Homotopy-theoretic properties of gyrations

Stephen Theriault

Vol. 344 (2026), No. 2, 431–448

DOI: 10.2140/pjm.2026.344.431

Abstract

Let $M$ be a simply connected $n$-dimensional manifold with the property that the attaching map for the $n$-cell is inert, which means that the inclusion of the $(n-1)$-skeleton has a right homotopy inverse after looping. We show that a type of surgery on $M\times S^{k-1}$ producing a manifold called a gyration preserves the inertness property. The method generalises to Poincaré duality complexes and also recovers by different means a known homotopy equivalence for the based loops of the gyration. The underlying method is quite general and should have applications elsewhere.

Keywords

gyration, inert attaching map, loop space decomposition

Mathematical Subject Classification

Primary: 55P35, 57N65

Milestones

Received: 14 March 2026

Revised: 5 June 2026

Accepted: 10 June 2026

Published: 8 September 2026