On a Possible Noncommutative Generalization of the Signature and Euler Topological Invariants

Noncommutative topological gravity arises in a similar manner as noncommutative Yang-Mills theories. We use the Seiberg-Witten map to show how such a theory based on a SL(2,C) complex connection can be constructed, and from which generalized noncommutative Euler and the signature invariants can be obtained. Finally, in the discussion, we speculate on the description of noncommutative gravitational instantons, as well as noncommutative local gravitational anomalies.