Cap product on manifolds

Definition

The cap product on manifolds is a fundamental operation in algebraic topology that combines a cohomology class with a homology class, resulting in a new cohomology class. This operation provides a way to 'cap' a homology class with a cohomology class, allowing for a deeper understanding of the relationships between these classes in the context of manifold theory. The cap product plays a crucial role in various topological invariants and intersection theory.

5 Must Know Facts For Your Next Test

1. The cap product is denoted as `∩`, and it takes a cohomology class from `H^k(M)` and a homology class from `H_n(M)` to yield a cohomology class in `H^{k-n}(M)`.
2. It is important in defining Poincaré duality, which establishes an isomorphism between homology and cohomology classes in compact oriented manifolds.
3. The cap product respects the cup product structure on cohomology, meaning it can be integrated with other operations in cohomological algebra.
4. In the case of a manifold, the cap product can provide geometric interpretations, such as finding intersections of cycles and co-cycles.
5. The operation is associative but not necessarily commutative; the order of the classes involved can affect the outcome.

Review Questions

• How does the cap product relate cohomology and homology classes on manifolds?
  • The cap product provides a way to link cohomology and homology classes by taking a cohomology class from `H^k(M)` and capping it with a homology class from `H_n(M)`. This operation produces a new cohomology class in `H^{k-n}(M)`, showcasing how these two types of algebraic structures interact. This connection helps reveal more about the topology of the manifold and its properties through algebraic means.
• Discuss how the cap product contributes to Poincaré duality in the context of manifolds.
  • The cap product is essential for demonstrating Poincaré duality, which asserts an isomorphism between homology and cohomology groups of a compact oriented manifold. By using the cap product, one can show that every homology class has a corresponding cohomology class that represents its dual. This relationship strengthens our understanding of the manifold's topology and highlights the symmetry between these two forms of topological invariants.
• Evaluate the implications of associativity in cap products and how this affects computations in algebraic topology.
  • The associativity of the cap product allows mathematicians to group operations freely without worrying about the order in which they are performed. This property simplifies many calculations involving multiple homology and cohomology classes, making it easier to derive important results and perform more complex analyses. However, since cap products are not generally commutative, careful attention must be paid to the order of classes when interpreting results, which can lead to different insights about intersections and geometric properties within manifolds.

Related terms

Cohomology:

A mathematical tool that assigns algebraic objects, such as groups or rings, to a topological space to study its properties and structures.

Homology:

An algebraic invariant that provides information about the shape or structure of a topological space by associating sequences of abelian groups to the space.

Intersection Theory:

A branch of algebraic geometry that studies the intersection of subvarieties and provides tools to compute intersection numbers and multiplicities.