The signature package on Witt spaces

Pierre Albin, Éric Leichtnam, Rafe Mazzeo, Paolo Piazza

Research output: Contribution to journalArticlepeer-review


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* 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) ⊗ ℚ → K*(C* rΓ) ⊗ ℚ is injective.

Original languageEnglish (US)
Pages (from-to)241-310
Number of pages70
JournalAnnales Scientifiques de l'Ecole Normale Superieure
Issue number2
StatePublished - Mar 1 2012

ASJC Scopus subject areas

  • General Mathematics


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