arXiv subjects

Paulo Carrillo Rouse

Publications and source records attributed to Paulo Carrillo Rouse.

17 recordsLinked to original sources

arXiv2026

Gysin maps and wrong way functoriality via geometric deformation groupoids

In this article we study the normal bundle and the deformation to the normal cone functors to get deformation Lie groupoids that allow us to construct pushforward maps in any suitable (co)homology theory for Lie groupoids (not only K-theory) and in a natural and geometric way. The main theorems being the functoriality for these pushforward maps which recovers, unifies and generalizes many previous cases. The main new example we develop in this paper is the wrong way functoriality for equivariant (twisted) Orbifold K-theory with respect to a groupoid action.

math.KT↗

arXiv2025

Groupoids, equivalence bibundles and bimodules for noncommutative solenoids

Let $p$ be a prime number and $\mathcal{S}_p$ the $p$-solenoid. For $α\in \mathbb{R}\times \mathbb{Q}_p$ we consider in this paper a naturally associated action groupoid $S_α:=\mathbb{Z} [1/p]\ltimes_α\mathcal{S}_p \rightrightarrows \mathcal{S}_p$ whose $C^*-$algebra is a model for the noncommutative solenoid $\mathcal{A}_α^\mathscr{S}$ studied by Latremolière and Packer. Following the geometric ideas of Connes and Rieffel to describe the Morita equivalences of noncommutative torus using the Kronecker foliation on the torus, we give an explicit description of the geometric/topologic equivalence bibundle for groupoids $S_α$ and $S_β$ whenever $α,β\in \mathbb{R}\times \mathbb{Q}_p$ are in the same orbit of the $GL_2(\mathbb{Z}[1/p])$ action by linear fractional transformations. As a corollary, for $α,β\in \mathbb{R}\times \mathbb{Q}_p$ as above we get an explicit description of the imprimitivity bimodules for the associated noncommutative solenoids.

math.OA↗

arXiv2025

Fredholm anomalies on manifold with corners of low codimensions and conormal corner cycles

Given a connected manifold with corners $X$ of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles, these conormal homology groups are denoted by $H^{cn}_*(X)$. Using our previous works we define an index morphism $$K^0(^bT^*X)\stackrel{Ind_{ev,cn}^X}{\longrightarrow}H_{ev}^{cn}(X)$$ for $X$ a manifold with corners of codimension less or equal to three and called here the even conormal index morphism. In the case that $X$ is compact and connected and $D$ is an elliptic $b-$pseudodifferential operator in the associated $b-$calculus of $X$ we know, by our previous works and other authors works, that, up to adding an identity operator, $D$ can be perturbed (with a regularizing operator in the calculus) to a Fredholm operator iff $Ind_{ev,cn}^X([σ_D])$ (where $[σ_D]\in K^0(^bT^*X)$ is the principal symbol class) vanishes in the even conormal homology group $H_{ev}^{cn}(X)$. The main result of this paper is the explicit computation of the even and odd conormal index morphisms $Ind_{ev/odd,cn}^X(σ)\in H_{ev/odd}^{cn}(X)$ for $X$ a manifold with corners of codimension less or equal to three. The coefficients of the conormal corner cycles $Ind_{ev/odd,cn}^X(σ)$ are given in terms of some suspended Atiyah-Singer indices of the maximal codimension faces of $X$ and in terms of some suspended Atiyah-Patodi-Singer indices of the non-maximal codimension faces of $X$. As a corollary we give a complete caracterization to the obstruction of the Fredholm perturbation property for closed manifolds with corners of codimension less or equal to three in terms of the above mentioned indices of the faces, this allows us as well to give such a characterization in terms of the respective topological indices.

math.KT↗

arXiv2022

Topological K-theory for discrete groups and Index theory

We give a complete solution, for discrete countable groups, to the problem of defining and computing a geometric pairing between the left hand side of the Baum-Connes assembly map, given in terms of geometric cycles associated to proper actions on manifolds, and cyclic periodic cohomology of the group algebra. Indeed, for any such group $Γ$ (without any further assumptions on it) we construct an explicit morphism from the Left-Hand side of the Baum-Connes assembly map to the periodic cyclic homology of the group algebra. This morphism, called here the Chern-Baum-Connes assembly map, allows to give a proper and explicit formulation for a Chern-Connes pairing with the periodic cyclic cohomology of the group algebra. Several theorems are needed to formulate the Chern-Baum-Connes assembly map. In particular we establish a delocalised Riemann-Roch theorem, the wrong way functoriality for periodic delocalised cohomology for $Γ$-proper actions, the construction of a Chern morphism between the Left-Hand side of Baum-Connes and a delocalised cohomology group associated to $Γ$ which is an isomorphism once tensoring with $\mathbb{C}$, and the construction of an explicit cohomological assembly map between the delocalised cohomology group associated to $Γ$ and the homology group $H_*(Γ,FΓ)$. We then give an index theoretical formula for the above mentioned pairing (for any $Γ$) in terms of pairings of invariant forms, associated to geometric cycles and given in terms of delocalized Chern and Todd classes, and currents naturally associated to group cocycles using Burghelea's computation. As part of our results we prove that left-Hand side group used in this paper is isomorphic to the usual analytic model for the left-hand side of the assembly map.

math.KT↗

arXiv2021

Longitudinal b-operators, Blups and Index theorems

Using recently introduced Debord-Skandalis Blup's groupoids we study index theory for a compact foliated manifold with boundary inducing a foliation in its boundary. For this we consider first a blup groupoid whose Lie algebroid has sections consisting of vector fields tangent to the leaves in the interior and tangent to the leaves of the foliation in the boundary. In particular the holonomy $b$-groupoid allows us to consider the appropriate pseudodifferential calculus and the appropriate index problems. We further use the blup groupoids as the one above, and in particular its functoriality properties, to actually get index theorems. In this situtation there are two index morphisms, one related to ellipticity and a second one related to fully ellipticity. For the first one, we are able to extend to this setting the longitudinal Connes-Skandalis index theorem and to use it to get that a $b$-longitudinal elliptic operator can be perturbed (up stable homotopy within elliptic operators) with a regularizing operator in the calculus to get a fully elliptic operator if and only if a certain boundary topological index vanishes. For example in the case of a fibration (family case) this topological obstruction is always zero. For the second index morphism, the one related to fully elliptic operators (families of Fredholm operators), we restrict ourselves to the case of families of manifolds with boundary and we prove a new K-theoretical index theorem, i.e. construct a topological index and prove the equality with the analytic-Fredholm index, and use it to get a cohomological index formula for every fully elliptic operator. In particular, for a perturbed family of generalized Dirac operators we can compare our formula with the one by Melrose-Piazza to get a new geometric expression for the eta form of the family.

math.KT↗

arXiv2019

On Fredholm boundary conditions on manifolds with corners I: Global corner's cycles obstructions

Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles. Our main theorem is that, for any manifold with corners $X$ of any codimension, there is a natural and explicit morphism $$K_*(\mathcal{K}_b(X)) \stackrel{T}{\longrightarrow} H^{pcn}_*(X,\mathbb{Q})$$ between the $K-$theory group of the algebra $\mathcal{K}_b(X)$ of $b$-compact operators for $X$ and the periodic conormal homology group with rational coeficients, and that $T$ is a rational isomorphism. As shown by the first two authors in a previous paper this computation implies that the rational groups $H^{pcn}_{ev}(X,\mathbb{Q})$ provide an obstruction to the Fredholm perturbation property for compact connected manifold with corners. The difference with respect to the previous article of the first two authors in which they solve this problem for low codimensions is that we overcome in the present article the problem of computing the higher spectral sequence K-theory differentials associated to the canonical filtration by codimension by introducing an explicit topological space whose singular cohomology is canonically isomorphic to the conormal homology and whose K-theory is naturally isomorphic to the $K-$theory groups of the algebra $\mathcal{K}_b(X)$.

math.KT↗

arXiv2018

Pseudodifferential Calculus, Twisted gerbes and twisted index theory for Lie groupoids

The goal of this paper is to construct a calculus whose higher indices are naturally elements in the twisted K-theory groups for Lie groupoids. Given a Lie groupoid $G$ and a $PU(H)$-valued groupoid cocycle, we construct an algebra of projective pseudodifferential operators. The subalgebra of regularizing operators identifies with the naturally associated smooth convolution algebra of the associated twisted gerbe. We develop the associated symbolic calculus, symbol short exact sequences and existence of parametrices. In particular the algebra of projective operators appears as a quantization of the twisted symbol algebra. As the (untwisted) Lie groupoid case that it encompasses, the negative order operators extend to the twisted $C^*$-algebra and the zero order operators act as bounded multipliers on it. We obtain an analytic index morphism in twisted K-theory associated in a classic way by the corresponding pseudodifferential extension. We prove that this index morphism only depends on the isomorphism class of the cocycle, {\it i.e.,} on the twisting as the associated class in $H^1(G;PU(H))$. We also show that this twisted analytic index morphism is compatible with the index we constructed in a previous work, in collaboration with Bai-Ling Wang, by means of the Connes tangent groupoid, obtaining as a consequence the analytic interpretation, in terms of projective pseudodifferential operators and ellipticity, of the twisted longitudinal Connes-Skandalis index theorem. Our construction encompasses and unifies several previous cases treated in the literature, we discuss in the final section some examples of classes of operators unified by our setting.

math.OA↗

arXiv2017

Geometric obstructions for Fredholm boundary conditions for manifolds with corners

For every connected manifold with corners we use a homology theory called conormal homology, defined in terms of faces and incidences and whose cycles correspond geometrically to corner's cycles. Its Euler characteristic (over the rationals, dimension of the total even space minus the dimension of the total odd space), $χ_{cn}:=χ_0-χ_1$, is given by the alternated sum of the number of (open) faces of a given codimension. The main result of the present paper is that for a compact connected manifold with corners $X$ given as a finite product of manifolds with corners of codimension less or equal to three we have that 1) If $X$ satisfies the Fredholm Perturbation property (every elliptic pseudodifferential b-operator on $X$ can be perturbed by a b-regularizing operator so it becomes Fredholm) then the even Euler corner character of $X$ vanishes, i.e. $χ_0(X)=0$. 2) If the even Periodic conormal homology group vanishes, i.e. $H_0^{pcn}(X)=0$, then $X$ satisfies the stably homotopic Fredholm Perturbation property (i.e. every elliptic pseudodifferential b-operator on $X$ satisfies the same named property up to stable homotopy among elliptic operators). 3) If $H_0^{pcn}(X)$ is torsion free and if the even Euler corner character of $X$ vanishes, i.e. $χ_0(X)=0$ then $X$ satisfies the stably homotopic Fredholm Perturbation property. For example for every finite product of manifolds with corners of codimension at most two the conormal homology groups are torsion free. The main theorem behind the above result is the explicit computation in terms of conormal homology of the $K-$theory groups of the algebra $\mathcal{K}_b(X)$ of $b$-compact operators for $X$ as above. Our computation unifies the only general cases covered before, for codimension zero (smooth manifolds) and for codimension 1 (smooth manifolds with boundary).

math.DG↗

arXiv2016

Multiplicative Structures and the Twisted Baum-Connes Assembly map

Using a combination of Atiyah-Segal ideas on one side and of Connes and Baum-Connes ideas on the other, we prove that the Twisted geometric K-homology groups of a Lie groupoid have an external multiplicative structure extending hence the external product structures for proper cases considered by Adem-Ruan in [1] or by Tu,Xu and Laurent-Gengoux in [24]. These Twisted geometric K-homology groups are the left hand sides of the twisted geometric Baum-Connes assembly maps recently constructed in [9] and hence one can transfer the multiplicative structure via the Baum-Connes map to the Twisted K-theory groups whenever this assembly maps are isomorphisms.

math.KT↗

arXiv2016

Geometric Baum-Connes assembly map for twisted Differentiable Stacks

We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant $PU(H)-$principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of the associated pushforward maps and to establish the functoriality. The main results in this paper are to define the geometric twisted K-homology groups and to construct the assembly map. Even in the untwisted case the fact that the geometric twisted K-homology groups and the geometric assembly map are well defined for Lie groupoids is new, as it was only sketched by Connes in his book for general Lie groupoids without any restrictive hypothesis, in particular for non Hausdorff Lie groupoids. We also prove the Morita invariance of the assembly map, giving thus a precise meaning to the geometric assembly map for twisted differentiable stacks. We discuss the relation of the assembly map with the associated assembly map of the $S^1$-central extension. The relation with the analytic assembly map is treated, as well as some cases in which we have an isomorphism. One important tool is the twisted Thom isomorphism in the groupoid equivariant case which we establish in the appendix.

math.KT↗

arXiv2013

A cohomological formula for the Atiyah-Patodi-Singer index on manifolds with boundary

We give a cohomological formula for the index of a fully elliptic pseudodifferential operator on a manifold with boundary. As in the classic case of Atiyah-Singer, we use an embedding into an euclidean space to express the index as the integral of a cohomology class depending in this case on a noncommutative symbol, the integral being over a $C^\infty$-manifold called the singular normal bundle associated to the embedding. The formula is based on a K-theoretical Atiyah-Patodi-Singer theorem for manifolds with boundary that is drawn from Connes' tangent groupoid approach.

math.OA↗

arXiv2010

Twisted longitudinal index theorem for foliations and wrong way functoriality

For a Lie groupoid G with a twisting (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of Connes-Skandalis longitudinal index theorem. When the foliation is given by fibers of a fibration, our index coincides with the one recently introduced by Mathai-Melrose-Singer. We construct the pushforward map in twisted K-theory associated to any smooth (generalized) map $f:W\longrightarrow M/F$ and a twisting $σ$ on the holonomy groupoid $M/F$, next we use the longitudinal index theorem to prove the functoriality of this construction. We generalize in this way the wrong way functoriality results of Connes-Skandalis when the twisting is trivial and of Carey-Wang for manifolds.

math.KT↗

arXiv2009

An index theorem for manifolds with boundary

In his book (II.5), Connes gives a proof of the Atiyah-Singer index theorem for closed manifolds by using deformation groupoids and appropiate actions of these on R^N. Following these ideas, we prove an index theorem for manifolds with boundary.

math.KT↗

arXiv2008

Higher localized analytic indices and strict deformation quantization

This paper is concerned with the localization of higher analytic indices for Lie groupoids. Let $\gr$ be a Lie groupoid with Lie algebroid $A\gr$. Let $τ$ be a (periodic) cyclic cocycle over the convolution algebra $\cg$. We say that $τ$ can be localized if there is a correspondence K^0(A^*\gr)\stackrel{Ind_τ}{\longrightarrow}\mathbb{C} satisfying $Ind_τ(a)=< ind D_a,τ>$ (Connes pairing). In this case, we call $Ind_τ$ the higher localized index associated to $τ$. In {Ca4} we use the algebra of functions over the tangent groupoid introduced in {Ca2}, which is in fact a strict deformation quantization of the Schwartz algebra $\sw(A\gr)$, to prove the following results: \item Every bounded continuous cyclic cocycle can be localized. \item If $\gr$ is {é}tale, every cyclic cocycle can be localized. We will recall this results with the difference that in this paper, a formula for higher localized indices will be given in terms of an asymptotic limit of a pairing at the level of the deformation algebra mentioned above. We will discuss how the higher index formulas of Connes-Moscovici, Gorokhovsky-Lott fit in this unifying setting.

math.KT↗

arXiv2008

Compactly supported analytic indices for Lie groupoids

For any Lie groupoid we construct an analytic index morphism taking values in a modified $K-theory$ group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in \cite{Ca2}. This allows in particular to prove a more primitive version of the Connes-Skandalis Longitudinal index Theorem for foliations, that is, an index theorem taking values in a group which pairs with Cyclic cocycles. As other application, for $D$ a $\gr-$PDO elliptic operator with associated index $ind D\in K_0(\ci_c (\gr))$, we prove that the pairing $$ ,$$ with $τ$ a bounded continuous cyclic cocycle, only depends on the principal symbol class $[σ(D)]\in K^0(A^*\gr)$. The result is completely general for {É}tale groupoids. We discuss some potential applications to the Novikov's conjecture.

math.KT↗

arXiv2008

A Schwartz type algebra for the Tangent Groupoid

We construct an algebra of smooth functions over the tangent groupoid associated to any Lie groupoid. This algebra is a field of algebras over the closed interval [0, 1] which fiber at zero is the algebra of Schwartz functions over the Lie algebroid, whereas any fiber out of zero is the convolution algebra of the initial groupoid. Our motivation comes from index theory for Lie groupoids. In fact, our construction gives an intermediate algebra between the enveloping C-algebra and the convolution algebra of compactly supported functions of the tangent groupoid; and it will allows us, in a further work, to define other analytic index morphisms as a sort of deformations.

math.DG↗

arXiv2006

An analytic index for Lie groupoids

For a Lie groupoid there is an analytic index morphism which takes values in the $K-$theory of the $C^*$-algebra associated to the groupoid. This is a good invariant but extracting numerical invariants from it, with the existent tools, is very difficult. In this work, we define another analytic index morphism associated to a Lie groupoid; this one takes values in a group that allows us to do pairings with cyclic cocycles. This last group is related to the compactly supported functions on the groupoid. We use the tangent groupoid to define our index as a sort of ''deformation''.

math.KT↗