Groupoids and an index theorem for conical pseudo-manifolds

Details Groupoids and an index theorem for conical pseudo-manifolds Groupoids and an index theorem for conical pseudo-manifolds

2006-09-15

arxiv.org/abs/math/0609438v2

J. Reine Angew Math 628 (2009) 1-35

Mathematics

Operator Algebras

Scientific paper

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.

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