Long exact sequence of a pair

Definition

The long exact sequence of a pair is a fundamental concept in algebraic topology that connects the cohomology groups of a topological space and its subspaces. This sequence arises from a pair of spaces, typically denoted as $(X, A)$, where $X$ is a topological space and $A$ is a subspace of $X$. The long exact sequence provides a systematic way to understand how the cohomology groups of the space $X$ relate to those of the subspace $A$ and the quotient space $X/A$, revealing deeper properties about their topological structure.

5 Must Know Facts For Your Next Test

1. The long exact sequence of a pair arises from the inclusion of a subspace $A$ into a space $X$, leading to a sequence involving the cohomology groups $H^*(A)$, $H^*(X)$, and $H^*(X/A)$.
2. This sequence is exact at each cohomology group, meaning that the image of one group is precisely the kernel of the next, which allows for significant deductions about these groups.
3. The long exact sequence helps to identify and calculate relative cohomology groups, which capture information about how a space is built from its subspaces.
4. The key applications of this sequence include proving important results like excision and establishing relationships between different types of cohomology theories.
5. The construction of the long exact sequence often utilizes the Universal Coefficient Theorem, linking homology and cohomology through the framework of pairs.

Review Questions

• How does the long exact sequence of a pair facilitate the understanding of relationships between cohomology groups?
  • The long exact sequence of a pair establishes a direct connection between the cohomology groups of a topological space and its subspace. By creating a sequence involving $H^*(A)$, $H^*(X)$, and $H^*(X/A)$, it shows how changes in one group affect others. This relationship allows mathematicians to derive important properties about these groups and understand how spaces interact with their subspaces.
• Discuss the role of the long exact sequence of a pair in proving important topological results like excision.
  • The long exact sequence of a pair plays an essential role in proving results such as excision by demonstrating how removing certain subspaces does not affect the overall topological structure. The exactness property reveals that if we have a certain condition met for our spaces, we can use this long exact sequence to argue that cohomological features are preserved under excision. This understanding ultimately leads to significant implications for calculating cohomology groups efficiently.
• Evaluate how the Universal Coefficient Theorem connects to the long exact sequence of a pair and its applications in algebraic topology.
  • The Universal Coefficient Theorem serves as a bridge linking homology and cohomology, providing critical insight into how these two theories interact. When applied alongside the long exact sequence of a pair, it allows mathematicians to derive relationships between relative and absolute cohomology groups more effectively. This connection facilitates various applications in algebraic topology, such as simplifying complex calculations and uncovering deeper relationships among spaces based on their structural components.

Related terms

Cohomology:

A mathematical tool used in algebraic topology that assigns a sequence of abelian groups or modules to a topological space, providing insight into its shape and structure.

Mayer-Vietoris Sequence: A powerful tool used to compute the cohomology of a space by breaking it down into simpler pieces, often leading to long exact sequences in specific situations.

Homotopy:

A concept in topology that studies when two continuous functions can be deformed into each other, important for understanding the relationships between spaces in cohomology.