Orientability of smooth manifolds

Azer Akhmedov – GJM, Volume 10, Issue 2 (2025), 28-45.

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• décembre 30, 2025
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This semi-expository note provides a pedagogical account of the notion of orientability in smooth manifolds. We describe various settings in which orientability is established and used. Sections 5, 11, and 12 contain original results. In these and other sections, we bring together various results that are otherwise buried in the literature. Section 5 discusses the orientability of manifolds with a finite atlas in terms of a finite graph associated with the given atlas. In Section 11, we determine which orientable 3-manifolds cover a non-orientable one, depending on which of the eight geometries (in the sense of Thurston) they admit. Section 12 is devoted to the question of when a given finitely presented group occurs as a fundamental group of a closed non-orientable manifold, and we provide both positive and negative results.

Milestones:

Received: January 09, 2025
Revised: July 20, 2025
Accepted: December 01, 2025
Published: December 21, 2025

Authors:

Azer Akhmedov
Department of Mathematics,
North Dakota State University, PO Box 6050,
Fargo, ND, 58108-6050, USA.

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