Proper Push-Forward

Definition

Proper push-forward is a mathematical operation that generalizes the concept of pushing forward classes from a space to another through a continuous map, specifically in the context of proper maps. This operation is significant in the study of Gysin homomorphisms and plays a vital role in the intersection theory and cohomology of manifolds, linking various spaces while preserving their topological features.

5 Must Know Facts For Your Next Test

1. The proper push-forward is defined for proper maps, which means that the preimage of every compact set under the map is compact.
2. When applying the proper push-forward to a cohomology class, it respects the orientation and dimension of the spaces involved.
3. In many cases, proper push-forward operations are used to relate the topological features of different spaces through their mappings.
4. The proper push-forward plays a crucial role in deriving Gysin sequences, which connect the cohomology of a manifold with its submanifolds.
5. This operation helps in understanding how classes transform under continuous maps, providing insights into invariants associated with topological spaces.

Review Questions

• How does a proper push-forward relate to Gysin homomorphisms in the context of topology?
  • A proper push-forward is essential for understanding Gysin homomorphisms as it allows for the transfer of cohomology classes between manifolds through proper maps. Gysin homomorphisms utilize this push-forward to link the cohomology of a manifold to that of its submanifold. The proper nature of these maps ensures that compactness is preserved, which is crucial when deriving relationships between different topological structures.
• Discuss how the concept of proper push-forward influences the understanding of intersection theory.
  • Proper push-forward significantly enhances our grasp of intersection theory by providing a way to translate cohomological information from one manifold to another. When analyzing intersections of submanifolds, proper push-forwards facilitate calculations by allowing us to work with reduced dimensionalities while maintaining essential topological properties. This connection aids in computing intersection numbers and understanding how various manifolds intersect through their images under continuous mappings.
• Evaluate the implications of using improper maps instead of proper maps when dealing with push-forwards in algebraic topology.
  • Using improper maps instead of proper maps in push-forwards can lead to complications such as failure to maintain compactness and issues with continuity in preimages. This could result in loss of information or invalid transformation of cohomology classes, making it difficult to apply results from algebraic topology correctly. Therefore, maintaining the property of being proper ensures that critical relationships and invariants are preserved, allowing for reliable interpretations in applications such as Gysin homomorphisms and intersection theory.

Related terms

Gysin Homomorphism: A specific type of homomorphism that arises in the context of push-forward maps, connecting the cohomology of a manifold to that of its submanifold.

Cohomology: A mathematical tool used to study topological spaces through algebraic invariants, often involving the construction of cochain complexes and their corresponding cohomology groups.

Push-Forward Map: A general operation that sends cohomology classes from one space to another via a continuous map, which can be proper or improper depending on the properties of the map.