9.1 Statement and significance of the Whitney Embedding Theorem
3 min read•august 9, 2024
The is a game-changer in differential topology. It says any smooth n-dimensional manifold can be smoothly embedded in . This means we can visualize and study complex manifolds in familiar Euclidean spaces.
While the theorem guarantees embeddings exist, it doesn't tell us how to construct them. It sets an upper limit on the needed dimensions, but some manifolds might fit in smaller spaces. This idea bridges abstract math with practical applications in physics and geometry.
Smooth Manifolds and Embeddings
Fundamental Concepts of Manifolds and Mappings
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- describes a topological space locally resembling
- Possesses a smooth structure allowing for calculus operations
- Examples include spheres, tori, and Möbius strips
- Embedding refers to a smooth injective immersion that maps a manifold into another space
- Preserves the topological and differential structure of the original manifold
- Allows visualization of abstract manifolds in more familiar spaces
- Immersion denotes a differentiable function between manifolds with injective derivative
- May have self-intersections unlike embeddings
- Examples include figure-eight curves and Klein bottles in 3D space
- Smooth map represents a function between manifolds with continuous derivatives of all orders
- Enables transfer of geometric and analytic properties between spaces
- Includes diffeomorphisms, which preserve both smoothness and invertibility
The Whitney Embedding Theorem
- Whitney Embedding Theorem states any smooth n-dimensional manifold can be smoothly embedded in
- Provides a powerful tool for visualizing and studying abstract manifolds
- Demonstrates the universality of Euclidean spaces for embedding manifolds
- Theorem guarantees existence of an embedding but does not provide a constructive method
- Optimal embeddings may exist in lower dimensions for specific manifolds
- Serves as an upper bound for the required embedding
- Applications extend to differential topology, algebraic geometry, and theoretical physics
- Facilitates study of manifold properties through their embeddings
- Enables representation of physical systems in higher-dimensional spaces
Euclidean Space and Dimensions
Properties of Euclidean Space
- Euclidean space represents the standard n-dimensional space
- Characterized by a flat geometry and intuitive distance measure
- Serves as the ambient space for many mathematical and physical models
- Dimension of a space denotes the number of independent parameters needed to specify a point
- Determines the degrees of freedom available in the space
- Examples include 3D physical space and 4D spacetime in relativity
- Codimension measures the difference in dimensions between a subspace and its ambient space
- Calculated as the dimension of the ambient space minus the dimension of the subspace
- Helps quantify how "much room" a subspace has within its containing space
Mappings and Injectivity
- Injective function maps distinct elements of the domain to distinct elements in the codomain
- Also known as one-to-one functions
- Crucial for preserving the structure of the original space in embeddings
- Relationship between injectivity and dimension in embeddings
- Generally easier to find injective maps into higher-dimensional spaces
- Whitney Embedding Theorem exploits this by using a target space of double the dimension
Topological Properties
Characteristics of Topological Embeddings
- defines a homeomorphism between a space and its image in another space
- Preserves topological properties such as connectedness and compactness
- Weaker notion than smooth embedding, requiring only continuity
- Relationship between topological and smooth embeddings
- Every smooth embedding induces a topological embedding
- Not all topological embeddings can be smoothed (wild embeddings)
- Applications of topological embeddings in knot theory and low-dimensional topology
- Study of knotted curves in 3D space as embeddings of circles
- Classification of surfaces through their embeddings in 3D and 4D spaces
Implications for Manifold Theory
- Topological invariants preserved under embeddings
- Fundamental group, homology groups, and homotopy type remain unchanged
- Allows study of abstract spaces through their concrete realizations
- Role of embeddings in proving theorems about manifolds
- Enables transfer of results from Euclidean spaces to more general manifolds
- Facilitates visualization and intuition for higher-dimensional phenomena
- Limitations and challenges of topological embeddings
- Not all topological properties of the ambient space transfer to the embedded subspace
- Complexity of determining whether two embeddings are equivalent up to ambient isotopy
Key Terms to Review (15)
Algebraic Topology: Algebraic topology is a branch of mathematics that uses algebraic methods to study topological spaces and their properties. By associating algebraic structures, like groups, to these spaces, it allows for a deeper understanding of their shape and connectivity. This approach is crucial for analyzing submanifolds and embeddings as well as understanding the implications of the Whitney Embedding Theorem.
Compact manifold: A compact manifold is a topological space that is both a manifold and compact, meaning it is closed and bounded. Compact manifolds have important properties that make them easier to work with, such as every open cover having a finite subcover, which leads to various significant results in differential topology and geometry.
Differentiable Structure: A differentiable structure on a manifold is a way of defining how to differentiate functions on that manifold, allowing us to consider smooth functions and smooth transitions between charts. This structure is crucial because it enables the application of calculus in more abstract settings, which can then be connected to important concepts like submanifolds, examples of manifolds, partitions of unity, and embedding theorems.
Dimension: Dimension refers to the minimum number of coordinates needed to specify a point within a mathematical space. It serves as a fundamental concept in topology and geometry, allowing us to classify spaces based on their complexity and structure. The concept of dimension connects various important features, such as the behavior of submanifolds, the intricacies of embeddings, and the properties of different types of manifolds like spheres and tori.
Euclidean Space: Euclidean space is a fundamental concept in mathematics, representing a flat, two-dimensional or three-dimensional space defined by points, lines, and planes. It serves as the classical model for geometry and provides the groundwork for understanding various mathematical structures and topologies.
Hassler Whitney: Hassler Whitney was a prominent mathematician known for his work in differential topology and geometry, particularly recognized for developing concepts related to transversality and the Whitney Embedding Theorem. His contributions laid the groundwork for understanding how manifolds can be embedded in Euclidean space and the conditions under which various geometric objects intersect nicely.
Immersions of Manifolds: Immersions of manifolds refer to smooth maps between differentiable manifolds that are locally injective and have a derivative that is injective at every point. This means that the manifold can be 'smeared' or 'twisted' into a higher-dimensional space while retaining its local structure, which is crucial for understanding how manifolds can be embedded within other geometric contexts. This concept lays the groundwork for deeper insights into manifold theory and relates closely to the Whitney Embedding Theorem, which addresses the conditions under which a manifold can be smoothly embedded in Euclidean space.
John Milnor: John Milnor is a prominent American mathematician known for his significant contributions to differential topology, particularly in the areas of manifold theory, Morse theory, and the topology of high-dimensional spaces. His work has fundamentally shaped the field and has broad implications for various topics within topology, including submersions, critical values, and cohomology groups.
Local flatness: Local flatness refers to a property of a manifold or submanifold where, around each point, there exists a neighborhood that resembles Euclidean space in terms of local geometry. This concept is crucial in understanding how manifolds can be embedded in higher-dimensional spaces, particularly in the context of the Whitney Embedding Theorem, which asserts that any smooth manifold can be embedded in a Euclidean space of sufficiently high dimension.
Riemannian Geometry: Riemannian geometry is a branch of differential geometry that studies smooth manifolds equipped with a Riemannian metric, which defines how distances and angles are measured on the manifold. This field plays a crucial role in understanding the geometric properties of curved spaces and has deep connections to physics, particularly in the theory of general relativity.
Smooth manifold: A smooth manifold is a topological space that is locally similar to Euclidean space and has a globally defined differential structure, allowing for the smooth transition of functions. This concept is essential in many areas of mathematics and physics, as it provides a framework for analyzing shapes, curves, and surfaces with differentiable structures.
Topological embedding: A topological embedding is a function between two topological spaces that preserves the structure of the spaces and allows for one space to be represented within another without any distortion. This means that the embedded space retains its properties, like open sets, and behaves like a subspace of the larger space, which is crucial for understanding the relationships between different geometric shapes and manifolds.
Transversality: Transversality is a concept in differential topology that describes the condition where two submanifolds intersect in a way that is 'nice' or 'generic', meaning they meet at a finite number of points and the tangent spaces at those points span the ambient space. This idea is essential for understanding the behavior of functions and their critical values, as well as the relationships between different geometric objects.
Whitney Embedding Theorem: The Whitney Embedding Theorem states that any smooth manifold can be embedded as a smooth submanifold of Euclidean space. This result is fundamental in differential topology because it shows how abstract manifolds can be realized in a more familiar geometric setting, allowing for the application of Euclidean tools to study their properties.
Whitney's Immersion Theorem: Whitney's Immersion Theorem states that any smooth manifold can be immersed in a Euclidean space of sufficiently high dimension. This means that for a manifold of dimension n, there exists an embedding into a space of dimension 2n, allowing for the representation of the manifold without self-intersections. The theorem is significant because it provides a way to understand the behavior of smooth manifolds in higher dimensions and has profound implications in topology and geometry.