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
7 - Whitney’s Embedding Theorems
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 prove Whitney’s Embedding Theorem, which states that every abstract smooth manifold can be embedded into Euclidean space. This establishes the equivalence of the two definitions of smooth manifolds presented in this book. We then show that the dimension of the Euclidean space required for an embedding can be reduced to twice the dimension of the manifold plus one. Two fundamental tools are introduced: smooth bump functions and partitions of unity. We prove the existence of partitions of unity for arbitrary subspaces of Euclidean space and for compact abstract smooth manifolds, providing essential techniques for later developments in differential topology.
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- Differential Topology , pp. 128 - 141Publisher: Cambridge University PressPrint publication year: 2026
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