Homotopy classification of $PD_4$-complexes relative an order relation

TL;DR

This paper introduces an order relation among oriented $PD_4$-complexes, establishing conditions under which two such complexes are homotopy equivalent based on isometries of their second homology groups.

Contribution

It defines a new order relation on $PD_4$-complexes and characterizes homotopy equivalence via isometries of second homology groups.

Findings

Homotopy equivalence characterized by isometry of second homology groups.

Introduction of an order relation among $PD_4$-complexes.

Analysis of minimal objects within this relation.

Abstract

We define an order relation among oriented $PD_4$-complexes. We show that with respect to this relation, two $PD_4$-complexes over the same complex are homotopy equivalent if and only if there is an isometry between the second homology groups. We also consider minimal objects of this relation.