Long exact sequence

Definition

A long exact sequence is a sequence of homology groups associated with a topological space and a pair of spaces that provides crucial information about the relationships between these groups. This concept is key to understanding how homology groups interact when considering pairs of spaces, particularly when applying excision and other techniques to compute homology. The long exact sequence helps in deriving relationships that connect various homological properties across different dimensions.

5 Must Know Facts For Your Next Test

1. The long exact sequence arises from a short exact sequence of chain complexes related to a pair of spaces, enabling connections between the homology groups of both spaces and their intersection.
2. In a long exact sequence, each term relates to the next one, showing how the homology groups change as you move through the sequence.
3. The connecting homomorphism in a long exact sequence plays a critical role in linking the homology of the subspace with that of the larger space.
4. The existence of a long exact sequence is essential for proving many important results in algebraic topology, such as the excision theorem and the Mayer-Vietoris theorem.
5. When working with pairs of spaces, the long exact sequence reveals how relative homology can be used to extract information about the topology of both spaces involved.

Review Questions

• How does the concept of a long exact sequence relate to the computation of homology groups for pairs of spaces?
  • A long exact sequence is crucial for computing homology groups in situations involving pairs of spaces because it connects the homology groups of the individual spaces with that of their intersection. By applying this concept, you can derive relations between these groups, helping simplify computations. The connecting homomorphism in this sequence specifically ties together the relative homology of the subspace with the total homology, highlighting how these groups influence one another.
• In what ways does the long exact sequence contribute to understanding excision in algebraic topology?
  • The long exact sequence illustrates how excision can be applied within algebraic topology by demonstrating that certain subspaces can be 'ignored' without affecting the overall homological properties. It shows how removing a subspace leads to relationships that remain valid among the homology groups involved. By connecting these groups through a long exact sequence, we see that despite changes made by excision, we still retain essential information about the topological structure.
• Evaluate how long exact sequences can be utilized in proving complex topological results like the Mayer-Vietoris theorem.
  • Long exact sequences are powerful tools in proving complex topological results such as the Mayer-Vietoris theorem because they provide a structured way to piece together information from smaller parts into larger constructs. In Mayer-Vietoris, you analyze two overlapping spaces and their intersection. The resulting long exact sequence links their respective homologies, allowing us to deduce properties about the union from these parts. This framework not only simplifies proofs but also clarifies how different topological features interact through their homological characteristics.

Related terms

Short Exact Sequence: A short exact sequence is a sequence of algebraic structures where the image of one map equals the kernel of the next, typically involving three objects, which is a building block for constructing long exact sequences.

Homology Groups: Homology groups are algebraic invariants that classify topological spaces based on their cycles and holes, providing a way to analyze their structure through algebraic means.

Pair of Spaces: A pair of spaces consists of a topological space along with a subspace, allowing for the study of how these spaces relate to each other through their respective homology groups.