Cohomology

Definition

Cohomology is a mathematical tool in algebraic topology that assigns a sequence of abelian groups or vector spaces to a topological space, capturing information about its shape and structure. This concept helps to analyze spaces by providing invariants that are useful for distinguishing between them and studying their properties through cochains, cocycles, and cohomology classes. Cohomology can be applied to various contexts such as spaces, pairs of spaces, and even spectral sequences, revealing deep connections among different areas of mathematics.

5 Must Know Facts For Your Next Test

1. Cohomology groups are denoted as $$H^n(X; G)$$ where $$X$$ is the space, $$n$$ indicates the dimension, and $$G$$ is the coefficient group.
2. One important property of cohomology is its ability to distinguish between spaces that may appear similar at first glance but have different topological characteristics.
3. The Universal Coefficient Theorem connects homology and cohomology, providing a way to compute cohomology groups using homology groups and additional data about coefficients.
4. Cohomology theories can vary based on the coefficients chosen, leading to various types such as singular cohomology, sheaf cohomology, and others that have specific applications.
5. The long exact sequence of a pair relates the cohomology of two spaces with that of their intersection, helping to understand how properties change when moving between different spaces.

Review Questions

• How does cohomology provide information about the structure of topological spaces?
  • Cohomology provides information about the structure of topological spaces by assigning groups or vector spaces that capture topological features like holes or voids in various dimensions. These cohomological invariants help mathematicians distinguish between different types of spaces based on their connectivity and dimensional properties. By examining these groups, one can gain insights into how a space behaves under continuous deformations and transformations.
• What is the significance of the long exact sequence in cohomology when studying pairs of topological spaces?
  • The long exact sequence in cohomology plays a crucial role when analyzing pairs of topological spaces because it connects the cohomological information of the entire space, the subspace, and their intersection. This sequence illustrates how changes in one part of the pair affect the overall topology, providing relationships among their cohomology groups. Understanding this sequence allows mathematicians to draw conclusions about both the individual spaces and their combined properties.
• Evaluate how cohomology theory can be applied using spectral sequences and what insights this provides for complex spaces.
  • Cohomology theory applied through spectral sequences offers powerful tools for computing the cohomology groups of complex topological spaces systematically. Spectral sequences help break down complicated calculations into manageable steps by working through a series of approximations and filtrations. This method not only simplifies the computation but also reveals deeper relationships among different dimensions and structures within the space, providing insights that may not be immediately obvious from traditional approaches.

Related terms

Homology:

Homology is a fundamental concept in algebraic topology that studies the topological features of spaces by associating a sequence of abelian groups, focusing on the number of holes of various dimensions.

Cohomology Ring:

A cohomology ring is an algebraic structure formed by the direct sum of cohomology groups of a space, equipped with a cup product that provides additional structure and relationships among cohomology classes.

Spectral Sequence:

A spectral sequence is an advanced tool used in homological algebra and topology to compute homology or cohomology groups by providing a systematic way to handle complex filtrations.