Pairs

Definition

In cohomology theory, pairs refer to a space $X$ together with a subspace $A$, typically denoted as the pair $(X, A)$. This concept is fundamental in understanding how the topology of the space $X$ interacts with that of the subspace $A$, particularly when analyzing the long exact sequence of cohomology groups that arise from a pair.

5 Must Know Facts For Your Next Test

1. The long exact sequence of a pair is derived from the short exact sequences associated with the pairs $(X, A)$ in cohomology.
2. When considering pairs, the relative cohomology groups $H^n(X, A)$ provide insight into how the topology of $A$ influences the topology of $X$.
3. The long exact sequence of cohomology for a pair connects the cohomology groups of $X$, $A$, and their relative groups, illustrating their relationships.
4. Pairs are crucial when applying excision in cohomology, allowing for simplification of calculations involving more complex spaces.
5. The concept of pairs aids in understanding concepts like Mayer-Vietoris sequences, which provide powerful tools for computing cohomology in complicated spaces.

Review Questions

• How do pairs $(X, A)$ contribute to our understanding of topological properties in cohomology?
  • Pairs $(X, A)$ play a significant role in understanding topological properties by allowing us to study how the subspace $A$ affects the overall structure of $X$. The long exact sequence associated with a pair provides relationships between the cohomology groups of both spaces, revealing insights about how features in $A$ are reflected in $X$. This connection helps mathematicians compute invariants and analyze complex topological behaviors more effectively.
• Discuss the importance of the long exact sequence in relation to pairs and how it can be applied in problem-solving.
  • The long exact sequence is essential when dealing with pairs because it establishes crucial links between different cohomology groups. When we have a pair $(X, A)$, the long exact sequence allows us to connect $H^n(X)$ and $H^n(A)$ with relative cohomology $H^n(X, A)$. This relationship is particularly useful for solving problems where direct computation is difficult; we can utilize known values from one group to infer information about another, significantly simplifying our calculations.
• Evaluate how the concept of pairs and their associated long exact sequences impacts our approach to excision in cohomology theory.
  • The concept of pairs greatly impacts our approach to excision by allowing us to redefine and simplify spaces involved in cohomological computations. When working with pairs $(X, A)$, we can often disregard certain parts of $X$ while still retaining essential topological features that influence its behavior. This leads to powerful simplifications using long exact sequences and relative groups. As such, excision becomes more manageable; we can isolate specific portions of our space without losing critical information about its overall structure, thus enhancing our ability to analyze complex topological phenomena.

Related terms

Cohomology: A mathematical tool used to study topological spaces through algebraic invariants, providing a way to classify and compare different spaces.

Exact Sequence: A sequence of abelian groups (or modules) and homomorphisms between them such that the image of one homomorphism is equal to the kernel of the next, indicating a relationship between their algebraic structures.

Homology: An algebraic structure that associates a sequence of abelian groups or modules to a topological space, measuring its 'holes' in various dimensions.