Cup product

Definition

The cup product is an operation in cohomology that combines two cohomology classes to produce a new cohomology class, allowing us to create a ring structure from the cohomology groups of a topological space. This operation plays a key role in understanding the algebraic properties of cohomology, connecting various concepts such as the cohomology ring, cohomology operations, and the Künneth formula.

5 Must Know Facts For Your Next Test

1. The cup product is bilinear, meaning it distributes over addition and respects scalar multiplication.
2. The cup product can be used to define the structure of a graded ring on the cohomology groups, where multiplication is given by the cup product.
3. If you have cohomology classes $\alpha \in H^p(X)$ and $\beta \in H^q(X)$, their cup product $\alpha \smile \beta$ will reside in $H^{p+q}(X)$.
4. The cup product satisfies certain properties such as commutativity and associativity, making it a well-behaved algebraic operation.
5. In relation to other products like the cap product, the cup product provides insight into duality and relationships between homology and cohomology.

Review Questions

• How does the cup product structure influence the properties of cohomology rings?
  • The cup product allows us to combine cohomology classes from different degrees to form new classes in a way that respects the algebraic structure of the rings. Specifically, this bilinear operation enables us to define a multiplication on cohomology groups that captures important topological features of spaces. The result is a cohomology ring where one can analyze relationships between different dimensions of cohomological information.
• Explain how the cup product connects with Poincaré duality and its implications for topological spaces.
  • The cup product is closely tied to Poincaré duality as it provides a framework for relating homology and cohomology classes through their algebraic structures. For closed oriented manifolds, Poincaré duality asserts an isomorphism between specific homology and cohomology groups. The cup product helps visualize this duality by showing how these groups interact under multiplication, revealing deeper insights about how geometric features correspond to algebraic properties.
• Analyze the role of the cup product in defining and understanding cohomology operations within algebraic topology.
  • The cup product plays a crucial role in establishing cohomology operations by providing a foundational operation that can generate more complex interactions between classes. By examining how classes combine through the cup product, we can develop higher-level operations that extend our understanding of cohomological structures. These operations not only help in computations but also reveal intricate relationships within topology, paving the way for advanced topics like characteristic classes and spectral sequences.

Related terms

Cohomology ring: A structure that combines the cohomology groups of a space with the cup product, forming an algebraic object that reflects the topological properties of the space.

Poincaré duality: A principle stating that for a closed oriented manifold, there is an isomorphism between the k-th cohomology group and the (dim-n-k)-th homology group, which is deeply related to the cup product.

Cohomology operations: Algebraic operations on cohomology classes that help in understanding their relationships and providing tools for calculations within cohomology theory.