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
13 - Intersection Theory Modulo 2
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
This chapter generalises the construction of the Brouwer degree to maps between smooth manifolds of varying dimensions and to submanifolds, leading to the notion of intersection numbers modulo 2. The construction relies on Thom’s transversality theorems. We compute intersection numbers in several examples, including the open Möbius band. Using intersection numbers, we show that real projective -space and the -sphere are not homotopy equivalent. From this and Brouwer’s Invariance of Domain for smooth maps, we deduce that no commutative division algebra structures exist on real Euclidean spaces of dimension greater than two. Exercises introduce the concept of cobordism and explore its relationship with intersection numbers.
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- Differential Topology , pp. 227 - 238Publisher: Cambridge University PressPrint publication year: 2026
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