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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 new ingredient in our proof is a non-commutative algebra that plays in our setting the role of  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.

Groupoids and an index theorem for conical pseudo-manifolds