For a fibration (with a connected base space) the Euler characteristic of the total space is the product of the Euler characteristics of the base and the fiber. The Euler characteristic is also additive on subcomplexes. Combining these we have a description of the Euler characteristic of a G-space in terms of the quotients of the isotropy subspaces. The generalizations of the Euler characteristic to fixed point invariants, the Lefschetz number and Reidemeister trace, are similarly additive and multiplicative. This enables a description of the equivariant Lefschetz number and Reidemeister trace in terms of the classical invariants of isotropy subspaces and their quotients.

Some of these results are well known, and they were established using a range of approaches. In fact, all of these results are consequences of a simple formal observation and some topological input. The formal observation is a generalization of the invariance of (the linear algebra) trace under change of basis.