Regular Value

Definition

A regular value is a point in the target space of a smooth map such that the preimage of that point consists only of points where the differential of the map is surjective. This concept is important for understanding how smooth maps behave and has applications in various areas, including the implicit function theorem, submersions, and determining properties like the degree of a map.

5 Must Know Facts For Your Next Test

1. For a point to be a regular value, it must not be an output of the map for which the differential fails to be surjective.
2. The preimage of a regular value under a smooth map is a submanifold, which allows for various topological and geometrical analyses.
3. In the context of the implicit function theorem, regular values help identify conditions under which certain equations can be solved locally.
4. The concept of regular values is crucial for defining the degree of a map, which measures how many times the map covers its target space.
5. Regular values relate closely to critical points; they help distinguish between generic behaviors and more complex phenomena in differential topology.

Review Questions

• How does the concept of regular values connect to the implicit function theorem and its applications?
  • The implicit function theorem states that if a point is a regular value of a smooth map, then around every point in its preimage, there exists a neighborhood where the level set can be described as a smooth manifold. This means that regular values help establish conditions for when we can solve equations locally and understand how variables relate through smooth maps. Therefore, recognizing regular values allows us to utilize the implicit function theorem effectively.
• Discuss how submersions are related to regular values and their significance in differential topology.
  • Submersions are specific types of smooth maps where the differential is surjective everywhere in their domain. Because every regular value corresponds to points where the submersion behaves nicely, this relationship means that if a map is a submersion, then all points in its image can be considered regular values. This connection helps us explore how submersions create lower-dimensional structures and plays an essential role in understanding manifolds.
• Evaluate the implications of identifying regular values on the degree of a map and its broader topological consequences.
  • Identifying regular values in a smooth map is fundamental for calculating its degree, which quantifies how many times the domain covers its range. When regular values are present, they ensure that the preimage consists solely of smooth manifolds, leading to consistent results when determining the degree. Understanding these relationships contributes significantly to algebraic topology and can impact various areas like homology and cohomology theories.

Related terms

Smooth Map: A function between differentiable manifolds that has continuous derivatives of all orders.

Submersion:

A smooth map whose differential is surjective at every point in its domain, meaning it locally resembles a projection onto a lower-dimensional space.

Level Set:

The set of points in the domain of a function where the function takes on a specific value.