Borel–Moore Spectral Sequence

Definition

The Borel–Moore spectral sequence is a tool in algebraic topology used to compute the homology of a space by associating it with the cohomology of a more manageable space, often constructed from a filtered space. It is particularly useful in situations where the underlying space has a nice stratification, allowing one to analyze complex topological structures in a systematic way. This sequence provides a means to connect different layers of cohomological information and is often employed in the study of sheaf cohomology and derived functors.

5 Must Know Facts For Your Next Test

1. The Borel–Moore spectral sequence arises from considering the derived functors associated with cohomology theories, allowing one to analyze spaces with singularities or stratifications.
2. This spectral sequence often computes the homology of the complement of a subspace by relating it to the cohomology of the subspace and its filtration.
3. The Borel–Moore spectral sequence can be used to obtain results about the connectivity and dimensions of spaces by tracking how information propagates through its strata.
4. In many cases, the first page of the Borel–Moore spectral sequence consists of cohomology groups that can be computed directly from the structure of the underlying filtered space.
5. It is crucial in applications involving intersection cohomology, where one wants to understand how singularities affect the topology of a space.

Review Questions

• How does the Borel–Moore spectral sequence relate to filtered spaces and their cohomological properties?
  • The Borel–Moore spectral sequence is fundamentally tied to filtered spaces because it uses their structure to break down complex topological features into more manageable parts. By considering a filtration on a space, one can derive a sequence that captures the essential cohomological data needed to compute homology. This relationship allows mathematicians to analyze how different layers within the filtration contribute to the overall topological characteristics of the space.
• Discuss how the Borel–Moore spectral sequence can be applied in the context of intersection cohomology.
  • In intersection cohomology, the Borel–Moore spectral sequence plays an important role by providing tools to study spaces with singularities. It enables one to relate the cohomology of a stratified space to that of its strata, allowing for an understanding of how these singularities affect overall topological features. This application is crucial for computing intersection cohomology groups that reflect both local and global characteristics of the underlying spaces.
• Evaluate the implications of using the Borel–Moore spectral sequence for computing homological invariants in algebraic topology.
  • Using the Borel–Moore spectral sequence significantly enhances our ability to compute homological invariants by establishing connections between various levels of cohomological information across different strata. This approach provides deep insights into the topology of complex spaces, especially those with intricate structures. The implications are profound as it allows for more nuanced analyses in various fields, including algebraic geometry and mathematical physics, revealing hidden relationships and properties that would otherwise remain obscured.

Related terms

Spectral Sequence: A computational tool in homological algebra that provides a way to compute homology or cohomology groups through a sequence of successive approximations.

Cohomology: A mathematical concept that studies the properties of spaces using cochains and cohomology groups, providing insights into the shape and structure of topological spaces.

Filtration: A process of breaking down a topological space into simpler pieces, often used to study the properties of the space through its layers or stages.