Functoriality of the Long Exact Sequence

Definition

Functoriality of the long exact sequence refers to the property that allows one to derive a long exact sequence in cohomology when given a pair of spaces, along with continuous maps between them. This concept highlights how the long exact sequence behaves under continuous mappings, meaning if you have a map between spaces, the induced sequences on cohomology groups will also preserve the structure of the long exact sequence. This is essential for understanding how different topological spaces relate to each other through cohomological methods.

5 Must Know Facts For Your Next Test

1. The long exact sequence arises from the existence of a pair of spaces, typically denoted as (X, A), where X is a topological space and A is a subspace.
2. Functoriality ensures that for any continuous map f: X → Y, there exists an induced map on the cohomology groups that respects the long exact sequence structure.
3. The long exact sequence in cohomology includes terms that represent relative cohomology groups, allowing for deeper analysis of how spaces interact.
4. Functoriality plays a crucial role in proving various results in algebraic topology, such as the excision theorem and the Mayer-Vietoris sequence.
5. Understanding functoriality helps in establishing important relationships between different types of topological invariants, facilitating further study and applications in geometry and topology.

Review Questions

• How does the functoriality of the long exact sequence relate to continuous mappings between pairs of spaces?
  • Functoriality shows that if there is a continuous map between two topological spaces, it induces corresponding maps on their cohomology groups that maintain the structure of the long exact sequence. This means that not only do we have a sequence associated with one pair, but we can also generate similar sequences with different pairs while preserving relationships. Thus, this property is crucial in understanding how cohomological invariants behave under continuous transformations.
• Discuss how functoriality contributes to the understanding and applications of the Mayer-Vietoris sequence.
  • The Mayer-Vietoris sequence is built on the concept of functoriality, allowing one to compute cohomology groups by breaking down a space into simpler pieces. Functoriality ensures that when you take open covers and apply continuous maps, the induced long exact sequences retain their form. This gives mathematicians a powerful tool for computing cohomological properties in complex spaces by piecing together information from simpler subspaces.
• Evaluate how an understanding of functoriality influences the study of relative cohomology and its applications.
  • An understanding of functoriality significantly enhances the study of relative cohomology as it establishes how different spaces are interconnected through continuous maps and their induced cohomological properties. This insight allows mathematicians to explore and prove results concerning homotopy invariance and excision, which are pivotal in many areas of topology and geometry. The ability to relate different topological constructs through functorial mappings opens avenues for further research and application, particularly in more advanced topics like spectral sequences or sheaf cohomology.

Related terms

Cohomology: A mathematical tool used in algebraic topology that provides a way to associate algebraic invariants to topological spaces.

Exact Sequence: A sequence of abelian groups and homomorphisms between them such that the image of one homomorphism is equal to the kernel of the next.

Pair of Spaces: A mathematical construct consisting of a topological space and a subspace, allowing for the examination of their relationship through cohomology.