Friday, December 02, 2005

[math/0512040] A pairing between super Lie-Rinehart and periodic cyclic homology

[math/0512040] A pairing between super Lie-Rinehart and periodic cyclic homology

From the arXiv

Authors: Tomasz Maszczyk,
Comments: 11 pages,
Subj-class: K-Theory and Homology; Mathematical Physics,
MSC-class: Primary 16E40, 17B35, 19K56, Secondary 46L87

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K0-group of a smooth noncommutative torus. It also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected Lie group. Finally we compare it with the characteristic map for the Hopf-cyclic cohomology.

Sunday, November 13, 2005

Piotr Nowak's Homepage

Piotr Nowak

I was going to reference just a couple of Piotr's articles but then I thought that I might as well point to his entire home page. Everything is good here! In particular he shows that coarse embeddability into a Hilbert space (or into ell-one) is not the same as property A. The example is devastatingly simple: take the disjoint union of n-fold products of copies of some finite group (e.g. the group of order 2). Notice that the spaces here are quasi-isometric to cubes in R^n with the ell-one metric. If one took the ell-two (Euclidean) metric instead, wouldn't one get Yu's old counterexample to coarse Baum-Connes? Something interesting seems to be going on here.

[Of course these aren't bg spaces. Is there a bg space with the same property?]

Monday, November 07, 2005

[math/0511098] K-Theory of pseudodifferential operators with semi-periodic symbols

[math/0511098] K-Theory of pseudodifferential operators with semi-periodic symbols

Authors: Severino T. Melo, Cintia C. Silva
Subj-class: Operator Algebras; K-Theory and Homology
MSC-class: 46L80; 47G30

Let A denote the C*algebra of bounded operators on L2(R) generated by: (i) all multiplications a(M) by functions a\in C[-\infty,+\infty], (ii) all multiplications by 2\pi-periodic continuous functions and (iii) all Fourier multipliers F^{-1}b(M)F, where F denotes the Fourier transform and b is in C[-\infty,+\infty]. The Fredholm property for operators in A is governed by two symbols, the principal symbol \sigma and an operator-valued symbol \gamma. We give two proofs of the fact that K0(A) and K1(A) are isomorphic, respectively, to Z and Z\oplus Z: by computing the connecting mappings in the standard K-theory six-term exact sequences associated to \sigma and to \gamma. For the second computation, we prove that the image of \gamma is isomorphic to the direct sum of two copies of the crossed product of C[-\infty,+\infty] by an automorphism, and then use the Pimsner-Voiculescu exact sequence to compute its K-theory.

Tuesday, October 11, 2005

Another Coarse Geometry correction

See this link for another correction to LCG, this time in the statement and proof of Rosenblatt's theorem in chapter 3. Thanks to Steve Ferrt and his students for spotting this one.

Best wishes


Wednesday, September 28, 2005

[math/0509526] An analogue of the Novikov Conjecture in complex algebraic geometry

[math/0509526] An analogue of the Novikov Conjecture in complex algebraic geometry

We're familiar with the idea that statements about positive scalar curvature metrics (Gromov-Lawson-Rosenberg conjecture) and statements about higher signatures (Novikov conjecture) have certain parallels - ultimately they involve the higher index theorem applied to different elliptic operators, Dirac in the first case and signature in the second.

In this new paper Jonathan Rosenberg proposes a further family of statements involving higher index theory for the Dolbeault operator. These are statements in complex algebraic geometry about "higher Todd genera" for varieties.

This could be a whole new playground for higher index theorists.

Wednesday, August 31, 2005

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.

Tuesday, August 09, 2005

[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 $\pi$.

Wednesday, July 27, 2005

[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.

Monday, July 18, 2005

Monday, July 11, 2005

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.

Friday, July 08, 2005

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

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

Wednesday, June 29, 2005

[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.

Tuesday, June 21, 2005

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.

Friday, June 17, 2005

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.

Wednesday, June 15, 2005

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.

Tuesday, June 14, 2005

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.

hgw.pdf (application/pdf Object)

hgw.pdf (application/pdf Object)

Guentner, Higson and Weinberger prove the Novikov conjecture for all linear groups. The proof goes via coarse embeddings into Hilbert space.

cime.pdf (application/pdf Object)

cime.pdf (application/pdf Object)

Nigel Higson's notes on C*-algebras, K-theory and the Baum-Connes conjecture. Includes discussion of the "spectral picture" of K-theory; the E-theory approach to the Baum Connes conjecture; some counter-example stuff.

Mendel-Naor - cotype

Metric Cotype

[This gives the definitive answer to the question when Lp is uniformly (coarsely) embeddable into Lq.]

Title: Metric Cotype
Authors: Manor Mendel, Assaf Naor
Comments: 42 pages
Subj-class: Functional Analysis; Metric Geometry
MSC-class: 46B20; 51F99
We introduce the notion of cotype of a metric space, and prove that
for Banach spaces it coincides with the classical notion of Rademacher
cotype. This yields a concrete version of Ribe's theorem, settling a
long standing open problem in the non-linear theory of Banach spaces.
We apply our results to several problems in metric geometry. Namely,
we use metric cotype in the study of uniform and coarse embeddings,
settling in particular the problem of classifying when L_p coarsely or
uniformly embeds into L_q. We also prove a non-linear analog of the
Maurey-Pisier theorem, and use it to answer a question posed by Arora,
Lovasz, Newman, Rabani, Rabinovich and Vempala, and to obtain
quantitative bounds in a metric Ramsey theorem due to Matousek.


Geometry and Topology, Volume 9 (2005) Paper no. 26, pages 1147--1185

Geometry of pseudocharacters

Jason Fox Manning


If G is a group, a pseudocharacter f: G-->R is a function which is
"almost" a homomorphism. [in a coarse-geometric sense: JR]
If G admits a nontrivial pseudocharacter f,
we define the space of ends of G relative to f and show that if the
space of ends is complicated enough, then G contains a nonabelian free
group. We also construct a quasi-action by G on a tree whose space of
ends contains the space of ends of G relative to f. This construction
gives rise to examples of "exotic" quasi-actions on trees.