  By studying an explicit form of the Eichler--Shimura map for ${\rm GL}_3$, we
describe a precise relation between critical values of the complete
$L$-function for the Rankin--Selberg convolution ${\rm GL}_3 \times {\rm GL}_2$
and the cohomological cup product of certain rational cohomology classes which
are uniquely determined up to rational scalar multiples from the cuspidal
automorphic representations under consideration. This refines rationality
results on critical values due to Raghuram et al.
