How to Define Orientation of a Topological Manifold via Sheaves

Question

I have just started reading orientation of a topological manifold from Hacther's Algebraic Topology.

It was hinted by one of my professors that orientation of a manifold can be looked at from the perspective of sheaves. So I want to read about orientations using the language of sheaves, hoping that the abstraction will be easily understandable as geometry will provide motivation for it. This way I also want to familiarize myself with basic concepts in sheaf theory.

I googled for quite a bit but I could not find a source where I can read about orientations using sheaves.

Can somebody please provide a reference. Thanks.

Answer 1

Sheaves on manifolds, by Masaki Kashiwara and Pierre Schapira.

In the edition of the book that I have, you can find the definition of the (relative) orientation sheaf on page 153 (Definition 3.3.3.) and a nice explicit (explicit modulo sheafification) definition on page 154 (Proposition 3.3.6. (i)).

Nevertheless, if you are just starting with sheaves, I don't think this book is a good self-introduction to begin with (see this discussion if you want to hear more opinions in this regard). If you are looking for some geometric motivation for sheaves, I would suggest to look for it in algebraic geometry. But this is already a personal opinion, and whether it is a good idea or not depends much on your taste.

In any case, Vakil's The Rising Sea: Foundations of Algebraic Geometry offers a very good introduction (chapter 2) and includes some convenient background in category theory (chapter 1).

And if algebraic geometry you like and you want to put more effort and self-study to it (and get more knowledge in exchange), the first section of the second chapter of Hartshorne's Algebraic Geometry may be a good choice. But this is perhaps more specific, as he considers only sheaves of abelian groups.