# 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. ### Geometry and TopologyGuest

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