Cohomology classes

Definition

Cohomology classes are equivalence classes of cochains, which provide algebraic invariants that help describe the topological structure of a space. They capture information about the global properties of spaces through the dual nature of cohomology, linking it closely with concepts like homology and the Gysin homomorphism. Understanding these classes is essential for applying push-forward maps, as they allow for the transfer of cohomological information across different spaces.

5 Must Know Facts For Your Next Test

1. Cohomology classes are formed by taking the kernel of coboundary maps in a cochain complex, leading to a well-defined algebraic structure.
2. These classes are important for defining invariants of topological spaces, such as Betti numbers and other features that help classify spaces.
3. Cohomology classes can be manipulated through operations like cup products, which allows for rich algebraic structures to be formed.
4. The Gysin homomorphism provides a way to relate the cohomology of a fiber bundle with the base space, showing how cohomology classes can change under different mappings.
5. Push-forward maps allow for the transfer of cohomology classes between spaces, enabling deeper analysis of topological properties across different dimensions.

Review Questions

• How do cohomology classes relate to the structure of a cochain complex?
  • Cohomology classes arise from cochain complexes, where they represent equivalence classes of cochains modulo coboundaries. This means that two cochains are considered equivalent if their difference is a coboundary, which helps capture essential features of the topological space. The structure of the cochain complex facilitates the calculation of these classes, providing tools to derive important invariants of the space.
• Discuss how the Gysin homomorphism connects cohomology classes to push-forward maps and their applications.
  • The Gysin homomorphism plays a critical role in relating the cohomology classes of a manifold and its submanifolds. It allows us to compute push-forward maps which transfer cohomological information from one space to another. This relationship is vital for understanding how certain topological properties are preserved or altered when we move between different dimensions or spaces within fiber bundles.
• Evaluate the significance of cup products in the manipulation and application of cohomology classes within topological spaces.
  • Cup products provide an essential operation for combining cohomology classes, enabling us to derive new classes that encapsulate more intricate topological features. By evaluating how these products behave under various conditions, we can glean insights into the structure of the underlying space. This operation not only enriches our understanding of individual cohomology classes but also helps us discover deeper connections between various topological properties and invariants.

Related terms

Cochain Complex: A sequence of abelian groups or modules linked by homomorphisms that are used to define cohomology.

Homology: A mathematical concept that studies topological spaces through sequences of abelian groups or modules related to chains.

Push-Forward Map: A map that transfers cohomology classes from one space to another, particularly useful in the context of fiber bundles and Gysin homomorphisms.