Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Introduction
- 1A Brief Introduction to Topological Spaces
- 2Smooth Manifolds
- 3The Inverse Function Theorem, Immersions, and Embeddings
- 4Submersions and Regular Values
- 5Transversality
- 6Abstract Smooth Manifolds
- 7Whitney’s Embedding Theorems
- 8Smooth Homotopy
- 9Manifolds with Boundary
- 10Brouwer Fixed Point Theorem
- 11The Brouwer Degree Modulo 2
- 12Tubular Neighbourhoods and Transversality
- 13Intersection Theory Modulo 2
- 14Orientation
- 15The Integer-Valued Brouwer Degree
- 16Pontryagin Construction and Hopf’s Degree Theorem
- 17Vector Fields and the Poincaré—Hopf Index Theorem
- AppendixSolutions to Selected Exercises
- References
- Index
15 - The Integer-Valued Brouwer Degree
Published online by Cambridge University Press: 08 August 2026
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Introduction
- 1A Brief Introduction to Topological Spaces
- 2Smooth Manifolds
- 3The Inverse Function Theorem, Immersions, and Embeddings
- 4Submersions and Regular Values
- 5Transversality
- 6Abstract Smooth Manifolds
- 7Whitney’s Embedding Theorems
- 8Smooth Homotopy
- 9Manifolds with Boundary
- 10Brouwer Fixed Point Theorem
- 11The Brouwer Degree Modulo 2
- 12Tubular Neighbourhoods and Transversality
- 13Intersection Theory Modulo 2
- 14Orientation
- 15The Integer-Valued Brouwer Degree
- 16Pontryagin Construction and Hopf’s Degree Theorem
- 17Vector Fields and the Poincaré—Hopf Index Theorem
- AppendixSolutions to Selected Exercises
- References
- Index
Summary
In this chapter, we use the concept of orientation to define an integer-valued version of the Brouwer degree for smooth maps between oriented manifolds of the same dimension. We prove a boundary theorem for the degree and compute it in numerous examples, exploring several applications of these computations. We then establish Hopf’s Degree Theorem in dimension one, which states that the homotopy class of a smooth self-map of the circle is completely determined by its Brouwer degree. Finally, we use the degree to define integer-valued linking numbers and the Hopf invariant and conclude by showing through direct computation that the Hopf invariant of the Hopf map equals one.
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- Differential Topology , pp. 261 - 277Publisher: Cambridge University PressPrint publication year: 2026
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