Noncommutative topological theories of gravity

The possibility of noncommutative topological gravity arising in the same manner as Yang-Mills theory is explored. We use the Seiberg-Witten map to construct such a theory based on a SL(2,C) complex connection, from which the Euler characteristic and the signature invariant are obtained. Finally, we speculate on the description of noncommutative gravitational instantons, as well as noncommutative local gravitational anomalies.