arXiv subjects

Jesús A. Álvarez López

Publications and source records attributed to Jesús A. Álvarez López.

At least 19 recordsLinked to original sources

arXiv2026

New Applications and Computations of the Lefschetz Number of Homeomorphisms and Open Maps

We show that the combinatorial Lefschetz number is a topological invariant. This is an important result in itself; in order to point it out, we will also work here several relevant consequences in different directions. The first of them is a significant simplification of the computations involved in obtaining the Lefschetz number of certain maps, as well as some new Lefschetz fixed-point theorems for unbounded spaces. Indeed, these ideas allow us to obtain a clear lower bound for the Nielsen number of a triad in some spaces, such as, for example, the connected sum of two p-tori (p greater than 2). Another consequence, in the case of homeomorphisms, is that, in the classical axiomatic definition of the Lefschetz number, the wedge-of-circles axiom and the cofibration axiom can be replaced by the single axiom of topological invariance of the combinatorial Lefschetz number. Using the invariance of the combinatorial Lefschetz number we generalize O'Neill's classical result about topological invariance of the fixed-point index. We also generalize the combinatorial Lefschetz number from homeomorphisms to open maps and we obtain a new fixed-point theorem.

math.AT↗

arXiv2025

The basic component of the mean curvature of Riemannian foliations

For a Riemannian foliation $F$ on a compact manifold $M$ with a bundle-like metric, the de Rham complex of $M$ is $C^{\infty}$-splitted as the direct sum of the basic complex and its orthogonal complement. Then the basic component $κ_{b}$ of the mean curvature form of $F$ is closed and defines a class $ξ(F)$ in the basic cohomology that is invariant under any change of the bundle-like metric. Moreover, any element in $ξ(F)$ can be realized as the basic component of the mean curvature of some bundle-like metric. It is also proved that $ξ(F)$ vanishes iff there exists some bundle-like metric on $M$ for which the leaves are minimal submanifolds. As a consequence, this tautness property is verified in any of the following cases: (a) when the Ricci curvature of the transverse Riemannian structure is positive, or (b) when $F$ is of codimension one. In particular, a compact manifold with a Riemannian foliation of codimension one has infinite fundamental group. A small correction of a lemma from the original manuscript is included as an addendum, written in collaboration with Ken Richardson.

math.DG↗

arXiv2025

A Combinatorial Study of the Fixed Point Index

We introduce a theory of integration with respect to the fixed point index, offering a substantial improvement over previous approaches based on the Lefschetz number. This framework eliminates several restrictive assumptions -- such as the need for definability, openness, or f-invariance of subspaces -- thereby allowing broader applicability. We also present a natural combinatorial adaptation of the fixed point index that extends the combinatorial Lefschetz number. This extension yields new topological and homotopical invariance results and facilitates the integration of real-valued functions with respect to fixed points.

math.AT↗

arXiv2024

Realization of manifolds as leaves using graph colorings

It is proved that any (repetitive) Riemannian manifold of bounded geometry can be realized as a leaf of some (minimal) Riemannian matchbox manifold without holonomy. Our methods can be adapted to achieve Cantor transversals or a prescribed holonomy covering, but then the manifold may not be realized as a dense leaf.

math.GT↗

arXiv2024

Regularized determinant formulas for the zeta functions of 3-dimensional Riemannian foliated dynamical systems

We prove a regularized determinant formula for the zeta functions of certain 3-dimensional Riemannian foliated dynamical systems, in terms of the infinitesimal operator induced by the flow acting on the reduced leafwise cohomologies. It is the formula conjectured by Deninger. The proof is based on relating the dynamical spectral $ξ$-functions, analogues of the $ξ$-function in analytic number theory, with the zeta function, by applying the distributional dynamical Lefschetz trace formula.

math.DS↗

arXiv2024

Zeta invariants of Morse forms

Let $η$ be a closed real 1-form on a closed Riemannian $n$-manifold $(M,g)$. Let $d_z$, $δ_z$ and $Δ_z$ be the induced Witten's type perturbations of the de~Rham derivative and coderivative and the Laplacian, parametrized by $z=μ+iν\in\mathbb C$ ($μ,ν\in\mathbb{R}$, $i=\sqrt{-1}$). Let $ζ(s,z)$ be the zeta function of $s\in\mathbb{C}$, defined as the meromorphic extension of the function $ζ(s,z)=\operatorname{Str}({η\wedge}\,δ_zΔ_z^{-s})$ for $\Re s\gg0$. We prove that $ζ(s,z)$ is smooth at $s=1$ and establish a formula for $ζ(1,z)$ in terms of the associated heat semigroup. For a class of Morse forms, $ζ(1,z)$ converges to some $\mathbf{z}\in\mathbb{R}$ as $μ\to+\infty$, uniformly on $ν$. We describe $\mathbf{z}$ in terms of the instantons of an auxiliary Smale gradient-like vector field $X$ and the Mathai-Quillen current on $TM$ defined by $g$. Any real 1-cohomology class has a representative $η$ satisfying the hypothesis. If $n$ is even, we can prescribe any real value for $\mathbf{z}$ by perturbing $g$, $η$ and $X$, and achieve the same limit as $μ\to-\infty$. This is used to define and describe certain tempered distributions induced by $g$ and $η$. These distributions appear in another publication as contributions from the preserved leaves in a trace formula for simple foliated flows, giving a solution to a problem of C.~Deninger.

math.DG↗

arXiv2024

Topology of the space of conormal distributions

Given a closed manifold $M$ and a closed regular submanifold $L$, consider the corresponding locally convex space $I=I(M,L)$ of conormal distributions, with its natural topology, and the strong dual $I'=I'(M,L)=I(M,L;Ω)'$ of the space of conormal densities. It is shown that $I$ is a barreled, ultrabornological, webbed, Montel, acyclic LF-space, and $I'$ is a complete Montel space, which is a projective limit of bornological barreled spaces. In the case of codimension one, similar properties and additional descriptions are proved for the subspace $K\subset I$ of conormal distributions supported in $L$ and for its strong dual $K'$. We construct a locally convex Hausdoff space $J$ and a continuous linear map $I\to J$ such that the sequence $0\to K\to I\to J\to 0$ as well as the transpose sequence $0\to J'\to I'\to K'\to 0$ are short exact sequences in the category of continuous linear maps between locally convex spaces. Finally, it is shown that $I\cap I'=C^\infty(M)$ in the space of distributions. In another publication, these results are applied to prove a Lefschetz trace formula for a simple foliated flow $ϕ=\{ϕ^t\}$ on a compact foliated manifold $(M,F)$. It describes a Lefschetz distribution $L_{\text{\rm dis}}(ϕ)$ defined by the induced action $ϕ^*=\{ϕ^{t\,*}\}$ on the reduced cohomologies $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$ of the complexes of leafwise currents that are conormal and dual-conormal at the leaves preserved by $ϕ$.

math.FA↗

arXiv2024

A trace formula for foliated flows

Let $F$ be a transversely oriented foliation of codimension 1 on a closed manifold $M$, and let $ϕ=\{ϕ^t\}$ be a foliated flow on $(M,F)$. Assume the closed orbits of $ϕ$ are simple and its preserved leaves are transversely simple. In this case, there are finitely many preserved leaves, which are compact. Let $M^0$ denote their union, $M^1=M\setminus M^0$ and $F^1=F|_{M^1}$. We consider two topological vector spaces, $I(F)$ and $I'(F)$, consisting of the leafwise currents on $M$ that are conormal and dual-conormal to $M^0$, respectively. They become topological complexes with the differential operator $d_{F}$ induced by the de~Rham derivative on the leaves, and they have an $\mathbb{R}$-action $ϕ^*=\{ϕ^{t\,*}\}$ induced by $ϕ$. Let $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$ denote the corresponding leafwise reduced cohomologies, with the induced $\mathbb{R}$-action $ϕ^*=\{ϕ^{t\,*}\}$. We define some kind of Lefschetz distribution $L_{\text{\rm dis}}(ϕ)$ of the actions $ϕ^*$ on both $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$, whose value is a distribution on $\mathbb{R}$. Its definition involves several renormalization procedures, the main one being the b-trace of some smoothing b-pseudodifferential operator on the compact manifold with boundary obtained by cutting $M$ along $M^0$. We also prove a trace formula describing $L_{\text{\rm dis}}(ϕ)$ in terms of infinitesimal data from the closed orbits and preserved leaves. This solves a conjecture of C.~Deninger involving two leafwise reduced cohomologies instead of a single one.

math.GT↗

arXiv2020

Derived heat trace asymptotics for the de Rham and Dolbeault complexes

We examine the derived heat trace asymptotics in both the real and the complex settings for a generalized Witten perturbation. If the dimension is even, in the real context we show the integral of the local density for the derived heat trace asymptotics is half the Euler characteristic of the underlying manifold. In the complex context, we assume the underlying geometry is Kähler and show the integral of the local density for the derived heat trace asymptotics defined by the Dolbeault complex is a characteristic number of the complex tangent bundle and the twisting vector bundle. We identify this characteristic number if the real dimension is $2$ or $4$. In both the real and complex settings, the local density differs from the corresponding characteristic class by a divergence term.

math.DG↗

arXiv2020

Limit aperiodic and repetitive colorings of graphs

Let $X$ be a (repetitive) infinite connected simple graph with a finite upper bound $Δ$ on the vertex degrees. The main theorem states that $X$ admits a (repetitive) limit aperiodic vertex coloring by $Δ$ colors. This refines a theorem for finite graphs proved by Collins and Trenk, and by Klavžar, Wong and Zhu, independently. It is also related to a theorem of Gao, Jackson and Seward stating that any countable group has a strongly aperiodic coloring by two colors, and to recent research on distinguishing number of graphs by Lehner, Pilśniak and Stawiski, and by Hüning et al. In our theorem, the number of colors is optimal for general graphs of bounded degree. We derive similar results for edge colorings, and for more general graphs, as well as a construction of limit aperiodic and repetitive tilings by finitely many prototiles. In a subsequent paper, this result is also used to improve the construction of compact foliated spaces with a prescribed leaf.

math.MG↗

arXiv2020

Analysis on Riemannian foliations of bounded geometry

A leafwise Hodge decomposition was proved by Sanguiao for Riemannian foliations of bounded geometry. Its proof is explained again in terms of our study of bounded geometry for Riemannian foliations. It is used to associate smoothing operators to foliated flows, and describe their Schwartz kernels. All of this is extended to a leafwise version of the Novikov differential complex.

math.DG↗

arXiv2020

Transversal hard Lefschetz theorem on transversely symplectic foliations

We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K.~Pak, "Transversal harmonic theory for transversally symplectic flows", J. Aust. Math. Soc. 84 (2008), 233--245) to the general transversely symplectic foliation.

math.DG↗

arXiv2019

Simple foliated flows

We describe transversely oriented foliations of codimension one on closed manifolds that admit simple foliated flows.

math.GT↗

arXiv2019

Molino's description and foliated homogeneity

The topological Molino's description of equicontinuous foliated spaces, studied by the first author and Moreira Galicia, gives conditions to reduce their study to the particular case where the holonomy pseudogroup can be represented by a pseudogroup on some local group $G$ generated by some of its local left translations (a $G$-foliated space). That description is sharpened in this paper by introducing a foliated action of a compact topological group on the resulting $G$-foliated space, like in the case of Riemannian foliations. Moreover a $C^\infty$ version is also studied. The triviality of this compact group characterizes compact minimal $G$-foliated spaces, which are also characterized by their foliated homogeneity in the $C^\infty$ case. We also give an example where the projection of the Molino's description is not a principal bundle, and another example of positive topological codimension where the foliated homogeneity cannot be checked by only comparing pairs of leaves---in the case of zero topological codimension, weak solenoids with this property were given by Fokkink and Oversteegen, and later by Dyer, Hurder and Lukina.

math.GT↗

arXiv2018

Witten's perturbation on strata with general adapted metrics

Let $M$ be a stratum of a compact stratified space $A$. It is equipped with a general adapted metric $g$, which is slightly more general than the adapted metrics of Nagase and Brasselet-Hector-Saralegi. In particular, $g$ has a general type, which is an extension of the type of an adapted metric. A restriction on this general type is assumed, and then $g$ is called good. We consider the maximum/minimun ideal boundary condition, $d_{\text{\rm max/min}}$, of the compactly supported de~Rham complex on $M$, in the sense of Brüning-Lesch. Let $H^*_{\text{\rm max/min}}(M)$ and $Δ_{\text{\rm max/min}}$ denote the cohomology and Laplacian of $d_{\text{\rm max/min}}$. The first main theorem states that $Δ_{\text{\rm max/min}}$ has a discrete spectrum satisfying a weak form of the Weyl's asymptotic formula. The second main theorem is a version of Morse inequalities using $H_{\text{\rm max/min}}^*(M)$ and what we call rel-Morse functions. An ingredient of the proofs of both theorems is a version for $d_{\text{\rm max/min}}$ of the Witten's perturbation of the de~Rham complex. Another ingredient is certain perturbation of the Dunkl harmonic oscillator previously studied by the authors using classical perturbation theory. Assume that $A$ is a stratified pseudomanifold, and consider its intersection homology $I^{\bar p}H_*(A)$ with perversity $\bar p$; in particular, the lower and upper middle perversities are denoted by $\bar m$ and $\bar n$, respectively. Then, for any perversity $\bar p\le\bar m$, there is an associated good adapted metric on $M$ satisfying the Nagase isomorphism $H^r_{\text{\rm max}}(M)\cong I^{\bar p}H_r(A)^*$ ($r\in\N$). If $M$ is oriented and $\bar p\ge\bar n$, we also get $H^r_{\text{\rm min}}(M)\cong I^{\bar p}H_r(A)$. Thus our version of the Morse inequalities can be described in terms of $I^{\bar p}H_*(A)$.

math.DG↗

arXiv2017

Generic coarse geometry of leaves

A compact Polish foliated space is considered. Part of this work studies coarsely quasi-isometric invariants of leaves in some residual saturated subset when the foliated space is transitive. In fact, we also use "equi-" versions of this kind of invariants, which means that the definition is satisfied with the same constants by some given set of leaves. For instance, the following properties are proved. Either all dense leaves without holonomy are equi-coarsely quasi-isometric to each other, or else there exist residually many dense leaves without holonomy such that each of them is coarsely quasi-isometric to meagerly many leaves. Assuming that the foliated space is minimal, the first of the above alternatives holds if and if the leaves without holonomy satisfy a condition called coarse quasi-symmetry. A similar dichotomy holds for the growth type of the leaves, as well as an analogous characterization of the first alternative in the minimal case, involving a property called growth symmetry. Moreover some classes of growth are shared, either by residually many leaves, or by meagerly many leaves. If some leaf without holonomy is amenable, then all dense leaves without holonomy are equi-amenable, and, in the minimal case, they satisfy a property called amenable symmetry. Residually many leaves have the same asymptotic dimension. If the foliated space is minimal, then any pair of nonempty open sets in the Higson coronas of the leaves with holonomy contain homeomorphic nonempty open subsets. Another part studies limit sets of leaves at points in their Higson corona, defined like the usual limit sets at their ends.

math.GT↗

arXiv2017

Non-reduction of relations in the Gromov space to Polish actions

It is shown that, in the Gromov space of isometry classes of pointed proper metric spaces, the equivalence relations defined by existence of coarse quasi-isometries or being at finite Gromov-Hausdorff distance, cannot be reduced to the equivalence relation defined by any Polish action.

math.GT↗

arXiv2017

A Perturbation of the Dunkl Harmonic Oscillator on the Line

Let $J_σ$ be the Dunkl harmonic oscillator on ${\mathbb{R}}$ ($σ>-\frac{1}{2}$). For $0 0$, it is proved that, if $σ>u-\frac{1}{2}$, then the operator $U=J_σ+ξ|x|^{-2u}$, with appropriate domain, is essentially self-adjoint in $L^2({\mathbb{R}},|x|^{2σ} dx)$, the Schwartz space ${\mathcal{S}}$ is a core of $\overline U^{1/2}$, and $\overline U$ has a discrete spectrum, which is estimated in terms of the spectrum of $\overline{J_σ}$. A generalization $J_{σ,τ}$ of $J_σ$ is also considered by taking dif\/ferent parameters $σ$ and $τ$ on even and odd functions. Then extensions of the above result are proved for $J_{σ,τ}$, where the perturbation has an additional term involving, either the factor $x^{-1}$ on odd functions, or the factor $x$ on even functions. Versions of these results on ${\mathbb{R}}_+$ are derived.

math.SP↗