Poincaré Complex Diagonals and the Bass trace Conjecture

TL;DR

The paper investigates when the diagonal map $\Delta:M\to M\times M$ of a finitely dominated Poincaré duality space $M$ admits a Poincaré embedding and links the diagonal obstruction $\mu_M$ to the Reidemeister characteristic via an equivariant Hopf invariant framework. It proves the central relation $\tilde{r}(M)=\tilde{\sigma}(\mu_M)$, so a diagonal embedding forces $\tilde{r}(M)=0$, and it otherwise derives vanishing results in several cases including even-dimensional, abelian 2-primary fundamental groups and cases where the Bass trace conjecture holds. The work further relates Wall finiteness obstructions, the Spivak tangent data, and dualizing spectra within a unified stable homotopy and $G$-equivariant context, and it formulates a conjecture that the diagonal embedding should always exist under certain hypotheses. The results extend known cases (simply connected or manifold-like PD spaces) and provide new criteria to certify diagonal embeddability by algebraic K-theory traces and Reidemeister-type invariants, with potential implications for higher-dimensional topology and surgery theory.

Abstract

For a finitely dominated Poincaré duality space $M$, we show how the author's total obstruction to the existence of a Poincaré 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 $π_1(M)$ is a finite direct product of cyclic groups of order two, then the diagonal map admits a Poincaré embedding.

Theorems & Definitions (43)

• Theorem 1
• Remark 1.1
• Theorem 2
• Corollary 3
• Remark 1.3
• Corollary 4
• Corollary 5
• Conjecture 6
• Remark 2.1
• Remark 2.2