Poincar\'e Complex Diagonals and the Bass trace Conjecture

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.