Searcharxiv⌕ Search

arXiv subjects

Moulay-Tahar Benameur

Publications and source records attributed to Moulay-Tahar Benameur.

At least 19 recordsLinked to original sources
arXiv2025

Admissible Higson-Roe sequences for transformation groupoids

Given a finitely generated discrete group Γ, we construct for any admissible crossed product completion and for any metrizable finite dimensional compact Γ-space X, a universal Higson-Roe six-term exact sequence for the transformation groupoid X\rtimes Γ. In particular, we generalize the maximal Higson- Roe sequence to such groupoids. In the case where the groupoid X\rtimes Γ satisfies the rectified Baum-Connes conjecture, this yields some rigidity consequences.

math.OA↗
arXiv2021

The Higson-Roe sequence for étale groupoids II. The universal sequence for equivariant families

This is the second part of our series about the Higson-Roe sequence for étale groupoids. We devote this part to the proof of the universal $K$-theory surgery exact sequence which extends the seminal results of N. Higson and J. Roe to the case of transformation groupoids. In the process, we prove the expected functoriality properties as well as the Paschke-Higson duality theorem.

math.KT↗
arXiv2020

The index of leafwise G-transversally elliptic operators on foliations

We introduce and study the index morphism for G-invariant leafwise G-transversally elliptic operators on smooth closed foliated manifolds which are endowed with leafwise actions of the compact group G. We prove the usual axioms of excision, multiplicativity and induction for closed subgroups. In the case of free actions, we relate our index class with the Connes-Skandalis index class of the corresponding leafwise elliptic operator on the quotient foliation. Finally we prove the compatibility of our index morphism with the Gysin Thom isomorphism and reduce its computation to the case of tori actions. We also construct a topological candidate for an index theorem using the Kasparov Dirac element for euclidean G-representations.

math.KT↗
arXiv2020

The relative L^2 index theorem for Galois coverings

Given a Galois covering of complete spin manifolds where the base metric has PSC near infinity, we prove that for small enough epsilon > 0, the epsilon spectral projection of the Dirac operator has finite trace in the Atiyah von Neumann algebra. This allows us to define the L2 index in the even case and we prove its compatibility with the Xie-Yu higher index. We also deduce L2 versions of the classical Gromov-Lawson relative index theorems. Finally, we briefly discuss some Gromov-Lawson L2 invariants.

math.OA↗
arXiv2018

The Higson-Roe sequence for étale groupoids. I. Dual algebras and compatibility with the BC map

We introduce the dual Roe algebras for proper étale groupoid actions and deduce the expected Higson-Roe short exact sequence. When the action is cocompact, we show that the Roe $C^*$-ideal of locally compact operators is Morita equivalent to the reduced $C^*$-algebra of our groupoid, and we further identify the boundary map of the associated periodic six-term exact sequence with the Baum-Connes map, via a Paschke-Higson map for groupoids. For proper actions on continuous families of manifolds of bounded geometry, we associate with any $G$-equivariant Dirac-type family, a coarse index class which generalizes the Paterson index class and also the Moore-Schochet Connes' index class for laminations.

math.KT↗
arXiv2018

Enlargeability, foliations, and positive scalar curvature

We extend the deep and important results of Lichnerowicz, Connes, and Gromov-Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold admits no PSC metric; that any metric of PSC on such a foliation is bounded by a multiple of the reciprocal of the foliation K-area of the ambient manifold; and that Connes' vanishing theorem for characteristic numbers of PSC foliations extends to a vanishing theorem for Haefliger cohomology classes.

math.DG↗
arXiv2018

Atiyah covering index theorem for riemannian foliations

We use the symbol calculus for foliations developed in our previous paper to derive a cohomological formula for the Connes-Chern character of the semi-finite spectral triple. The same proof works for the Type I spectral triple of Connes-Moscovici. The cohomology classes of the two Connes-Chern characters induce the same map on the image of the maximal Baum-Connes map in K-theory, thereby proving an Atiyah $L^2$ covering index theorem.

math.GT↗
arXiv2018

Transverse noncommutative geometry of foliations

We define an $L^2$-signature for proper actions on spaces of leaves of transversely oriented foliations with bounded geometry. This is achieved by using the Connes fibration to reduce the problem to the case of Riemannian bifoliations where we show that any transversely elliptic first order operator in an appropriate Beals-Greiner calculus, satisfying the usual axioms, gives rise to a semi-finite spectral triple over the crossed product algebra of the foliation by the action, and hence a periodic cyclic cohomology class through the Connes-Chern character. The Connes-Moscovici hypoelliptic signature operator yields an example of such a triple and gives the differential definition of our "$L^2$-signature". For Galois coverings of bounded geometry foliations, we also define an Atiyah-Connes semi-finite spectral triple which generalizes to Riemannian bifoliations the Atiyah approach to the $L^2$-index theorem. The compatibility of the two spectral triples with respect to Morita equivalence is proven, and by using an Atiyah-type theorem proven in [BH17b], we deduce some integrality results for Riemannian foliations with torsion-free monodromy groupoids.

math.GT↗
arXiv2017

The higher twisted index theorem for foliations

Given a gerbe $L$, on the holonomy groupoid $\mathcal G$ of the foliation $(M, \mathcal F)$, whose pull-back to $M$ is torsion, we construct a Connes $Φ$-map from the twisted Dupont-Sullivan bicomplex of $\mathcal G$ to the cyclic complex of the $L$-projective leafwise smoothing operators on $(M, \mathcal F)$. Our construction allows to couple the $K$-theory analytic indices of $L$-projective leafwise elliptic operators with the twisted cohomology of $B\mathcal G$ producing scalar higher invariants. Finally by adapting the Bismut-Quillen superconnection approach, we compute these higher twisted indices as integrals over the ambiant manifold of the expected twisted characteristic classes.

math.KT↗
arXiv2014

The Higson-Roe exact sequence and $\ell^2$ eta invariants

The goal of this paper is to solve the problem of existence of an $\ell^2$ relative eta morphism on the Higson-Roe structure group. Using the Cheeger-Gromov $\ell^2$ eta invariant, we construct a group morphism from the Higson-Roe maximal structure group constructed in [HiRo:10] to the reals. When we apply this morphism to the structure class associated with the spin Dirac operator for a metric of positive scalar curvature, we get the spin $\ell^2$ rho invariant. When we apply this morphism to the structure class associated with an oriented homotopy equivalence, we get the difference of the $\ell^2$ rho invariants of the corresponding signature operators. We thus get new proofs for the classical $\ell^2$ rigidity theorems of Keswani obtained in [Ke:00].

math.KT↗
arXiv2013

Conformal invariants of twisted Dirac operators and positive scalar curvature

For a closed, spin, odd dimensional Riemannian manifold $(Y,g)$, we define the rho invariant $ρ_{spin}(Y,E,H, g)$ for the twisted Dirac operator $D^E_H$ on $Y$, acting on sections of a flat hermitian vector bundle $E$ over $Y$, where $H = \sum i^{j+1} H_{2j+1}$ is an odd-degree closed differential form on $Y$ and $H_{2j+1}$ is a real-valued differential form of degree ${2j+1}$. We prove that it only depends on the conformal class of the pair $[H,g]$. In the special case when $H$ is a closed 3-form, we use a Lichnerowicz-Weitzenbock formula for the square of the twisted Dirac operator, to show that whenever $Y$ is a closed spin manifold, then $ρ_{spin}(Y,E,H, g)= ρ_{spin}(Y,E, g)$ for all $|H|$ small enough, whenever g is a Riemannian metric of positive scalar curvature. When $H$ is a top-degree form on an oriented three dimensional manifold, we also compute $ρ_{spin}(Y,E,H, g)$.

math.DG↗
arXiv2012

Higher spectral flow and an entire bivariant JLO cocycle

Given a smooth fibration of closed manifolds and a family of generalised Dirac operators along the fibers, we define an associated bivariant JLO cocycle. We then prove that, for any $\ell \geq 0$, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the $C^{\ell+1}$ topology and functions on the base manifold with the $C^\ell$ topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

math.KT↗
arXiv2011

Leafwise homotopies and Hilbert-Poincare complexes. I. Regular HP-complexes and leafwise pull-back maps

This paper is a first of a series of three papers which study eta invariants for laminations. In this first paper, we extend the results of Higson and Roe to deal with regular (unbounded) operators and more importantly to take into account Morita equivalence of underlying $C^*$-algebras. Given an oriented leafwise map between leafwise oriented laminations on compact spaces, satisfying some natural assumptions, we construct a pull-back morphism between the de Rham HP complexes and prove their expected functoriality. As a byproduct, when two leafwise oriented laminations are leafwise homotopy equivalent, our construction allows to deduce an explicit path joining the leafwise signature operators and hence the famous Large Time Path appearing in the proof of the homotopy invariance of the measured Cheeger-Gromov number.

math.KT↗
arXiv2011

Foliated group actions and cyclic cohomology

We prove a cyclic Lefschetz formula for foliations. To this end, we define a notion of equivariant cyclic cohomology and show that its expected pairing with K-theory is well defined. This enables to associate to any invariant transverse current on a compact foliated manifold, a Lefschetz formula for leafwise preserving isometries.

math.KT↗
arXiv2010

The higher fixed point theorem for foliations I. Holonomy invariant currents

In this paper, we prove a higher Lefschetz formula for foliations in the presence of a closed Haefliger current. We associate with such a current an equivariant cyclic cohomology class of Connes' C*-algebra of the foliation, and compute its pairing with the localized equivariant K-theory in terms of local contributions near the fixed points. As special cases, we recover a number of classical results, and since we may use any closed Haefliger current on the foliation, we get new and very interesting formulae.

math.KT↗
arXiv2009

The Twisted Higher Harmonic Signature for Foliations

We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation of a compact Riemannian manifold with coefficients in a leafwise U(p,q)-flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences for the Novikov conjecture for foliations and for groups are investigated. Replaces The Higher Harmonic Signature for Foliations I: The Untwisted Case, and contains significant improvements.

math.KT↗