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
11 - The Brouwer Degree Modulo 2
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
This chapter introduces the Brouwer degree modulo 2 for smooth maps between manifolds of the same dimension, an invariant of fundamental importance in differential topology. We construct the invariant and prove it is well defined using the Isotopy Lemma. The Brouwer degree is then applied as an obstruction to extending a map from the boundary of a smooth manifold. We compute the degree for self-maps of the circle. Then we prove the Isotopy Lemma. Building on the Brouwer degree, we define further invariants, including the winding number, linking number, and the Hopf invariant modulo 2. We prove the Borsuk—Ulam Theorem on the degree of odd self-maps of spheres and conclude by computing the Hopf invariant modulo 2 for the Hopf map.
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- Differential Topology , pp. 177 - 199Publisher: Cambridge University PressPrint publication year: 2026
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