A major reason the Euler characteristic is a fantastic topological invariant is because it is additive on subcomplexes. It is also a fixed point invariant - it is a signed count of the number of fixed points of the identity map. (Replace the map bya homotopic map with isolated fixed points.) These two observations raise questions about how to think about fixed point invariants. We'll describe an approach to fixed point invariants that, while initially motivated by classical invariants, is reaching toward prioritizing additivity.
Reading List:
 R. Geoghegan, \Nielsen Fixed Point Theory" pp. 499 - 521 in R.J. Daverman
and R.B. Sher (eds) Handbook of Geometric Topology, (2001), Elsevier B.V.
 K. Ponto and M. Shulman, \Traces in symmetric monoidal categories", Expo-
sitiones Mathematicae, Vol. 32, Issue 3, 2014. pp. 248 { 273.