Gysin Homomorphism

Definition

The Gysin homomorphism is a fundamental concept in algebraic topology that arises from the intersection theory of cohomology classes. It provides a way to relate the cohomology groups of a manifold and its submanifolds, allowing for computations involving cap products and pushforward operations. This homomorphism helps to analyze the behavior of cohomological operations when applied to fiber bundles and other geometric structures.

5 Must Know Facts For Your Next Test

1. The Gysin homomorphism is denoted as $$g: H^*(M) \to H^{*+2}(N)$$, where M is a manifold and N is its submanifold, typically a fiber over a point.
2. It arises naturally from the intersection of cohomology classes in the context of fiber bundles, particularly in situations involving smooth maps.
3. The Gysin homomorphism can be used to compute the Euler class of a vector bundle, highlighting its connection to characteristic classes.
4. In practical applications, it often relates to the notion of duality in cohomology, reflecting the geometric relationships between spaces and their subspaces.
5. Understanding the Gysin homomorphism is crucial for computations involving cap products and can simplify complex algebraic topology problems.

Review Questions

• How does the Gysin homomorphism connect with the concept of cap products in cohomology?
  • The Gysin homomorphism directly relates to cap products as it involves combining cohomology classes from a manifold and its submanifold. Specifically, when you apply the Gysin homomorphism to a cohomology class, it yields a new class that can be interpreted through cap products with appropriate homology classes. This connection highlights how intersection theory helps derive information about the topology of spaces through algebraic operations.
• Discuss how the Gysin homomorphism can be utilized to compute characteristic classes such as the Euler class of vector bundles.
  • The Gysin homomorphism provides a systematic way to compute characteristic classes by relating them to intersections of cycles in manifolds. For example, when analyzing vector bundles, applying the Gysin homomorphism allows you to obtain the Euler class by considering the relationship between the bundle's total space and its base space. By leveraging this relationship, one can effectively compute important topological invariants that capture essential geometric properties of the bundle.
• Evaluate the implications of the Gysin homomorphism on understanding duality in cohomology and its role in broader algebraic topology.
  • The Gysin homomorphism has significant implications for understanding duality in cohomology as it encapsulates how various topological spaces relate through their submanifolds. By evaluating how classes transform under this homomorphism, one can gain insights into the interplay between different dimensions of spaces and their topological features. This understanding contributes to broader algebraic topology by illuminating how complex relationships between spaces can be managed algebraically, revealing deeper symmetries and properties that may not be immediately apparent.

Related terms

Cap Product: The cap product is an operation in cohomology that combines a cohomology class with a homology class, yielding a new cohomology class. It plays a key role in defining the Gysin homomorphism.

Pushforward Map: A pushforward map is a homomorphism between the cohomology groups of two spaces induced by a continuous map, crucial for understanding how classes behave under mappings.

Cohomology Ring: The cohomology ring is an algebraic structure formed by the cohomology classes of a space, equipped with an intersection product that allows for operations like the cap product.