Oriented cohomology

Definition

Oriented cohomology is a refined version of cohomology theory that takes into account the orientation of manifolds and other topological spaces. It allows for the definition of cohomology classes that are sensitive to the orientation, meaning it can distinguish between different orientations of the same underlying space. This is particularly important when studying phenomena like Gysin homomorphisms and push-forward maps, as these concepts rely on how one can relate the cohomology of a manifold to that of its submanifolds or images under continuous maps.

5 Must Know Facts For Your Next Test

1. Oriented cohomology is essential for defining and understanding intersection theory, where the orientation of submanifolds plays a key role in determining intersection numbers.
2. The Gysin homomorphism arises from oriented cohomology and allows one to relate the cohomology of a manifold to its submanifolds, incorporating information about their orientations.
3. Push-forward maps in oriented cohomology can change depending on whether the map between spaces preserves or reverses orientation.
4. Oriented cohomology can often be computed using tools such as characteristic classes, which help connect geometric and topological properties of manifolds.
5. The concept of orientation is crucial in applications such as cobordism theory, where it helps classify manifolds based on their ability to bound other manifolds.

Review Questions

• How does oriented cohomology enhance our understanding of intersection theory?
  • Oriented cohomology plays a crucial role in intersection theory by allowing for the definition of intersection numbers that depend on the orientations of submanifolds. When two oriented manifolds intersect, the orientation dictates how we count their intersection points, leading to meaningful algebraic results. This orientation-sensitive approach provides a deeper understanding of how manifolds interact and intersect in various dimensions.
• What is the significance of the Gysin homomorphism in relation to oriented cohomology?
  • The Gysin homomorphism is significant because it provides a systematic way to relate the cohomology groups of a manifold with those of its submanifolds, taking into account their orientations. This means that when you have a submanifold sitting inside a larger manifold, you can 'push forward' cohomological information from the submanifold to the ambient manifold. The Gysin homomorphism effectively encodes how these orientations affect the overall topology and geometry between spaces.
• In what ways does oriented cohomology impact the classification of manifolds in cobordism theory?
  • In cobordism theory, oriented cohomology is pivotal as it allows mathematicians to classify manifolds based on whether they can bound other manifolds while preserving orientation. This classification hinges on the idea that two manifolds can be considered equivalent if they share a common boundary that respects their orientations. By analyzing how oriented cohomology groups behave under such conditions, researchers gain insights into the deeper relationships between different types of manifolds and their topological properties.

Related terms

Chern classes:

Topological invariants associated with vector bundles that provide information about their curvature and allow for the calculation of oriented cohomology.

Gysin homomorphism: A map in cohomology that describes how to push forward cohomology classes from a submanifold to the ambient manifold, incorporating orientation information.

Push-forward map:

A way of transferring cohomology classes from one space to another, usually involving a continuous map between the two spaces and respecting their orientations.