Given an $n$-manifold $M$ (say), we can talk about its Euler characteristic $\chi(M)$.

This can be generalized to the Euler number of any $n$-dimensional bundle ${\mathcal V}$. Or indeed, the Euler class of any $k$-dimensional oriented bundle.

Or, given a map $f:M\to M$, we can talk about its Lefschetz number, where $\chi(M)$ is the Lefschetz number of the identity.

Or, we can compute $M$'s Betti numbers, and see $\chi(M)$ as the alternating sum.

I know how to combine 2 & 3, replacing the Betti number by $Tr(f|_{H^i(M)})$.

Are 1 & 2 or 1 & 3 combinable? Or 1, 2, and 3?

I don't have an application in mind, just the usual mathematician's hankering to take LCMs wherever possible.