Topology Seminar: Equivariant Lefschetz invariants via analysis

Equivariant Lefschetz invariants have already appeared in algebraic topology. Here I will show how to approach them using the so-called equivariant KK-theory of Kasparov — the main tool of the new field of noncommutative geometry. I will sketch the construction of Lefschetz invariants for equivariant self-maps of a G-space, where G is a discrete group, and then define them for more general objects than just self-maps, called correspondences. There are always many interesting equivariant self-correspondences of a space with a group action, even if the group is not discrete. The case of compact connected groups seems in particular quite interesting. We state an equivariant version of the Lefschetz fixed-point formula for this situation. In the resulting formula, the geometric side is based on equivariant index theory of elliptic operators, while the global algebraic side involves the module trace of Hattori and Stallings. Computing the relevant traces seems to be a problem belonging properly to algebraic geometry