allow us to construct iterated multiple HNN extensions which exhibit
    similar isoperimetric behavior in higher dimensions. In particular,
    for each positive integer k and rational sgeq(k+1)/k, there exists a
    group with k-dimensional Dehn function x^s. Similar isoperimetric
    inequalities are obtained for fillings modeled on arbitrary manifold
    pairs (M,partial M) addition to (B^k+1,S^k).
     
    Geometry and Topology, Oct 22, 2008
    #1
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