Configuration space and Casson invariant

In this talk, we define an invariant of the isotopy classes of diffeomorphisms between a rational homology 3-sphere by using the graph of maps. We also show that it is equal to the Casson-Walker invariant when the map is identity. By comparing the constructions of the degree one term of the Kontsevich-Kuperberg-Thurston invariant Z of rational homology 3-spheres and our invariant, we can see a direct relation between them.

The invariant Z was constructed by Kontsevich by using the configuration space integrals, and Lescop proved that the degree one term of Z coincides with the Casson-Walker invariant. Roughly speaking, the complement of the graph of a map, which is used to define our invariant, corresponds to the two-point configuration space.