Comparison techniques on inert top cell attachments

TL;DR

This paper develops criteria to determine the inertness of top cell attachments in Poincaré duality complexes using algebraic and homotopy-theoretic methods, supported by examples and open problems.

Contribution

It introduces new criteria for inertness of top cell attachments in Poincaré duality complexes, combining algebraic and homotopy-theoretic approaches.

Findings

Established criteria for inertness using nonzero degree maps and intersection theory.

Provided numerous examples including surgery, homogeneous spaces, and low-dimensional manifolds.

Proposed eight open problems for further research.

Abstract

We establish various criteria for the inertness of the top cell attachments of Poincar\'{e} duality complexes through nonzero degree maps, algebraic intersection theory and various types of homotopy fibrations. Many examples are provided, including specific surgery, homogeneous spaces and low dimensional manifolds. Additionally, we propose eight open problems.