Cohomology

Definition

Cohomology is a mathematical concept used in algebraic topology to study the properties of topological spaces through algebraic invariants, specifically cochains and cocycles. It provides a way to classify spaces by examining the relationships between their differential forms and their structures, offering insights into manifold properties and cobordism theory. By using cohomology, one can relate different spaces and understand how they can be transformed or deformed without losing essential features.

5 Must Know Facts For Your Next Test

1. Cohomology groups are defined using cochains, which are functions that assign values to the chains of a topological space, allowing us to analyze the space's properties.
2. The most common type of cohomology is singular cohomology, which uses singular simplices to define cochains and cocycles.
3. Cohomology provides not just topological information but also has connections to other areas of mathematics, such as algebraic geometry and differential geometry.
4. Cohomological operations, like the cup product, allow us to combine different cohomology classes to gain deeper insights into the structure of manifolds.
5. Cohomology theories can sometimes reveal obstructions to certain properties being true for manifolds, helping mathematicians understand complex relationships between spaces.

Review Questions

• How does cohomology relate to differential forms on manifolds?
  • Cohomology is closely linked to differential forms on manifolds through the De Rham cohomology theory. This approach identifies cohomology classes with equivalence classes of closed differential forms. By analyzing these forms, one can obtain information about the manifold's topology and geometric structure, leading to insights about how these spaces behave under various transformations.
• Discuss how cohomology theory aids in understanding cobordism theory.
  • Cohomology theory plays a significant role in cobordism theory by providing tools for classifying manifolds up to cobordism equivalence. Cohomological invariants can be used to distinguish between different cobordism classes and analyze how manifolds can be related or transformed. This connection helps mathematicians study the relationships between various manifolds and their boundaries in a rigorous way.
• Evaluate the importance of Poincaré Duality in connecting homology and cohomology theories.
  • Poincaré Duality is crucial because it establishes a deep connection between homology and cohomology theories for closed oriented manifolds. This duality shows that for such manifolds, there is an isomorphism between the k-th cohomology group and the (n-k)-th homology group, where n is the dimension of the manifold. This relationship allows for powerful cross-connections between these two fundamental concepts, facilitating a deeper understanding of manifold topology and enhancing the techniques available in both homological and cohomological analyses.

Related terms

Homology:

A fundamental concept in algebraic topology that studies topological spaces by associating chains, cycles, and boundaries, providing a way to measure the 'holes' in a space.

Cohomology Ring: An algebraic structure formed by the cohomology groups of a space, equipped with a cup product operation that combines cohomology classes.

Poincaré Duality:

A theorem stating that for a closed, oriented manifold, the k-th cohomology group is isomorphic to the (n-k)-th homology group, connecting the topological properties of manifolds.