Smooth manifolds

Definition

Smooth manifolds are mathematical spaces that locally resemble Euclidean space and are equipped with a smooth structure, allowing for the differentiation of functions. They provide a setting where concepts from calculus can be applied in a more generalized context, enabling the study of geometric and topological properties. This concept is crucial for understanding how Gysin homomorphisms and push-forward maps operate in algebraic topology, particularly in the context of smooth mappings between manifolds.

5 Must Know Facts For Your Next Test

1. Smooth manifolds can be described using charts and atlases that provide local coordinates, allowing for the definition of smooth functions.
2. The dimension of a smooth manifold is determined by the number of independent directions available at any point in the manifold.
3. Gysin homomorphisms arise when considering smooth maps between manifolds and provide an important tool for computing push-forwards in cohomology.
4. Push-forward maps allow one to translate differential forms and cohomological information from one manifold to another under smooth mappings.
5. The study of smooth manifolds is fundamental in various fields, including physics, particularly in the theory of general relativity and differential geometry.

Review Questions

• How do smooth manifolds facilitate the understanding of concepts like Gysin homomorphisms and push-forward maps?
  • Smooth manifolds create a structured environment where local Euclidean properties can be analyzed, allowing for the application of calculus. Gysin homomorphisms utilize the smooth structure to relate cohomology groups of different manifolds through smooth maps. Push-forward maps also leverage this structure by enabling the transfer of differential forms and cohomological data from one manifold to another, making computations and transformations manageable within this framework.
• Discuss the role of differentiable maps in the context of smooth manifolds and their importance in algebraic topology.
  • Differentiable maps between smooth manifolds are essential because they maintain the differentiable structure necessary for applying calculus concepts. In algebraic topology, these maps help in defining notions such as homotopy and homology classes, which are pivotal for understanding topological properties. They also play a significant role in Gysin homomorphisms, as these mappings can lead to rich interactions between different cohomological theories and ultimately inform the relationships between various topological spaces.
• Evaluate how the concept of tangent spaces at points on smooth manifolds contributes to the broader implications of geometry and physics.
  • Tangent spaces at points on smooth manifolds provide insights into how curves and surfaces behave locally around those points, serving as a foundation for defining derivatives in a higher-dimensional context. This notion not only enriches differential geometry but also has profound implications in physics, especially in general relativity where spacetime is modeled as a smooth manifold. Understanding tangent spaces allows physicists to apply geometric techniques to analyze motion, forces, and other phenomena in a rigorous way, bridging mathematical theory with physical reality.

Related terms

Tangent Space: The tangent space at a point on a manifold consists of all possible directions in which one can tangentially pass through that point, forming a vector space.

Differentiable Maps: Functions between manifolds that preserve the smooth structure, allowing for differentiation and the application of calculus.

Submanifold: A subset of a manifold that is itself a manifold, inheriting its smooth structure from the larger manifold.