🔁Elementary Differential Topology Unit 9 – Whitney Embedding Theorem in Topology

Key Concepts and Definitions

• Manifolds are topological spaces that locally resemble Euclidean space near each point
  • Smooth manifolds have a differentiable structure compatible with the topology
  • Examples include spheres, tori, and projective spaces
• Embeddings are injective continuous functions that map a topological space into another space
  • Smooth embeddings are embeddings that are also smooth functions between smooth manifolds
• Immersions are smooth functions between manifolds whose derivative is everywhere injective
  • Immersions may not be one-to-one, allowing for self-intersections
• Whitney regularity conditions ensure that a map between manifolds is an embedding
  • These conditions involve the behavior of the map and its derivatives

Historical Context and Significance

• Hassler Whitney introduced the concept of embeddings and immersions in the 1930s
• The Whitney Embedding Theorem was a groundbreaking result in differential topology
  • It established a fundamental connection between abstract manifolds and their representations in Euclidean space
• The theorem has implications for the study of submanifolds and the classification of manifolds
  • It allows for the study of intrinsic properties of manifolds using extrinsic methods
• Whitney's work laid the foundation for further developments in differential topology and geometry
  • It influenced the study of characteristic classes, cobordism theory, and Morse theory

Statement of Whitney Embedding Theorem

• The strong Whitney Embedding Theorem states that any smooth $n$-dimensional manifold can be smoothly embedded into $\mathbb{R}^{2n}$
  • This means that every abstract manifold has a concrete realization in Euclidean space
• The weak Whitney Embedding Theorem states that any smooth $n$-dimensional manifold can be smoothly embedded into $\mathbb{R}^{2n+1}$
  • This version requires one additional dimension compared to the strong theorem
• The theorem holds for both compact and non-compact manifolds
  • Compact manifolds can be embedded into a bounded subset of Euclidean space

Proof Outline and Main Ideas

• The proof of the Whitney Embedding Theorem relies on several key ideas and techniques
  • It involves the construction of a smooth map from the manifold to Euclidean space
  • The map is shown to be an immersion and then modified to be an embedding
• The proof uses the concept of Whitney regularity conditions
  • These conditions ensure that the map is locally an embedding
• Partitions of unity are employed to globalize the local embeddings
  • Partitions of unity allow for the construction of a global embedding from local data
• The proof also relies on the compactness of the manifold or the existence of a proper exhaustion function
  • Compactness ensures that the local embeddings can be combined into a global embedding

Applications in Differential Topology

• The Whitney Embedding Theorem has numerous applications in differential topology
• It allows for the study of submanifolds of Euclidean space
  • Submanifolds inherit properties from the ambient Euclidean space
  • The theorem guarantees the existence of smooth embeddings of submanifolds
• The theorem is used in the classification of manifolds
  • Embeddings provide a way to compare and distinguish different manifolds
• Embeddings are crucial in the study of vector bundles and characteristic classes
  • Vector bundles can be constructed using embeddings of manifolds into Euclidean space
• The theorem is also relevant to the study of Morse functions and critical points
  • Morse functions can be constructed using embeddings and projections

Related Theorems and Results

• The Whitney Embedding Theorem is related to several other important results in differential topology
• The Nash Embedding Theorem extends the Whitney Embedding Theorem to Riemannian manifolds
  • It states that any Riemannian manifold can be isometrically embedded into Euclidean space
• The Takens Embedding Theorem concerns the embedding of dynamical systems into Euclidean space
  • It has applications in the study of chaos theory and time series analysis
• The Mostow Embedding Theorem deals with the embedding of hyperbolic manifolds
  • It relates the geometry of hyperbolic manifolds to their topology
• The Whitney Approximation Theorem is a related result on the approximation of continuous functions by smooth functions
  • It is used in the proof of the Whitney Embedding Theorem

Examples and Illustrations

• The circle $S^1$ can be smoothly embedded into $\mathbb{R}^2$ using the map $(x,y) = (\cos\theta, \sin\theta)$
  • This is an example of a compact 1-dimensional manifold embedded in 2-dimensional Euclidean space
• The torus $T^2$ can be smoothly embedded into $\mathbb{R}^4$ using the map $(x,y,z,w) = ((\cos\theta)(R+r\cos\phi), (\sin\theta)(R+r\cos\phi), r\sin\phi\cos\theta, r\sin\phi\sin\theta)$
  • This is an example of a compact 2-dimensional manifold embedded in 4-dimensional Euclidean space
• The real projective plane $\mathbb{RP}^2$ can be smoothly embedded into $\mathbb{R}^4$
  • This is an example of a non-orientable 2-dimensional manifold embedded in 4-dimensional Euclidean space
• The Klein bottle is a non-orientable 2-dimensional manifold that cannot be embedded in $\mathbb{R}^3$
  • However, it can be smoothly embedded into $\mathbb{R}^4$

Common Misconceptions and Pitfalls

• It is important to distinguish between embeddings and immersions
  • Embeddings are always injective, while immersions may allow for self-intersections
• The Whitney Embedding Theorem does not guarantee a unique embedding
  • There may be many different ways to embed a manifold into Euclidean space
• The theorem does not provide an explicit construction of the embedding
  • The proof is existential and does not give a specific formula for the embedding
• The dimension of the Euclidean space in the theorem is optimal for smooth embeddings
  • However, continuous embeddings may exist in lower dimensions (e.g., the strong Whitney Embedding Theorem for continuous embeddings states that any $n$-dimensional manifold can be continuously embedded into $\mathbb{R}^{2n-1}$)
• The Whitney Embedding Theorem does not apply to all topological spaces
  • It specifically deals with smooth manifolds and their embeddings into Euclidean space