Poincar\'e Complex Diagonals and the Bass trace Conjecture

TL;DR

This paper explores the relationship between Poincaré complex diagonals and the Bass trace conjecture, establishing conditions for Poincaré embeddings based on algebraic invariants and group structures.

Contribution

It connects the total obstruction to Poincaré embeddings with the Reidemeister trace and provides new conditions for embeddings when the fundamental group has a specific structure.

Findings

Obstruction relates to Reidemeister trace of the identity map.

Diagonal admits Poincaré embedding for even-dimensional cases with specific fundamental groups.

Conditions established for embeddings based on the dimension and group structure.

Abstract

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.