Ordinary cohomology

Definition

Ordinary cohomology is a mathematical tool that assigns algebraic invariants to topological spaces, allowing for the classification and comparison of these spaces. It helps in understanding the properties of manifolds and can be seen as a way to 'measure' the shape and structure of spaces by capturing information about their holes at different dimensions. This concept plays a significant role in various advanced topics such as spectral sequences and connections with characteristic classes.

5 Must Know Facts For Your Next Test

1. Ordinary cohomology is defined using singular cochains, which are functions on singular simplices that satisfy certain conditions.
2. Cohomology groups are graded by dimension, meaning each dimension has its own group that corresponds to different 'types' of holes in the space.
3. The universal coefficient theorem relates ordinary cohomology groups with homology groups, providing a powerful tool for calculations.
4. Ordinary cohomology is a functor, which means it behaves nicely with respect to continuous maps between spaces, preserving the structure of cohomology.
5. Ordinary cohomology can be used to derive important results such as Poincaré duality, which connects the cohomology of a manifold with its homology.

Review Questions

• How does ordinary cohomology relate to the study of topological spaces and what kind of information can it provide about these spaces?
  • Ordinary cohomology provides essential algebraic invariants that classify and compare topological spaces by measuring their properties at various dimensions. It reveals information about the existence of holes in the space, which can help determine if two spaces are homeomorphic. By assigning groups to different dimensions, ordinary cohomology allows mathematicians to analyze the structure of manifolds and understand their geometric characteristics.
• What role does ordinary cohomology play in the construction of spectral sequences, particularly in relation to the Atiyah-Hirzebruch spectral sequence?
  • In constructing spectral sequences like the Atiyah-Hirzebruch spectral sequence, ordinary cohomology serves as a foundational tool for organizing and computing cohomological data. The spectral sequence is an iterative process that reveals deeper relationships between different cohomology groups and can simplify complex calculations. The initial page of this spectral sequence often comprises ordinary cohomology groups, which are then related through differentials to yield results about K-theory and other invariants.
• Evaluate how ordinary cohomology connects with characteristic classes and how this relationship enhances our understanding of vector bundles.
  • Ordinary cohomology is intricately linked with characteristic classes, which serve as topological invariants associated with vector bundles. This connection allows us to use cohomological methods to compute and understand these classes, offering insights into the geometry and topology of vector bundles over manifolds. By applying ordinary cohomology techniques, mathematicians can derive important results about how these classes behave under various operations, ultimately leading to a deeper understanding of the interaction between topology and geometry in mathematical contexts.

Related terms

Cohomology groups: Algebraic structures that capture the topological features of a space, organized by dimension, helping to classify spaces based on their 'holes'.

Exact sequence: A sequence of algebraic objects and morphisms that provides a way to connect different cohomology groups, often used to study their relationships.

Characteristic classes:

Invariants associated with vector bundles that reflect the geometry and topology of the underlying space, often linked with cohomological techniques.