Generalized cohomology

Definition

Generalized cohomology is a broad concept in algebraic topology that extends the traditional notion of cohomology theories to capture more complex topological features. It encompasses various cohomology theories such as singular cohomology, K-theory, and bordism, each providing different insights into the structure of topological spaces. This flexibility makes generalized cohomology a powerful tool in both geometry and topology.

5 Must Know Facts For Your Next Test

1. Generalized cohomology theories are designed to provide invariants that can be used to differentiate between topological spaces that may be indistinguishable by classical methods.
2. The Atiyah-Hirzebruch spectral sequence is a specific spectral sequence that arises in the computation of generalized cohomology groups, linking them with other types of homological information.
3. In many contexts, generalized cohomology theories satisfy the Eilenberg-Steenrod axioms, which provide a foundational framework for understanding their properties.
4. Examples of generalized cohomology theories include stable homotopy, bordism, and equivariant cohomology, each serving unique purposes in various areas of mathematics.
5. The relationship between generalized cohomology and characteristic classes is crucial for understanding how these theories apply in differential geometry and algebraic topology.

Review Questions

• How do generalized cohomology theories expand upon classical cohomology theories?
  • Generalized cohomology theories broaden the scope of classical cohomology by introducing additional structures and properties that capture more intricate topological features. They allow mathematicians to study spaces that might be indistinguishable through classical means by offering new invariants. This expansion includes various specific theories like K-theory and bordism, each designed to highlight different aspects of topological structures.
• Discuss the significance of the Atiyah-Hirzebruch spectral sequence in the context of generalized cohomology.
  • The Atiyah-Hirzebruch spectral sequence is significant because it provides a systematic method for computing generalized cohomology groups from simpler data. It establishes connections between different cohomology theories and allows for an organized approach to understanding the relationships between them. By applying this spectral sequence, one can derive important invariants that reveal deep geometric and topological insights about spaces.
• Evaluate how generalized cohomology theories interact with characteristic classes and their implications in algebraic topology.
  • Generalized cohomology theories have a profound interaction with characteristic classes, which are tools used to describe vector bundles over manifolds. This interaction is crucial because it allows us to study geometric structures using algebraic invariants derived from generalized cohomology. The implications are significant as they provide a way to understand properties such as curvature and topology of manifolds, leading to deeper results in differential geometry and algebraic topology.

Related terms

Cohomology Theory: A mathematical framework used to study topological spaces by associating algebraic invariants, allowing for the classification of spaces based on their structure.

Spectral Sequence: A computational tool used in algebraic topology and homological algebra to systematically compute homology or cohomology groups through a series of approximations.

K-Theory: A branch of algebraic topology that studies vector bundles and their classes, providing a generalized cohomology theory related to the structure of vector bundles over topological spaces.