The signature package on Witt spaces

Abstract

A. -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 selfadjoint 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 * r Γ 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 * r Γ. 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 mapR. -Dans cet article nous prouvons plusieurs résultats pour l'opérateur de la signature sur un espace de Witt X compact orienté quelconque. Nous construisons une paramétrix de l'opérateur de la signature de X en raisonnant par récurrence sur la profondeur de X et en utilisant une analyse très fine de l'opérateur normal (près d'une strate). Ceci nous permet de montrer que le domaine maximal de l'opérateur de la signature est compactement inclus dans l'espace L 2 correspondant. On peut alors (re)démontrer que l'opérateur de la signature est essentiellement self-adjoint et a un spectre L 2 discret de multiplicité finie de sorte que son indice est bien défini. Nous donnons donc une nouvelle démonstration de certains résultats dus à Jeff Cheeger. Nous considérons ensuite le cas où X est muni d'un revêtement galoisien de groupe Γ. Nous utilisons alors nos constructions pour définir la classe d'indice de signature analytique à valeurs dans le groupe de K-théorie K * (C * r Γ). Nous généralisons dans cette situation singulière la plupart des résultats connus dans le cas où X est lisse. C'est ce qu'on appelle le « forfait signature ». En particulier, nous prouvons un nouveau théorème, purement topologique, qui permet de prouver l'invariance par homotopie stratifiée des hautes signatures de X (définies à l'aide de la L−classe homologique de X) pourvu que l'application d'assemblement rationnelle K * (BΓ) ⊗ Q → K * (C * r Γ) ⊗ Q soit injective.

The Yamabe problem on stratified spaces

“…Fortunately we need very few consequences of such a calculus and can deduce these from a somewhat primitive parametrix construction. This is carried out in more detail in [4]. Recall that we wish to construct an operator G such that, with L equal to the conformal Laplacian for g, GL = I − Q where Q maps into A ν (M ), which has mapping properties analogous to (3.6), (3.7), (3.8), and finally, so that the commutator [∂ y , G] enjoys the same mapping properties.…”

“…Further details can be found in in the foundational monograph of Verona [31] and the exposition by Pflaum [24]. Basic definitions vary between sources, and the recent paper [4] provides a clarification and unified presentation of some of this material; we follow the notation and development of [4, §2] and refer to it for all further details, in particular, for a proof that this class of spaces coincides with the class of iterated edge spaces considered by Cheeger [11], cf. also [19].…”

“…A smoothly stratified space X is a pseudomanifold if the stratum of maximal dimension is dense in X. In distinction to [4], we allow X to have strata of codimension one, but since hypersurface boundaries play a somewhat different role in our main theorem, we say that X is an iterated edge space (or smoothly stratified pseudomanifold) with boundary in this case. • Any element X ∈ I k has a well-defined dimension, where in the decomposition above, dim…”

“…This extends the definition to spaces of depth j + 1. (This smoothness hypothesis excludes many spaces that constitute the standard broader class of stratified spaces, where these local trivializations are only required to be continuous; see [4] for more on this.) A smoothly stratified space X is a pseudomanifold if the stratum of maximal dimension is dense in X.…”

An Obata singular theorem for stratified spaces

“…We refer to α has the cone angle of the stratum Σ n−2 . On a stratified space we can consider an iterate edge metric, as defined in [3] or [1], which is a smooth Riemannian metric on the regular set Ω, and define the usual tools of geometric analysis.…”