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
9 - Manifolds with Boundary
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 smooth manifolds with boundary, allowing us to treat the domains of smooth homotopies as manifolds. We show that the boundary of a smooth manifold with boundary is itself a smooth manifold and define tangent spaces and derivatives in this setting. We extend key results to manifolds with boundary, proving the Preimage Theorem and its generalisation to transverse intersections, as well as Sard’s Theorem. Exercises include introducing the Möbius band, determining its boundary, and showing the existence of inward- and outward-pointing normal vectors.
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- Differential Topology , pp. 156 - 166Publisher: Cambridge University PressPrint publication year: 2026
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