Relative cohomology

Definition

Relative cohomology is a concept in algebraic topology that deals with the study of cohomology groups associated with a pair of topological spaces, typically a space and a subspace. This approach allows mathematicians to understand the features of the larger space by examining how it behaves relative to the smaller subspace, which is especially useful in situations where both spaces are of interest. The relative cohomology groups provide a way to compute cohomology in a way that captures important information about the relationships between spaces.

5 Must Know Facts For Your Next Test

1. Relative cohomology is often denoted as \( H^*(X, A) \), where \( X \) is a topological space and \( A \) is a subspace.
2. The computation of relative cohomology can often be facilitated using long exact sequences derived from pairs of spaces.
3. In the context of the Atiyah-Hirzebruch spectral sequence, relative cohomology helps in calculating the stable cohomological properties of fiber bundles.
4. Relative cohomology groups can provide insights into how certain properties of the space change when considering its relationship with subspaces.
5. The Universal Coefficient Theorem for cohomology connects relative cohomology with homology, allowing for computations in different contexts.

Review Questions

• How does relative cohomology provide insights into the relationships between a space and its subspace?
  • Relative cohomology allows mathematicians to analyze how features of a larger space relate to a smaller subspace. By studying the cohomology groups of both spaces together, one can uncover properties that might be obscured when looking at each space in isolation. This perspective is particularly useful in applications involving inclusions of spaces or when exploring deformation retractions.
• Discuss the role of long exact sequences in calculating relative cohomology and their importance in algebraic topology.
  • Long exact sequences are crucial in calculating relative cohomology because they establish connections between various cohomology groups associated with pairs of spaces. These sequences arise from applying functoriality to short exact sequences and allow for the extraction of information about relative cohomology by relating it to simpler known cohomology groups. This method helps streamline complex calculations and facilitates deeper understanding in algebraic topology.
• Evaluate how relative cohomology is utilized within the framework of the Atiyah-Hirzebruch spectral sequence and its implications for stable cohomological properties.
  • In the context of the Atiyah-Hirzebruch spectral sequence, relative cohomology is employed to compute stable cohomological properties of fiber bundles. This framework facilitates an understanding of how fibers interact with base spaces, leading to insights about their collective behavior and relationships. By harnessing relative cohomology, mathematicians can derive valuable information about complex structures in algebraic K-theory and gain deeper insights into topological invariants.

Related terms

Cohomology: A mathematical tool for associating a sequence of abelian groups or vector spaces to a topological space, capturing its shape and structure.

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.

Homology: A concept closely related to cohomology, it provides a way to associate algebraic structures (like groups) with topological spaces, focusing on the 'holes' or voids within those spaces.