Relative Cohomology Groups

Definition

Relative cohomology groups are a fundamental concept in algebraic topology, defined as the cohomology groups of a pair consisting of a topological space and a subspace. These groups capture how the topology of a space behaves with respect to the subspace, enabling us to understand the 'difference' between the entire space and the subspace. This concept is crucial for examining how inclusions of subspaces induce cohomomorphisms and is closely linked to the excision theorem, which allows for simplified calculations in specific scenarios involving relative cohomology.

5 Must Know Facts For Your Next Test

1. Relative cohomology groups are denoted as $$H^n(X, A)$$, where $$X$$ is the space and $$A$$ is the subspace.
2. The long exact sequence in cohomology allows for the relationship between relative cohomology groups and ordinary cohomology groups to be understood through an inclusion map.
3. The excision theorem states that if a subspace can be 'removed' without affecting the overall topology, then the relative cohomology remains unchanged.
4. Relative cohomology provides insight into homotopy properties by examining how spaces deform with respect to their subspaces.
5. These groups help in computing cohomology of complex spaces by breaking them down into simpler components using relative relationships.

Review Questions

• How do relative cohomology groups illustrate the relationship between a topological space and its subspace?
  • Relative cohomology groups show how the overall topology of a space interacts with that of its subspace by measuring the differences in their structure. Specifically, they can capture what features or elements exist in the larger space that do not appear in the subspace. This understanding is key for deriving relationships and constructing long exact sequences that connect ordinary and relative cohomology, revealing deeper insights into their properties.
• Discuss how induced cohomomorphisms relate to relative cohomology groups and their computations.
  • Induced cohomomorphisms arise from inclusion maps that connect the cohomology of a space with that of its subspace. When considering relative cohomology groups, these induced maps help form exact sequences that link various cohomological dimensions. They make it possible to compute relative cohomology by relating it back to known structures in ordinary cohomology, thus simplifying complex calculations.
• Evaluate how the excision theorem impacts the use of relative cohomology groups in algebraic topology.
  • The excision theorem fundamentally alters how we utilize relative cohomology groups by allowing us to ignore certain parts of a space when they do not influence the overall topological properties. This means that if we have a situation where we can remove a subspace without affecting homotopical features, we can use relative cohomology to simplify our analysis. It opens doors to computing cohomological information more efficiently by focusing on essential parts of spaces and leveraging relationships established through relative group definitions.

Related terms

Cohomology Groups:

Cohomology groups are algebraic structures that provide a way to classify and measure topological spaces, representing information about their shape and features.

Exact Sequence:

An exact sequence is a sequence of abelian groups and homomorphisms between them such that the image of one homomorphism equals the kernel of the next, capturing the relationships between cohomology groups.

Inclusion Map: An inclusion map is a function that takes an element from a subspace and maps it into a larger space, playing a key role in understanding how relative cohomology groups are induced.