Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Introduction
- 1 A Brief Introduction to Topological Spaces
- 2 Smooth Manifolds
- 3 The Inverse Function Theorem, Immersions, and Embeddings
- 4 Submersions and Regular Values
- 5 Transversality
- 6 Abstract Smooth Manifolds
- 7 Whitney’s Embedding Theorems
- 8 Smooth Homotopy
- 9 Manifolds with Boundary
- 10 Brouwer Fixed Point Theorem
- 11 The Brouwer Degree Modulo 2
- 12 Tubular Neighbourhoods and Transversality
- 13 Intersection Theory Modulo 2
- 14 Orientation
- 15 The Integer-Valued Brouwer Degree
- 16 Pontryagin Construction and Hopf’s Degree Theorem
- 17 Vector Fields and the Poincaré—Hopf Index Theorem
- Appendix Solutions to Selected Exercises
- References
- Index
14 - Orientation
Published online by Cambridge University Press: 08 August 2026
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Introduction
- 1 A Brief Introduction to Topological Spaces
- 2 Smooth Manifolds
- 3 The Inverse Function Theorem, Immersions, and Embeddings
- 4 Submersions and Regular Values
- 5 Transversality
- 6 Abstract Smooth Manifolds
- 7 Whitney’s Embedding Theorems
- 8 Smooth Homotopy
- 9 Manifolds with Boundary
- 10 Brouwer Fixed Point Theorem
- 11 The Brouwer Degree Modulo 2
- 12 Tubular Neighbourhoods and Transversality
- 13 Intersection Theory Modulo 2
- 14 Orientation
- 15 The Integer-Valued Brouwer Degree
- 16 Pontryagin Construction and Hopf’s Degree Theorem
- 17 Vector Fields and the Poincaré—Hopf Index Theorem
- Appendix Solutions to Selected Exercises
- References
- Index
Summary
In this chapter, we use the fact that the real numbers form an ordered field to define the concept of orientation for smooth manifolds. We introduce pullback orientation along smooth maps and apply it to show that real projective -space is orientable if and only if n is odd. We also demonstrate that the Möbius band is not orientable. Next, we consider the induced orientation on the boundary of a smooth manifold with boundary and study how orientations behave under preimages, particularly the relationship between boundary orientation and orientation as a preimage. We compute the orientation of the fibres of the Hopf map and conclude by proving that simply connected smooth manifolds are orientable.
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- Differential Topology , pp. 239 - 260Publisher: Cambridge University PressPrint publication year: 2026
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