  For a finitely dominated Poincar\'e duality space $M$, we show how the
author's total obstruction to the existence of a Poincar\'e embedding of the
diagonal map $M \to M \times M$ relates to the Reidemeister trace of the
identity map of $M$.
  We also show that if the dimension of $M$ is even and at least four, and if
$\pi_1(M)$ is a finite direct product of cyclic groups of order two, then the
diagonal map admits a Poincar\'e embedding.
