Relative cup product

Definition

The relative cup product is an operation in cohomology that combines classes from the relative cohomology groups of a pair of spaces, such as a topological space and a subspace. This product allows for the construction of new cohomology classes and plays a critical role in understanding the relationships between the topology of a space and its subspaces, particularly when examining long exact sequences and exact pairs.

5 Must Know Facts For Your Next Test

1. The relative cup product is defined for elements in the relative cohomology groups, denoted as $H^*(X, A)$, where $X$ is a space and $A$ is a subspace.
2. This operation is bilinear, meaning it satisfies linearity in each argument separately, allowing combinations of classes from different degrees.
3. The relative cup product contributes to the structure of the long exact sequence in cohomology, which connects various cohomology groups of spaces and their subspaces.
4. In practice, computing the relative cup product often involves specific techniques like using representatives or triangulations of the spaces involved.
5. The relative cup product can be used to derive important topological invariants and to establish relationships between different cohomology theories.

Review Questions

• How does the relative cup product relate to the long exact sequence in cohomology?
  • The relative cup product is crucial for understanding the long exact sequence in cohomology because it connects the cohomology groups of a space and its subspace. When considering a pair $(X,A)$, the relative cohomology group $H^*(X,A)$ provides insight into how cohomology classes interact. The long exact sequence links these groups together, showing how elements in $H^*(X)$ relate to those in $H^*(A)$ through the application of the relative cup product.
• Discuss the significance of bilinearity in the context of the relative cup product and its applications.
  • Bilinearity is essential for the relative cup product because it allows us to combine multiple classes from relative cohomology groups in a coherent manner. This property means that if you have two classes from $H^*(X,A)$ and you perform the relative cup product, you can manipulate these classes independently. This feature is particularly useful when calculating products across different degrees or when examining how complex structures within topological spaces can interact through their subspaces.
• Evaluate how the relative cup product enhances our understanding of topological invariants within algebraic topology.
  • The relative cup product enhances our understanding of topological invariants by providing a way to explore relationships between different spaces through their subspaces. By applying this operation, we can derive new invariants and gain deeper insights into the structure of spaces. For instance, when computing products that yield non-trivial elements in cohomology, we can uncover essential characteristics about how spaces are formed or deformed. This analysis contributes significantly to broader discussions within algebraic topology about continuity, connectivity, and dimension.

Related terms

Cohomology: A mathematical tool for studying topological spaces that associates algebraic invariants, called cohomology groups, to a space, capturing its topological properties.

Exact sequence: A sequence of algebraic objects (like groups or modules) and morphisms between them where the image of one morphism equals the kernel of the next, revealing relationships between different structures.

Cup product: An operation on cohomology classes that takes two classes from a given cohomology group and produces a new class in a higher degree, providing insight into the topology of the space.