Groupoids and an index theorem for conical pseudo-manifolds

Claire Debord (1) , Jean-Marie Lescure (1) , Victor Nistor (2)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 Departement of Mathematics

We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold $M$. A main ingredient is a non-commutative algebra that plays in our setting the role of $C_0(T^*M)$. We prove a Thom isomorphism between non-commutative algebras which gives a new example of wrong way functoriality in $K$-theory. We then give a new proof of the Atiyah-Singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds.

Mots clés: K-theory; Noncommutative geometry; index theory; singular manifolds; Lie groupoids

Domaines: Algèbres d'opérateurs [math.OA]

Claire Debord, Jean-Marie Lescure, Victor Nistor. Groupoids and an index theorem for conical pseudo-manifolds. Journal für die reine und angewandte Mathematik, 2009, 628, pp.1--35. ⟨hal-00094949v2⟩