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Eric Leichtnam

Publications and source records attributed to Eric Leichtnam.

At least 19 recordsLinked to original sources
arXiv2025

The G-signature Theorem on Witt spaces

Let G be a compact Lie group and let X be an oriented Witt G-pseudomanifold. Using intersection cohomology it is possible to define Sign(G,X) in R(G), the G-signature of X. Let g be an element in G. Assuming that the inclusion of the fixed point set associated to g is normally non-singular, we prove a formula for Sign(g,X), the G-signature of X computed at g, thus extending to Witt G-pseudomanifolds the fundamental result proved by Atiyah, Segal and Singer on smooth compact G-manifolds. Along the way, we give a detailed study of the fixed point set of a Thom-Mather G-space X and our main result in this direction is a sufficient condition ensuring that the fixed point set associated to G is included in X in a normally non-singular manner. This latter result provides many examples where our formula applies.

math.DG↗
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

Théorie Quasicristalline des Nombres: Recherche d'une Théorie de Drinfeld-Hayes en Charactéristique Zéro

This article develops the structure necessary for the formulation of a version of Drinfeld-Hayes theory in characteristic zero, using the arithmetic of quasicrystal rings attached to a number field. -- -- Cet article développe la structure nécessaire à la formulation d'une version de la théorie de Drinfeld-Hayes en caractéristique nulle, en utilisant la théorie liée à l'arithmétique des anneaux quasicristallins attachés aux corps de nombres.

math.NT↗
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

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↗
arXiv2019

Simple foliated flows

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

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↗
arXiv2016

The Novikov conjecture on Cheeger spaces

We prove the Novikov conjecture on oriented Cheeger spaces whose fundamental group satisfies the strong Novikov conjecture. A Cheeger space is a stratified pseudomanifold admitting, through a choice of ideal boundary conditions, an L2-de Rham cohomology theory satisfying Poincare duality. We prove that this cohomology theory is invariant under stratified homotopy equivalences and that its signature is invariant under Cheeger space cobordism. Analogous results, after coupling with a Mishchenko bundle associated to any Galois covering, allow us to carry out the analytic approach to the Novikov conjecture: we define higher analytic signatures of a Cheeger space and prove that they are stratified homotopy invariants whenever the assembly map is rationally injective. Finally we show that the analytic signature of a Cheeger space coincides with its topological signature as defined by Banagl.

math.DG↗
arXiv2015

Hodge theory on Cheeger spaces

We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary operators and the notion of mezzoperversity, which intermediates between the standard lower and upper middle perversities in intersection theory, as interpreted in this de Rham setting, and show that the de Rham operator with these boundary conditions is Fredholm and has compact resolvent. We also prove an isomorphism between the resulting Hodge and L2 de Rham cohomology groups, and that these are independent of the choice of iterated edge metric. On spaces which admit ideal boundary conditions of this type which are also self-dual, which we call `Cheeger spaces', we show that these Hodge/de Rham cohomology groups satisfy Poincare Duality.

math.DG↗
arXiv2014

Riemannian foliations of bounded geometry

Continuing the study of bounded geometry for Riemannian foliations, begun by Sanguiao, we introduce a chart-free definition of this concept. Our main theorem states that it is equivalent to a condition involving certain normal foliation charts. For this type of charts, it is also shown that the derivatives of the changes of coordinates are uniformly bounded, and there are nice partitions of unity. Applications to a trace formula for foliated flows will given in a forthcoming paper.

math.GT↗
arXiv2013

Refined intersection homology on non-Witt spaces

We develop a generalization to non-Witt spaces of the intersection homology theory of Goresky-MacPherson. The second author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a generalization of Cheeger's L2 de Rham cohomology. In this paper we extend both of these cohomologies by describing all sheaf complexes in the derived category of constructible sheaves that are compatible with middle perversity intersection cohomology, though not necessarily self-dual. On Thom-Mather stratified spaces this refined intersection cohomology theory coincides with the analytic de Rham theory.

math.GT↗
arXiv2013

On the analogy between L-functions and Atiyah-Bott-Lefschetz trace formulas for foliated spaces

This paper is motivated by Deninger's programme. First we prove, using Alvarez Lopez-Kordyukov results, an Atiyah-Bott-Lefschetz trace formula for the cohomology groups associated to a ramified leafwise flat line bundle on a riemannian foliation. Then we argue, by precise computations, that cohomology groups associated to ramified leafwise flat vector bundles on a suitable foliated space (whose existence is still unknown) might be useful to study arithmetic L-functions via Atiyah-Bott-Lefschetz trace formulas.

math.NT↗
arXiv2013

Commuting and noncommuting infinitesimals

Infinitesimals are natural products of the human imagination. Their history goes back to the Greek antiquity. Their role in the calculus and analysis has seen dramatic ups and downs. They have stimulated strong opinions and even vitriol. Edwin Hewitt developed hyperreal fields in the 1940s. Abraham Robinson's infinitesimals date from the 1960s. A noncommutative version of infinitesimals, due to Alain Connes, has been in use since the 1990s. We review some of the hyperreal concepts, and compare them with some of the concepts underlying noncommutative geometry.

math.HO↗
arXiv2011

The signature package on Witt spaces

In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction, which is inductive over the `depth' of the singularity, is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index -- the analytic signature of X -- is well-defined. This provides an alternate approach to some well-known results due to Cheeger. We then prove some new results. By coupling this parametrix construction to a C*_rΓ-Mishchenko bundle associated to any Galois covering of X with covering group Γ, we prove analogues of the same analytic results, from which it follows that one may define an analytic signature index class as an element of the K-theory of C*_rΓ. We go on to establish in this setting and for this class the full range of conclusions which sometimes goes by the name of the signature package. In particular, we prove a new and purely topological theorem, asserting the stratified homotopy invariance of the higher signatures of X, defined through the homology L-class of X, whenever the rational assembly map K_* (BΓ)\otimes\bbQ \to K_*(C*_r Γ)\otimes \bbQ is injective.

math.DG↗
arXiv2009

The signature package on Witt spaces, II. Higher signatures

This is a sequel to the paper "The signature package on Witt spaces, I. Index classes" by the same authors. In the first part we investigated, via a parametrix construction, the regularity properties of the signature operator on a stratified Witt pseudomanifold, proving, in particular, that one can define a K-homology signature class. We also established the existence of an analytic index class for the signature operator twisted by a C^*_rΓMischenko bundle and proved that the K-homology signature class is mapped to the signature index class by the assembly map. In this paper we continue our study, showing that the signature index class is invariant under rational Witt bordisms and stratified homotopies. We are also able to identify this analytic class with the topological analogue of the Mischenko symmetric signature recently defined by Banagl. Finally, we define Witt-Novikov higher signatures and show that our analytic results imply a purely topological theorem, namely that the Witt-Novikov higher signatures are stratified homotopy invariants if the assembly map in K-theory is rationally injective.

math.DG↗
arXiv2009

The signature package on Witt spaces, I. Index classes

We give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction is inductive. It is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index -- the analytic signature of X -- is well-defined. We then show how to couple this construction to a C^*_r(Gamma) Mischenko bundle associated to any Galois covering of X with covering group Gamma. The appropriate analogues of these same results are then proved, and it follows that we may define an analytic signature class as an element of the K-theory of C^*_r(Gamma). In a sequel to this paper we establish in this setting the full range of conclusions for this class which sometimes goes by the name of the signature package.

math.DG↗