Nine papers published or posted by Geometry & Topology Publications
Geometry and Topology
(1) Algebraic & Geometric Topology 9 (2009) 2247-2309
Tangle analysis of difference topology experiments:
Applications to a Mu protein-DNA complex
by Isabel K Darcy, John Luecke and Mariel Vazquez
URL: http://www.msp.warwick.ac.uk/agt/2009/09-04/p078.xhtml
DOI: 10.2140/agt.2009.9.2247
(2) Algebraic & Geometric Topology 9 (2009) 2311-2347
Bordism groups of solutions to differential relations
by Rustam Sadykov
URL: http://www.msp.warwick.ac.uk/agt/2009/09-04/p079.xhtml
DOI: 10.2140/agt.2009.9.2311
(3) Algebraic & Geometric Topology 9 (2009) 2349-2360
Thompson's group F and uniformly finite homology
by Daniel Staley
URL: http://www.msp.warwick.ac.uk/agt/2009/09-04/p080.xhtml
DOI: 10.2140/agt.2009.9.2349
(4) Algebraic & Geometric Topology 9 (2009) 2361-2390
Orientation reversal of manifolds
by Daniel Muellner
URL: http://www.msp.warwick.ac.uk/agt/2009/09-04/p081.xhtml
DOI: 10.2140/agt.2009.9.2361
(5) Algebraic & Geometric Topology 9 (2009) 2391-2441
Permutative categories, multicategories and algebraic K-theory
by A D Elmendorf and M A Mandell
URL: http://www.msp.warwick.ac.uk/agt/2009/09-04/p082.xhtml
DOI: 10.2140/agt.2009.9.2391
Four papers have been posted in preview by Geometry & Topology
(6) Geometry & Topology 14 (2010) 393-433
Plane sextics via dessins d'enfants
by Alex Degtyarev
URL: http://www.msp.warwick.ac.uk/gt/2010/14-01/p008.xhtml
DOI: 10.2140/gt.2010.14.393
(7) Geometry & Topology 14 (2010) 435-472
Density of isoperimetric spectra
by Noel Brady and Max Forester
URL: http://www.msp.warwick.ac.uk/gt/2010/14-01/p009.xhtml
DOI: 10.2140/gt.2010.14.435
(8) Geometry & Topology 14 (2010) 473-519
Kleinian groups of small Hausdorff dimension are classical Schottky groups. I
by Yong Hou
URL: http://www.msp.warwick.ac.uk/gt/2010/14-01/p010.xhtml
DOI: 10.2140/gt.2010.14.473
(9) Geometry & Topology 14 (2010) 521-571
A cartesian presentation of weak n-categories
by Charles Rezk
URL: http://www.msp.warwick.ac.uk/gt/2010/14-01/p011.xhtml
DOI: 10.2140/gt.2010.14.521
Abstracts follow
(1) Tangle analysis of difference topology experiments:
Applications to a Mu protein-DNA complex
by Isabel K Darcy, John Luecke and Mariel Vazquez
We develop topological methods for analyzing difference topology
experiments involving 3-string tangles. Difference topology is a
novel technique used to unveil the structure of stable protein-DNA
complexes. We analyze such experiments for the Mu protein-DNA
complex. We characterize the solutions to the corresponding tangle
equations by certain knotted graphs. By investigating planarity
conditions on these graphs we show that there is a unique biologically
relevant solution. That is, we show there is a unique rational tangle
solution, which is also the unique solution with small crossing
number.
(2) Bordism groups of solutions to differential relations
by Rustam Sadykov
In terms of category theory, the Gromov homotopy principle for a set
valued functor F asserts that the functor F can be induced from a
homotopy functor. Similarly, we say that the bordism principle for an
abelian group valued functor F holds if the functor F can be induced
from a (co)homology functor.
We examine the bordism principle in the case of functors given by
(co)bordism groups of maps with prescribed singularities. Our main
result implies that if a family J of prescribed singularity types
satisfies certain mild conditions, then there exists an infinite loop
space Omega^infty B_J such that for each smooth manifold W the
cobordism group of maps into $ with only J-singularities is isomorphic
to the group of homotopy classes of maps [W, Omega^infty B_J]. The
spaces Omega^infty B_J are relatively simple, which makes explicit
computations possible even in the case where the dimension of the
source manifold is bigger than the dimension of the target manifold.
(3) Thompson's group F and uniformly finite homology
by Daniel Staley
We use the uniformly finite homology developed by Block and Weinberger
to study the geometry of the Cayley graph of Thompson's group F.
In particular, a certain class of subgraph of F is shown to be
nonamenable (in the Folner sense). This shows that if the Cayley
graph of F is amenable, these subsets, which include every finitely
generated submonoid of the positive monoid of F, must necessarily have
measure zero.
(4) Orientation reversal of manifolds
by Daniel Muellner
We call a closed, connected, orientable manifold in one of the
categories TOP, PL or DIFF chiral if it does not admit an
orientation-reversing automorphism and amphicheiral
otherwise. Moreover, we call a manifold strongly chiral if it does not
admit a self-map of degree -1. We prove that there are strongly
chiral, smooth manifolds in every oriented bordism class in every
dimension greater than two. We also produce simply-connected, strongly
chiral manifolds in every dimension greater than six. For every
positive integer k, we exhibit lens spaces with an
orientation-reversing self-diffeomorphism of order 2^k but no self-map
of degree -1 of smaller order.
(5) Permutative categories, multicategories and algebraic K-theory
by A D Elmendorf and M A Mandell
We show that the K-theory construction of our paper [Adv. Math 205
(2006) 163-228], which preserves multiplicative structure, extends
to a symmetric monoidal closed bicomplete source category, with the
multiplicative structure still preserved. The source category of
[op cit], whose objects are permutative categories, maps fully and
faithfully to the new source category, whose objects are (based)
multicategories.
(6) Plane sextics via dessins d'enfants
by Alex Degtyarev
We develop a geometric approach to the study of plane sextics with a
triple singular point. As an application, we give an explicit
geometric description of all irreducible maximal sextics with a type
E_7 singular point and compute their fundamental groups. All groups
found are finite; one of them is nonabelian.
(7) Density of isoperimetric spectra
by Noel Brady and Max Forester
We show that the set of k-dimensional isoperimetric exponents of
finitely presented groups is dense in the interval inR|tgeq1 for k
geq 2. Hence there is no higher-dimensional analogue of Gromov's gap
(1,2) in the isoperimetric spectrum.
(8) Kleinian groups of small Hausdorff dimension are classical Schottky groups. I
by Yong Hou
It has been conjectured that the Hausdorff dimensions of nonclassical
Schottky groups are strictly bounded from below. In this first part
of our work on this conjecture, we prove that there exists a universal
positive number lambda > 0 such that any 2-generated nonelementary
Kleinian group with limit set of Hausdorff dimension < lambda is a
classical Schottky group.
(9) A cartesian presentation of weak n-categories
by Charles Rezk
We propose a notion of weak (n+k,n)-category, which we call
(n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely
the fibrant objects of a certain model category structure on the
category of presheaves of simplicial sets on Joyal's category
Theta_n. This notion is a generalization of that of complete Segal
spaces (which are precisely the (infty,1)-Theta-spaces). Our
main result is that the above model category is cartesian.