Ordinary cohomology

Definition

Ordinary cohomology is a mathematical tool that associates a sequence of abelian groups or vector spaces with a topological space, allowing for the study of its properties through algebraic means. It provides insights into the structure and features of spaces by capturing information about their holes and topological features, often revealing relationships between different spaces. This concept plays a significant role in homological algebra, linking algebraic invariants to geometric intuition.

5 Must Know Facts For Your Next Test

1. Ordinary cohomology can be computed using various techniques, including simplicial, singular, and Čech methods, all leading to isomorphic cohomology groups for well-behaved spaces.
2. It is an invariant under homeomorphisms, meaning if two spaces are homeomorphic, their ordinary cohomology groups will be isomorphic.
3. The rank of the ordinary cohomology groups provides information about the number of holes in the corresponding dimension of a space.
4. In algebraic topology, the use of coefficients in ordinary cohomology (such as integers or fields) affects the structure and properties of the resulting cohomology groups.
5. Ordinary cohomology is crucial for applications such as Poincaré duality and the computation of characteristic classes in differential geometry.

Review Questions

• How does ordinary cohomology help in understanding the properties of a topological space?
  • Ordinary cohomology helps reveal important properties of a topological space by associating algebraic structures (abelian groups or vector spaces) with it. This relationship allows mathematicians to capture information about the space's holes and other topological features, translating geometric intuitions into algebraic terms. Through this process, ordinary cohomology provides a powerful tool for distinguishing between different topological spaces and understanding their inherent structures.
• Compare and contrast ordinary cohomology with singular cohomology. What are their respective roles in homological algebra?
  • Ordinary cohomology and singular cohomology are closely related but differ in their approach. While ordinary cohomology can be defined using various techniques, singular cohomology specifically utilizes singular simplices to construct its groups. Both types serve similar purposes in homological algebra by providing invariants that describe topological spaces. However, singular cohomology often serves as a foundational framework from which ordinary cohomology can be computed and understood in broader contexts.
• Evaluate the significance of ordinary cohomology in applications such as Poincaré duality and characteristic classes. How does it shape our understanding of topology?
  • Ordinary cohomology plays a crucial role in applications like Poincaré duality and the computation of characteristic classes. Poincaré duality establishes a deep connection between homology and cohomology for oriented manifolds, illustrating how these two areas inform one another within topology. Characteristic classes further utilize ordinary cohomology to classify vector bundles over manifolds, influencing areas like differential geometry and algebraic topology. Together, these applications demonstrate how ordinary cohomology shapes our understanding by bridging algebraic concepts with geometric insights.

Related terms

Singular Cohomology: A type of cohomology that uses singular simplices to define cohomology groups, providing a way to compute ordinary cohomology for topological spaces.

Cech Cohomology: A method of computing cohomology that uses open covers of a space, allowing for different approaches to determine the same topological invariants as ordinary cohomology.

Homology:

A related concept that focuses on the study of chains and cycles within a space, providing complementary information to cohomology about the structure of topological spaces.