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Heath Emerson

Publications and source records attributed to Heath Emerson.

At least 19 recordsLinked to original sources
arXiv2021

Zeta functions and topology of Heisenberg cycles for linear ergodic flows

Placing a Dirac-Schrödinger operator along the orbit of a flow on a compact manifold \(M\) defines an \(\R\)-equivariant spectral triple over the algebra of smooth functions on \(M\). We study some of the properties of these triples, especially their zeta functions, which have the form \(\trace (fH^{-s})\) with \(f\) the restriction to \(\R\) of a function on \(M\) and \(H = -\frac{\partial^2}{\partial x^2} + x^2\) the harmonic oscillator. The meromorphic continuation property and pole structure of these zeta functions is related to ergodic time averages in dynamics. The construction reproduces the `Heisenberg cycles' of Lesch and Moscovici, in the case of the periodic flow on the circle, where it produces a spectral triple over the smooth irrational torus in the irrational rotation algebra \(A_\h\). We strengthen a result of these authors, showing that the zeta function \(\trace (aH^{-s})\) extend mermomorphically for any element \(a\) of the C*-algebra \(A_\h\). Another variant of the construction produces a spectral cycle for \(A_\h\otimes A_{1/\h}\) and a spectral triple over a suitable subalgebra with the meromorphic continuation property if \(\h\) satisfies a Diophantine condition. The class of this cycle defines a fundamental class in the sense that it determines a KK-duality. We employ the Local Index Theorem of Connes and Moscovici in order to elaborate an index theorem of Connes for certain classes of differential operators on the line and compute the intersection form on K-theory induced by the fundamental class.

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arXiv2020

Baum-Connes and the Fourier-Mukai transform

The Baum-Connes map for finitely generated free abelian groups is a K-theoretic analogue of the Fourier-Mukai transform from algebraic geometry. We describe this K-theoretic transform in the language of topological correspondences, and compute its action on K-theory (of tori) described geometrically in terms of Baum-Douglas cocycles, showing that the Fourier-Mukai transform maps the class of a subtorus to the class of a suitably defined dual torus. We deduce the Fourier-Mukai inversion formula. We use these results to give a purely geometric description of the Baum-Connes assembly map for free abelian groups.

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arXiv2020

Transversals, duality, and irrational rotation

An early result of Noncommutative Geometry was Connes' observation in the 1980's that the Dirac-Dolbeault cycle for the $2$-torus $\mathbb{T}^2$, which induces a Poincaré self-duality for $\mathbb{T}^2$, can be 'quantized' to give a spectral triple and a K-homology class in $KK_0(A_θ\otimes A_θ, \mathbb{C})$ providing the co-unit for a Poincaré self-duality for the irrational rotation algebra $A_θ$ for any $θ\in \mathbb{R}\setminus \mathbb{Q}$. This spectral triple has been extensively studied since. Connes' proof, however, relied on a K-theory computation and does not supply a representative cycle for the unit of this duality. Since such representatives are vital in applications of duality, we supply such a cycle in unbounded form in this article. Our approach is to construct, for any non-trivial element $g$ of the modular group, a finitely generated projective module $\mathcal{L}_g$ over $A_θ\otimes A_θ$ by using a reduction-to-a-transversal argument of Muhly, Renault, and Williams, applied to a pair of Kronecker foliations along lines of slope $θ$ and $g(θ)$, using the fact that these flows are transverse to each other. We then compute Connes' dual of $[\mathcal{L}_g]$ for $g$ upper triangular, and prove that we obtain an invertible in $KK_0(A_θ, A_θ)$, represented by what one might regard as a noncommutative bundle of Dirac-Schrödinger operators. An application of $\mathbb{Z}$-equivariant Bott Periodicity proves that twisting the module by the family gives the requisite spectral cycle for the unit, thus proving self-duality for $A_θ$ with both unit and co-unit represented by spectral cycles.

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arXiv2019

The class of a fibre in Noncommutative Geometry

This paper studies the K-homology of a crossed product of a discrete group acting smoothly on a manifold, with a better understanding of the noncommutative geometry of the crossed-product as the primary goal, and the Baum-Connes apparatus as the main tool. Examples suggest that the correct notion of the `Dirac class' of such a noncommutative space is the image under the equivalence determined by Baum-Connes of the fibre of the fibration of the Borel space associated to the action and a smooth model for the classifying space of the group. We give a systematic study of such fibre, or `Dirac classes,' with applications to the construction of interesting spectral triples and computation of their K-theory functionals, and we prove in particular that both the well-known deformation of the Dolbeault operator on the noncommutative torus, and the class of the boundary extension of a hyperbolic group, are both Dirac classes in this sense and therefore can be treated topologically in the same way.

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arXiv2015

K-homological finiteness and hyperbolic groups

Motivated by classical facts concerning closed manifolds, we introduce a strong finiteness property in K-homology. We say that a C*-algebra has uniformly summable K-homology if all its K-homology classes can be represented by Fredholm modules which are finitely summable over the same dense subalgebra, and with the same degree of summability. We show that two types of C*-algebras associated to hyperbolic groups - the C*-crossed product for the boundary action, and the reduced group C*-algebra - have uniformly summable K-homology. We provide explicit summability degrees, as well as explicit finitely summable representatives for the K-homology classes.

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arXiv2013

Localization techniques in circle-equivariant KK-theory

Let T be the circle and A be a T-C*-algebra. Then the T-equivariant K-theory of A is a module over the representation ring of the circle. The latter is a Laurent polynomial ring. Using the support of the module as an invariant, and techniques of Atiyah, Bott and Segal, we deduce that there are examples of T-C*-algebras A not KK^T-equivalent to any commutative T-C*-algebra. This is in contrast to the non-equivariant situation, in which any C*-algebra in the boostrap category is KK-equivalent to a commutative one. Our examples arise from dynamics, and include Cuntz-Krieger algebras with their usual circle actions. Using similar techniques, we also prove an equivariant version of the Lefschetz fixed-point formula. This is a special case of a result with Ralf Meyer that applies to general compact connected groups. The Lefschetz theorem equates the module trace of the module map of the T-equivariant K-theory of a smooth compact manifold induced by an equivariant self-correspondence of the manifold, with an appropriate Kasparov product; the Kasparov product is the T-equivariant index of the Dirac operator on a suitable `coincidence manifold' of the correspondence. Finally, we prove several results related to localization and the Kunneth and universal coefficient theorems, and give an essentially complete description of the T-equivariant K-theory of compact spaces, by combining localization techniques of Atiyah and Segal and results of Paul Baum and Alain Connes for equivariant K-theory of finite group actions.

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arXiv2013

An equivariant Lefschetz fixed-point formula for correspondences

We compute the trace of an endomorphism in equivariant bivariant K-theory for a compact group G in several ways: geometrically using geometric correspondences, algebraically using localisation, and as a Hattori-Stallings trace. This results in an equivariant version of the classical Lefschetz fixed-point theorem, which applies to arbitrary equivariant correspondences, not just maps.

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arXiv2012

Finitely summable Fredholm modules for boundary actions of hyperbolic groups

We construct a family of odd, finitely summable Fredholm modules over the crossed product C*-algebra $C(\bd \G)\rtimes \G$ associated to the action of a non-elementary hyperbolic group $\G$ on its Gromov boundary $\bd \G$. These Fredholm modules all represent the same, distinguished class in K-homology, namely that of the `boundary extension' of $C(\bd \G)\rtimes \G$ associated to the Gromov compactification of $\G$, and is typically nonzero. Their summability is closely related to the Hausdorff dimension of the boundary. We use these results to compute the Connes-Chern character of the boundary extension in cyclic cohomology.

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arXiv2011

A functorial equivariant K-theory spectrum and an equivariant Lefschetz formula

We construct a symmetric spectrum representing the G-equivariant K-theory of C*-algebras for a compact group or a proper groupoid G. Our spectrum is functorial for equivariant *-homomorphisms. We use this to establish the additivity of the canonical traces for endomorphisms of strongly dualisable objects in the bootstrap class in equivariant KK, in analogy to previous results for traces in stable homotopy theory. As an application, we prove an equivariant analogue of the Lefschetz trace formula for Hodgkin Lie groups.

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arXiv2010

Structure and K-theory of crossed products by proper actions

We study the C*-algebra crossed product $C_0(X)\rtimes G$ of a locally compact group $G$ acting properly on a locally compact Hausdorff space $X$. Under some mild extra conditions, which are automatic if $G$ is discrete or a Lie group, we describe in detail, and in terms of the action, the primitive ideal space of such crossed products as a topological space, in particular with respect to its fibring over the quotient space $G\backslash X$. We also give some results on the $\K$-theory of such C*-algebras. These more or less compute the $\K$-theory in the case of isolated orbits with non-trivial (finite) stabilizers. We also give a purely $\K$-theoretic proof of a result due to Paul Baum and Alain Connes on (\K)-theory with complex coefficients of crossed products by finite groups.

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arXiv2010

Duality, correspondences and the Lefschetz map in equivariant KK-theory: a survey

We survey work by the author and Ralf Meyer on equivariant KK-theory. Duality plays a key role in our approach. We organize the survey around the objective of computing a certain homotopy-invariant of a space equipped with a proper action of a group or groupoid called the Lefschetz map. The Lefschetz map associates an equivariant K-homology class to an equivariant Kasparov self-morphism of a space X admitting a dual. We want to describe it explicitly in the setting of bundles of smooth manifolds over the base space of a proper groupoid, in which groupoid elements act by diffeomorphisms between fibres. To get the required description we describe a topological model of equivariant KK-theory by way of a theory of correspondences, building on ideas of Paul Baum, Alain Connes and Georges Skandalis that appeared in the 1980's. This model agrees with the analytic model for bundles of smooth manifolds under some technical conditions related to the existence of equivariant vector bundles. Subject to these conditions we obtain a computation of the Lefschetz map in purely topological terms.

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arXiv2010

The Baum-Connes conjecture, noncommutative Poincare duality and the boundary of the free Group

Every hyperbolic group acts continuously on its Gromov boundary. One can form the corresponding cross-product C*-algebra A. We show that there always exists a canonical Poincare duality map from the K-theory of A to the K-homology of A. We show that this map is an isomorphism when the group in question is the free group on two generators. There is a direct connection between our constructions and the Baum-Connes Conjecture, and we use the latter to deduce our result.

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arXiv2009

Bivariant K-theory via correspondences

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. We replace smooth K-oriented maps by a class of K-oriented normal maps, which are maps together with a certain factorisation. Our construction does not use any special features of equivariant K-theory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories. We formulate necessary and sufficient conditions for certain duality isomorphisms in the geometric bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, both bivariant K-theories agree if there is such a duality isomorphism.

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arXiv2009

Equivariant embedding theorems and topological index maps

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a factorisation. These factorisations are models for the topological index map. Under some assumptions concerning the existence of equivariant vector bundles, any smooth map admits a normal factorisation, and two such factorisations are unique up to a certain notion of equivalence. To prove this, we generalise the Mostow Embedding Theorem to spaces equipped with proper groupoid actions. We also discuss orientations of normally non-singular maps with respect to a cohomology theory and show that oriented normally non-singular maps induce wrong-way maps on the chosen cohomology theory. For K-oriented normally non-singular maps, we also get a functor to Kasparov's equivariant KK-theory. We interpret this functor as a topological index map.

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arXiv2009

Dualities in equivariant Kasparov theory

We study several duality isomorphisms between equivariant bivariant K-theory groups, generalising Kasparov's first and second Poincare duality isomorphisms. We use the first duality to define an equivariant generalisation of Lefschetz invariants of generalised self-maps. The second duality is related to the description of bivariant Kasparov theory for commutative C*-algebras by families of elliptic pseudodifferential operators. For many groupoids, both dualities apply to a universal proper G-space. This is a basic requirement for the dual Dirac method and allows us to describe the Baum-Connes assembly map via localisation of categories.

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arXiv2009

Lefschetz numbers for C*-algebras

Using Poincare duality, we formulate a formula of Lefschetz type which computes the Lefschetz number of an endomorphism of a separable, nuclear C*-algebra satisfying Poincare duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on K-theory tensored with the complex numbers, as in the classical case.) We then consider endomorphisms of Cuntz-Krieger algebras O_A. An endomorphism has an invariant, which is a permutation of an infinite set, and the contracting and expanding behavior of this permutation describes the Lefschetz number of the endomorphism. Using this description we derive a closed polynomial formula for the Lefschetz number depending on the matrix A and the presentation of the endomorphism.

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arXiv2008

Equivariant representable K-theory

We interpret certain equivariant Kasparov groups as equivariant representable K-theory groups. We compute these groups via a classifying space and as K-theory groups of suitable sigma-C*-algebras. We also relate equivariant vector bundles to these sigma-C*-algebras and provide sufficient conditions for equivariant vector bundles to generate representable K-theory. Mostly we work in the generality of locally compact groupoids with Haar system.

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