Smooth map

Definition

A smooth map is a function between differentiable manifolds that has continuous derivatives of all orders. This concept is essential in understanding the behavior of maps in differential topology, as smooth maps preserve the structure of manifolds and allow for analysis using calculus tools. The properties of smooth maps play a critical role in determining the characteristics of immersions, analyzing the differential of a map, exploring transversality, and defining degrees of mappings.

5 Must Know Facts For Your Next Test

1. Smooth maps are defined by the property that all their derivatives exist and are continuous, which means they can be differentiated any number of times without issues.
2. In the context of immersions, a smooth map can be classified as an immersion if its differential is injective at every point, preserving local structure.
3. The differential of a smooth map is a linear transformation that captures how the map behaves infinitesimally, allowing for the study of its properties.
4. Transversality involves examining how two smooth maps intersect or relate to each other in a way that is also governed by the properties of smoothness.
5. The degree of a smooth map can be computed using tools like homology or cohomology theories, highlighting its importance in algebraic topology.

Review Questions

• How does the concept of smooth maps relate to immersions, and what properties must hold for a smooth map to be considered an immersion?
  • Smooth maps are foundational for understanding immersions because an immersion requires the map's differential to be injective at every point. This means that not only must the map be smooth, but it must also reflect local structure without collapsing dimensions. The ability to maintain distinct tangent spaces ensures that the manifold's essence is preserved, allowing for a proper embedding into another space.
• Explain how the differential of a smooth map contributes to understanding its behavior in differential topology.
  • The differential of a smooth map captures how the map changes at each point by providing a linear approximation. It allows mathematicians to analyze local properties such as whether the map is an immersion or a submersion. By studying these differentials, one can gain insights into critical points, singularities, and overall mapping behaviors between manifolds.
• Evaluate the importance of smooth maps in defining the degree of a mapping and how this concept interacts with transversality.
  • Smooth maps are crucial in defining the degree of a mapping because they ensure that the structure and continuity needed for counting intersections are preserved. When examining transversality, which deals with how two smooth maps intersect, the smoothness ensures that these intersections can be counted accurately and meaningfully. The degree itself then becomes a topological invariant, helping to classify mappings based on their behavior relative to their targets.

Related terms

Differentiable manifold: A differentiable manifold is a topological space that locally resembles Euclidean space and allows for the definition of smooth functions and differentiation.

Immersion:

An immersion is a smooth map that reflects the local structure of its domain into its codomain in a way that is injective at every point.

Degree of a map: The degree of a map is an integer that represents the number of times a manifold wraps around another manifold, reflecting how the map behaves topologically.