Whitney's Theorem

Definition

Whitney's Theorem states that for a smooth manifold, the set of immersions into Euclidean space can be characterized by certain properties related to the dimensions of the manifold and the ambient space. It highlights the relationship between immersions, transversality, and the critical values of smooth functions, establishing key connections in differential topology that are essential for understanding the behavior of mappings between manifolds.

5 Must Know Facts For Your Next Test

1. Whitney's Theorem establishes that the set of immersions is dense in the space of smooth mappings from a manifold into Euclidean space.
2. This theorem implies that any smooth function can be approximated by immersions when considering certain dimensional constraints.
3. The critical values of immersions are closely related to the concept of transversality, as immersions that are transverse to a submanifold have significant implications for their topology.
4. The theorem provides a framework for understanding how changes in dimensionality affect immersion properties and related structures.
5. In applications, Whitney's Theorem helps to analyze and construct examples of smooth manifolds that exhibit desired immersion characteristics.

Review Questions

• How does Whitney's Theorem relate to the concept of immersions and what implications does this have for smooth mappings?
  • Whitney's Theorem highlights that immersions are dense within the space of smooth mappings between manifolds. This means that given any smooth function, one can find an immersion arbitrarily close to it in terms of topology. This relationship allows us to better understand how different mappings behave and how they can approximate more complex structures.
• Discuss the role of critical values in relation to Whitney's Theorem and transversality. Why are these concepts important?
  • Critical values play a crucial role in Whitney's Theorem as they indicate points where smooth functions may behave irregularly or lose surjectivity. In relation to transversality, immersions that are transverse to submanifolds avoid these critical values, ensuring smoother intersections and richer geometric structures. Understanding these relationships is vital for analyzing manifold behaviors under various mappings.
• Evaluate how Whitney's Theorem can influence the study of higher-dimensional manifolds and their immersions into Euclidean spaces.
  • Whitney's Theorem provides essential insights into how higher-dimensional manifolds can be represented within Euclidean spaces through immersions. By establishing conditions under which immersions are possible or approximate existing functions, it lays groundwork for exploring complex manifold structures and their applications. This evaluation enables mathematicians to draw connections between abstract topological properties and practical modeling in higher dimensions.

Related terms

Immersion: An immersion is a smooth function between manifolds where the differential is injective at every point, meaning it behaves like an embedding locally but may not be globally one-to-one.

Critical Value: A critical value of a smooth function is a point in the target space where the derivative fails to be surjective, indicating potential issues with regularity or structure in the mapping.

Transversality: Transversality is a condition that describes how two submanifolds intersect in a manifold; specifically, it ensures that their tangent spaces at intersection points span the ambient space.