Generalized Lefschetz numbers of pushout maps defined on non-connected spaces
Let A, $X_1$ and $X_2$ be topological spaces and let $i_1 : A → X_1$, $i_2: A → X_2$ be continuous maps. For all self-maps $f_A: A → A$, $f_1: X_1 → X_1$ and $f_2: X_2 → X_2$ such that $f_1i_1 = i_1f_A$ and $f_2i_2=i_2f_A$ there exists a pushout map f defined on the pushout space $X_1 ⊔_A X_2$. In [F] we proved a formula relating the generalized Lefschetz numbers of f, $f_A$, $f_1$ and $f_2$. We had to assume all the spaces involved were connected because in the original definition of the generalized Lefschetz number given by Husseini in [H] the space was assumed to be connected. So, to extend the result of [F] to the not necessarily connected case, a definition of generalized Lefschetz number for a map defined on a not necessarily connected space is given; it reduces to the original one when the space is connected and it is still a trace-like quantity. It allows us to prove the pushout formula in this more general case and therefore to get a tool for computing Nielsen and generalized Lefschetz numbers in a wide class of spaces.