Singular cohomology

Definition

Singular cohomology is a mathematical tool in algebraic topology that assigns a sequence of abelian groups or vector spaces to a topological space, providing a way to study its shape and structure. This concept extends the idea of singular homology by incorporating the duality of the spaces through the use of cochains, allowing for a deeper analysis of topological properties. It plays a crucial role in connecting various mathematical disciplines, including differential geometry and algebraic geometry, while adhering to the foundational axioms that define cohomology theories.

5 Must Know Facts For Your Next Test

1. Singular cohomology is defined using singular cochains, which are functions mapping singular simplices to abelian groups.
2. The main tool for computing singular cohomology is the cup product, which allows for the combination of cohomology classes.
3. One of the key results related to singular cohomology is the Universal Coefficient Theorem, which relates homology and cohomology theories.
4. Singular cohomology satisfies several important properties such as functoriality, excision, and the existence of long exact sequences.
5. The relationship between singular cohomology and de Rham cohomology can be established through the de Rham isomorphism, showing their equivalence on smooth manifolds.

Review Questions

• How does singular cohomology relate to homology, and what role do singular simplices play in this connection?
  • Singular cohomology builds on the ideas from homology by focusing on dual aspects. While homology studies cycles and boundaries in a topological space through singular simplices, singular cohomology uses these same simplices to define cochains that map into abelian groups. This dual approach allows for different insights into topological properties, emphasizing how structures interact with one another.
• What are some key axioms that singular cohomology must satisfy, and why are these axioms important in establishing its foundational properties?
  • Singular cohomology must adhere to several essential axioms: the axioms of identity, additivity, and homotopy invariance. These axioms ensure that singular cohomology behaves consistently under various conditions and transformations of topological spaces. They establish a solid foundation for understanding how cohomological properties relate across different spaces and provide a framework for further developments in algebraic topology.
• Discuss the significance of the relationship between singular cohomology and de Rham cohomology, particularly in the context of smooth manifolds.
  • The relationship between singular cohomology and de Rham cohomology is pivotal because it showcases how different mathematical approaches can yield equivalent results. On smooth manifolds, the de Rham isomorphism demonstrates that both theories capture similar topological features despite differing methodologies—one relying on differential forms and the other on singular simplices. This equivalence enriches our understanding by bridging topology with differential geometry, illustrating how various mathematical disciplines interconnect.

Related terms

Cohomology: A method in algebraic topology used to associate a sequence of groups or rings to a topological space, providing insight into its structure and properties.

De Rham Cohomology: A specific type of cohomology theory that uses differential forms and integrates them over singular chains to study smooth manifolds.

Homology: A related concept that focuses on the study of topological spaces by assigning sequences of abelian groups based on the presence of cycles and boundaries within those spaces.