Projection Formula

Definition

The projection formula is a crucial concept in K-Theory that relates the Gysin homomorphism to push-forward maps when dealing with a proper map between spaces. It allows one to compute the push-forward of a cohomology class via the Gysin map, bridging the gap between cohomology classes on a manifold and its submanifolds. This formula is significant because it helps in understanding how topological properties can be transferred through mappings.

5 Must Know Facts For Your Next Test

1. The projection formula expresses how the Gysin homomorphism can be used to compute the push-forward of a cohomology class from a submanifold to the ambient manifold.
2. This formula is particularly useful in situations involving smooth maps between varieties or manifolds, where it facilitates integration along fibers.
3. It illustrates how intersection theory is connected with K-Theory, allowing for computations involving characteristic classes.
4. The projection formula can be applied to compute the push-forward of classes under various geometric contexts, including toric varieties and more complex algebraic varieties.
5. It plays a critical role in understanding how topological invariants behave under morphisms, thereby linking different spaces through their cohomological features.

Review Questions

• How does the projection formula connect the Gysin homomorphism with push-forward maps in K-Theory?
  • The projection formula establishes a relationship between the Gysin homomorphism and push-forward maps by showing that one can compute the push-forward of a cohomology class using the Gysin map. Essentially, it provides a way to translate information from a submanifold into the larger manifold via proper maps, making it easier to analyze and understand how topological properties interact across different spaces.
• Discuss the implications of the projection formula on intersection theory and characteristic classes within K-Theory.
  • The projection formula has profound implications for intersection theory as it allows mathematicians to relate cohomological classes through intersection products. It shows how characteristic classes can be computed using the Gysin homomorphism and highlights how these classes behave under proper maps. This connection enhances our understanding of how topology and geometry influence one another, particularly when examining spaces with complex structures.
• Evaluate how the projection formula can be applied in different geometric contexts, such as toric varieties, and what this reveals about their topology.
  • The application of the projection formula in contexts like toric varieties showcases its versatility in connecting geometric objects through their cohomological properties. By leveraging this formula, one can compute push-forwards and understand how classes change as they move through different dimensional spaces. This not only reveals information about individual varieties but also contributes to broader insights about how various geometries interrelate, ultimately enhancing our comprehension of topological features in algebraic geometry.

Related terms

Gysin Homomorphism: A homomorphism that arises in K-Theory and relates the cohomology of a space to the cohomology of its subspaces, providing a way to push forward classes.

Push-Forward Map: A map that allows one to take a class in the cohomology of one space and 'push' it forward to another space under a continuous map, often used in the context of integrating over fibers.

Cohomology Class: An equivalence class of cochains in cohomology theory, which serves as a fundamental tool for distinguishing topological spaces based on their properties.