The signature package on Witt spaces

Albin, Pierre; Leichtnam, Éric; Mazzeo, Rafe; Piazza, Paolo

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 2, pp. 241-310.

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 $\Gamma$. Nous utilisons alors nos constructions pour définir la classe d’indice de signature analytique à valeurs dans le groupe de $K$-théorie $K_{*}\left(C_r^{*}\Gamma \right)$. 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 $K_{*}\left(B\Gamma \right)\otimes \mathbb{Q}\to K_{*}\left(C_r^{*}\Gamma \right)\otimes \mathbb{Q}$ soit injective.

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 $C_r^{*}\Gamma$ Mishchenko bundle associated to any Galois covering of $X$ with covering group $\Gamma$, 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 $C_r^{*}\Gamma$. 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 $K_{*}\left(B\Gamma \right)\otimes \mathbb{Q}\to K_{*}\left(C_r^{*}\Gamma \right)\otimes \mathbb{Q}$ is injective.

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