🔁Elementary Differential Topology Unit 7 – Partitions of Unity & Bump Functions

Key Concepts

• Partitions of unity are a fundamental tool in differential topology used to construct smooth functions with desired local properties
• Consist of a collection of smooth functions that sum to 1 on a given domain and have compact support
• Enable the extension of local properties to global ones by "gluing together" local information
• Bump functions are special cases of partitions of unity with compact support and values in the interval [0, 1]
  • Useful for constructing smooth functions with specific behavior in a localized region
• Partitions of unity and bump functions play a crucial role in proving important theorems and constructing objects in differential topology (smooth approximation theorem, existence of Riemannian metrics)
• Closely related to the concept of a manifold, as they are often used to define and work with smooth structures on manifolds
• Understanding the properties and construction of partitions of unity is essential for solving problems in differential topology

Definition and Properties

• A partition of unity subordinate to an open cover $\{U_i\}_{i \in I}$ of a topological space $X$ is a collection of continuous functions $\{\varphi_i: X \to [0, 1]\}_{i \in I}$ satisfying:
  1. $\text{supp}(\varphi_i) \subseteq U_i$ for each $i \in I$
  2. The family $\{\text{supp}(\varphi_i)\}_{i \in I}$ is locally finite
  3. $\sum_{i \in I} \varphi_i(x) = 1$ for all $x \in X$
• If $X$ is a smooth manifold and each $\varphi_i$ is smooth, the partition of unity is called a smooth partition of unity
• Partitions of unity are always subordinate to an open cover, meaning each function has support contained within a corresponding open set
• The local finiteness property ensures that at any point, only finitely many functions are non-zero
• The sum of all functions in a partition of unity equals 1 at every point in the domain

Construction of Partitions of Unity

• The construction of partitions of unity relies on the existence of bump functions and the paracompactness of the space
• Given a locally finite open cover $\{U_i\}_{i \in I}$ of a paracompact space $X$, the construction proceeds as follows:
  1. For each $i \in I$, choose a bump function $\psi_i: X \to [0, 1]$ with $\text{supp}(\psi_i) \subseteq U_i$
  2. Define $\varphi_i(x) = \frac{\psi_i(x)}{\sum_{j \in I} \psi_j(x)}$ for each $i \in I$ and $x \in X$
• The resulting family $\{\varphi_i\}_{i \in I}$ is a partition of unity subordinate to the open cover $\{U_i\}_{i \in I}$
• If $X$ is a smooth manifold and the bump functions $\psi_i$ are smooth, the constructed partition of unity is also smooth
• The paracompactness of the space ensures that every open cover has a locally finite refinement, which is necessary for the construction
• The normalization step (dividing by the sum of bump functions) ensures that the sum of the partition of unity functions equals 1 at every point

Bump Functions: Basics and Examples

• A bump function is a smooth function $\psi: \mathbb{R}^n \to [0, 1]$ with compact support
• Bump functions are used to construct partitions of unity and to localize properties in differential topology
• A basic example of a bump function is the standard bump function $\psi(x) = \begin{cases} e^{-\frac{1}{1-\|x\|^2}} & \text{if } \|x\| < 1 \\ 0 & \text{if } \|x\| \geq 1 \end{cases}$
  • This function is smooth, has compact support within the unit ball, and attains a maximum value of 1 at the origin
• Bump functions can be constructed using various methods, such as mollification or the composition of smooth functions
• The support of a bump function can be adjusted by scaling and translating the function (e.g., $\psi(\frac{x-a}{r})$ has support within a ball of radius $r$ centered at $a$)
• Bump functions are essential for proving the existence of smooth functions with prescribed local behavior

Applications in Differential Topology

• Partitions of unity and bump functions have numerous applications in differential topology, including:
  1. Proving the existence of Riemannian metrics on smooth manifolds
  2. Constructing smooth functions with desired properties (e.g., Morse functions, Lyapunov functions)
  3. Defining the smooth structure on a topological manifold
  4. Proving the smooth approximation theorem (approximating continuous functions by smooth ones)
• In the construction of Riemannian metrics, partitions of unity are used to glue together local metrics into a globally defined metric
• Bump functions can be used to construct smooth functions with specific behavior in a localized region, which is useful for studying the local properties of manifolds
• Partitions of unity are employed in the definition of the smooth structure on a manifold, ensuring that the transition maps between charts are smooth
• The smooth approximation theorem relies on partitions of unity to approximate continuous functions by smooth ones, which is a key result in differential topology

Relationship to Manifolds

• Partitions of unity and bump functions are closely related to the concept of a manifold in differential topology
• A topological manifold is a space that locally resembles Euclidean space, and a smooth manifold is a topological manifold with a smooth structure
• Partitions of unity are used to define the smooth structure on a topological manifold by ensuring that the transition maps between charts are smooth
• Bump functions are employed to construct smooth functions on manifolds with desired local properties
• The existence of partitions of unity on a manifold is equivalent to the manifold being paracompact, which is a crucial property in differential topology
• Many important constructions in differential topology, such as vector bundles and differential forms, rely on the existence of partitions of unity on the underlying manifold
• Understanding the relationship between partitions of unity, bump functions, and manifolds is essential for working with smooth structures and solving problems in differential topology

Problem-Solving Techniques

• When solving problems involving partitions of unity and bump functions, it is essential to:
  1. Identify the relevant open cover of the space or manifold
  2. Construct suitable bump functions subordinate to the open cover
  3. Use the bump functions to create a partition of unity
  4. Apply the partition of unity to localize the problem or glue together local solutions
• In problems involving the construction of smooth functions with specific properties, bump functions can be used to define the desired behavior in a localized region
• Partitions of unity are useful for extending local properties to global ones, such as constructing a global Riemannian metric from local ones
• When working with manifolds, it is crucial to consider the relationship between partitions of unity, bump functions, and the smooth structure of the manifold
• Exploiting the properties of partitions of unity, such as local finiteness and the sum being equal to 1, can simplify problem-solving and lead to more elegant solutions

Advanced Topics and Extensions

• There are several advanced topics and extensions related to partitions of unity and bump functions in differential topology, including:
  1. Partitions of unity on infinite-dimensional manifolds (e.g., Banach manifolds, Fréchet manifolds)
  2. Equivariant partitions of unity for spaces with group actions
  3. Partitions of unity in the context of sheaf theory and cohomology
  4. Applications of partitions of unity in geometric analysis and partial differential equations
• Infinite-dimensional manifolds require a more careful treatment of partitions of unity due to the lack of certain properties (e.g., paracompactness) that hold in the finite-dimensional case
• Equivariant partitions of unity are useful when working with spaces equipped with group actions, as they respect the symmetries of the space
• Sheaf theory provides a more abstract framework for studying partitions of unity and their relationship to cohomology and other algebraic invariants
• Partitions of unity find applications in geometric analysis and PDEs, such as in the study of elliptic operators and the construction of Green's functions
• Exploring these advanced topics and extensions can deepen one's understanding of partitions of unity and their role in differential topology and related fields