Discrete Morse theory and graph braid groups

AGT 5 (2005) Paper 44 (Abstract)

Discrete Morse theory and graph braid groups
Daniel Farley, Lucas Sabalka
Abstract. If Gamma is any finite graph, then the unlabelled configuration space of n points on Gamma, denoted UC^n(Gamma), is the space of n-element subsets of Gamma. The braid group of Gamma on n strands is the fundamental group of UC^n(Gamma). We apply a discrete version of Morse theory to these UC^n(Gamma), for any n and any Gamma, and provide a clear description of the critical cells in every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space UC^n(Gamma) strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in Gamma of degree at least 3 (and k is thus independent of n).

Subelliptic spinc Dirac operators

Charlie Epstein kindly gave me permission to publish this paper on index theory for generalizations of the d-bar Neumann problem and its relationship to index theory for Fourier integral operators.

[math/0505622] Free construction of CAT(1) spaces

[math/0505622] Free construction of CAT(1) spaces

[math/0508135] On the generalized Nielsen realization problem

[math/0508135] On the generalized Nielsen realization problem

Authors: Jonathan Block, Shmuel Weinberger
Subj-class: Geometric Topology
MSC-class: 57N

The main goal of this paper is to give the first examples of equivariant aspherical Poincare complexes, that are not realized by group actions on closed aspherical manifolds $M$. These will also provide new counterexamples to the Nielsen realization problem about lifting homotopy actions of finite groups to honest group actions. Our examples show that one cannot guarantee that a given action of a finitely generated group $\pi$ on Euclidean space extends to an action of $\Pi$, a group containing $\pi$ as a subgroup of finite index, even when all the torsion of $\Pi$ lives in $\pi$.

GT Vol 9 (2005) Paper 34 Hadamard spaces with isolated flats

GT Vol 9 (2005) Paper 34 (Abstract)

By Hruska and Kleiner with Hindawi

[math/0507542] A new kind of index theorem

[math/0507542] A new kind of index theorem

This is Ron Douglas' talk at the Kaminker retirement conference.

[hep-th/0507206] Dixmier traces on noncompact isospectral deformations

[hep-th/0507206] Dixmier traces on noncompact isospectral deformations

Ruan's lectures on orbifolds

Ruan's lectures at MSRI

[math/0502388] Quotients of Standard Hilbert Modules

[math/0502388] Quotients of Standard Hilbert Modules

Kind of a synthesis of BDF and algebraic geometry. By Bill Arveson.

A Comparison of Leibniz and Cyclic Homologies

Authors: Jerry Lodder
Comments: 14 pages
Subj-class: K-Theory and Homology; Algebraic Topology
MSC-class: 19D55; 17B66; 17A32

We relate Leibniz homology to cyclic homology by studying a map from a long exact sequence in the Leibniz theory to the ISB periodicity sequence in the cyclic theory. This provides a setting by which the two theories can be compared via the 5-lemma. We then show that the Godbillon-Vey invariant, as detected by the Leibniz homology of formal vector fields, maps to the Godbillon-Vey invariant as detected by the cyclic homology of the universal enveloping algebra of these vector fields.

Geometry of K\"ahler Metrics and Foliations by Holomorphic Discs

I've always been interested in holomorphic foliations...

[math/0506547] Coarse dimensions and partitions of unity

[math/0506547] Coarse dimensions and partitions of unity

More development of "coarse dimension theory" by professional dimension-theorists.

a space that can't embed in any uniformly convex Banach space

http://arxiv.org/abs/math/0506178

[math/0506544] Isometries, rigidity, and universal covers

[math/0506544] Isometries, rigidity, and universal covers

Farb and Weinberger classify aspherical Riemannian manifolds whose universal cover has "extra symmetry"

Coarse Homotopy - Trung's Thesis

This is a small advertising memo about part of Viet-Trung Luu's forthcoming thesis, which contains a very elegant unified treatment of various theorems about the 'coarse homotopy' or 'Lipschitz homotopy' invariance of the K-theory of C*(X). It formalizes the basic idea of the paper by Nigel and me in the Transactions, found here. One needs "C*(X) with coefficients", defined using Hilbert modules. Trung's approach has a geometric and an analytic part. The geometric part is to show that a coarse homotopy gives an element of C*(X; C[0,1]). The analytic part is to show that there is a natural pairing between the K-theory of C*(X;D) and the K-homology of D, with values in the K-theory of C*(X). Now one uses the known homotopy invariance of K-homology - the fact that the inclusions of 0 and 1 into [0,1] induce isomorphisms on K-homology - to conclude the coarse homotopy invariance of K_*(C*(X)).

[math/0506361] Property (T) and rigidity for actions on Banach spaces

[math/0506361] Property (T) and rigidity for actions on Banach spaces

Here the authors study variations of property T for actions on Banach spaces, especially Lp. When p is not 2 the equivalence between the original version of property T (isolation of the trivial representation) and the fixed point version F (affine isometric actions) does not hold.
They prove that every property T group has T(L^p) for all p, and F(L^p) for p<2+\epsilon.

Corrections to "Lectures on Coarse Geometry"

Corrections to "Lectures on Coarse Geometry"

Bernd Grave pointed out a couple of errors in Lectures on Coarse Geometry, explained here.

One is a foolish mistake I made with Urysohn's lemma. Given two disjoint closed subsets of a compact Hausdorff space, Urysohn's lemma allows one to find a continuous function equal to zero on one set, one on the other. But in general the set of points where said function is zero will be *larger* than the originally given closed set. Only in metric spaces can one expect to arrange equality.

Property A and nuclearity of Roe algebras (application/pdf Object)

propAnuclear.pdf (application/pdf Object)

By Brodzki, Niblo and Wright. One knows that for a discrete group G, property A is equivalent to exactness/nuclearity of its translation algebra. But the same thing is not known for a general (uniformly discrete) coarse space. In this paper is a measure of the 'grouplikeness' of a coarse space in terms of "partial translation structures". If X is uniformly embeddable in a discrete group, then it has a "free partial translation structure" and the expected equivalence holds.

Comments: Gromov's counterexample construction gives a way of building an 'almost' injective embedding of a more-or-less arbitrary coarse space into a discrete group. Relate that to this paper!

[math/0403059] A K-Theoretic Proof of Boutet de Monvel's Index Theorem

[math/0403059] A K-Theoretic Proof of Boutet de Monvel's Index Theorem

We study the C*-closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a compact connected manifold X with non-empty boundary. We find short exact sequences in K-theory
0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K denotes the compact ideal and T*X' the cotangent bundle of the interior of X. Using only simple K-theoretic arguments and the Atiyah-Singer Index Theorem, we show that the Fredholm index of an elliptic element in A is given as the composition of the topological index with mapping K_1(A/K)->K_0(C_0(T*X')) defined above. This relation was first established by Boutet de Monvel by different methods.

wall.pdf (application/pdf Object)

wall.pdf (application/pdf Object)

Cambridge lecture notes on differential topology (ca. 1960) by C.T.C. Wall.

Detailed treatment of foundational theorems.