Gysin Homomorphism

Definition

The Gysin homomorphism is a fundamental concept in algebraic topology and K-theory that provides a way to relate the cohomology of a space to the cohomology of its submanifolds, particularly when dealing with fiber bundles and push-forward maps. This homomorphism captures how the inclusion of a submanifold affects the overall topological structure of the manifold, allowing mathematicians to translate geometric information into algebraic data.

5 Must Know Facts For Your Next Test

1. The Gysin homomorphism arises in the context of inclusions of smooth submanifolds and is crucial for calculating characteristic classes.
2. It can be viewed as a homomorphism from the cohomology of the base space to the cohomology of the total space in a fibration.
3. The Gysin homomorphism is often denoted by 'g' and has properties that make it compatible with operations in K-theory.
4. This homomorphism is instrumental in deriving results like the Thom isomorphism, which connects the homology of a manifold with its cohomology.
5. In practical applications, the Gysin homomorphism can be used to compute the push-forward of classes under proper maps, linking local data to global properties.

Review Questions

• How does the Gysin homomorphism facilitate the understanding of relationships between different topological spaces?
  • The Gysin homomorphism allows mathematicians to bridge the gap between the cohomologies of a space and its submanifolds by translating inclusion data into algebraic information. By establishing a connection between these two spaces, it helps in identifying how geometric properties influence algebraic structures. This interaction is especially useful in areas like fiber bundles and studying characteristic classes, where understanding the behavior of submanifolds within a larger context is essential.
• Discuss how the Gysin homomorphism interacts with push-forward maps in K-theory.
  • In K-theory, the Gysin homomorphism plays a critical role when dealing with push-forward maps, as it allows for the transfer of classes from a submanifold to its ambient manifold. This interaction can simplify complex computations by enabling a consistent method for pushing forward cohomological information. Specifically, it preserves algebraic structures while translating geometric insights, making it easier to analyze global properties using local data provided by the submanifold.
• Evaluate the significance of the Gysin homomorphism in relation to characteristic classes and its implications for manifold theory.
  • The Gysin homomorphism is significant in manifold theory because it forms an essential component in the computation of characteristic classes, which are vital invariants in differential topology. It establishes a framework for relating local geometric data to global topological features through its ability to connect different cohomological structures. This relationship not only enhances our understanding of manifold topology but also has far-reaching implications in various fields, such as algebraic geometry and theoretical physics, where these invariants play crucial roles.

Related terms

Push-Forward Map: A map that takes cohomology classes from one space and translates them into another space, often associated with a continuous function between manifolds.

Cohomology: A mathematical tool used to study topological spaces through algebraic structures called cochains, which allow for the computation of various invariants.

Pullback: An operation that allows one to transfer cohomology classes from one space back along a continuous map, providing a means to connect different topological spaces.