For a Serre fibration
$$
F\to E \to B ,
$$
with $F,E,B$ having the homotopy type of finite complexes, it is known that the Euler characteristic is multiplicative:
$$
\chi(E) = \chi(F)\chi(B) .
$$
However, if we more generally assume that $B$ and $F$ are finitely dominated spaces, then does multiplicativity hold as well? (Recall that a finitely dominated space is a retract of a homotopy finite one.)
If true I am looking for a reference. If false, please explain.
Added later: it's true if the base is homotopy finite since we can take the fiberwise double suspension to obtain a fibration with homotopy finite fibers having the same Euler characteristic. So we only need to consider the case when the base is a finitely dominated and the fibers are homotopy finite.
Second Addition: I can solve the problem in general if I can solve it in the following case: Let $\tilde B \to B$ be a finite, regular, $n$-sheeted covering space, where $B$ is finitely dominated. Then $\chi(\tilde B) = n\chi(B)$.
(Note that $\tilde B$ is again finitely dominated, since there is a finite covering $\tilde B\times S^1 \to B\times S^1$ and by a theorem of Mather, a space $X$ is finitely dominated if and only if $X\times S^1$ is homotopy finite. But since the base $B\times S^1$ is homotopy finite, we can put a finiteness structure on the total space as well.)
Third Addition: The case alluded to in my "Second Addition" holds, by the second answer I gave below.
Best Answer
Note Added March 1, 2022:
I now think there is a gap in deducing multiplicativity of the Euler characteristic from the Pedersen-Taylor result on the finiteness obstruction. I think the argument I give in my other answer more-or-less fills that gap.
Ben Wieland has provided a reference which answers my question.
Pedersen, Erik Kjaer; Taylor, Lawrence R. The Wall finiteness obstruction for a fibration. Amer. J. Math. 100 (1978), no. 4, 887–896.
The authors identify the image in $K_0(\Bbb Z[\pi_1(B)])$ of the Wall finiteness obstruction $\sigma(E) \in K_0(\Bbb Z[\pi_1(E)])$. The Euler characteristic is the image of this class $K_0(\Bbb Z)$.