Ordinary cohomology

Definition

Ordinary cohomology is a mathematical tool used in algebraic topology that assigns a sequence of abelian groups or vector spaces to a topological space, capturing its shape and structure. It provides a way to classify spaces up to homeomorphism and helps in understanding their properties through the use of cohomological invariants. This concept plays a vital role in various mathematical areas, including the study of spectral sequences and the relationship between topology and algebra.

5 Must Know Facts For Your Next Test

1. Ordinary cohomology is typically denoted as $H^n(X)$ for a space $X$ and integer $n$, representing the $n$-th cohomology group.
2. Cohomology groups can be computed using various tools, such as singular cohomology, simplicial cohomology, or cellular cohomology, depending on the type of topological space being studied.
3. The Universal Coefficient Theorem relates cohomology with homology, allowing one to compute cohomology groups from known homology groups.
4. Cohomology theories can be defined with different coefficients, such as integers, rationals, or other rings, which can lead to different invariants.
5. Ordinary cohomology plays a central role in the Atiyah-Hirzebruch spectral sequence, which provides a systematic way to compute the cohomology of complex projective spaces and other related topological spaces.

Review Questions

• How does ordinary cohomology help in classifying topological spaces, and what role does it play in algebraic topology?
  • Ordinary cohomology serves as an important tool for classifying topological spaces by assigning sequences of abelian groups or vector spaces that reflect their shape and structure. By examining the properties of these groups, mathematicians can discern whether two spaces are homeomorphic. This classification is crucial in algebraic topology because it allows for deeper insights into the relationships between different spaces and their corresponding invariants.
• Discuss how the Universal Coefficient Theorem connects ordinary cohomology with homology and its implications for computations.
  • The Universal Coefficient Theorem establishes a relationship between the homology and cohomology of a space, stating that one can derive cohomology groups from known homology groups. This connection allows mathematicians to utilize computations from homology to infer properties about cohomology without directly calculating them. It effectively broadens the tools available for studying topological spaces and enhances our understanding of their structure.
• Evaluate the significance of ordinary cohomology within the framework of the Atiyah-Hirzebruch spectral sequence and its application in algebraic K-theory.
  • Ordinary cohomology is fundamentally significant in the context of the Atiyah-Hirzebruch spectral sequence because it provides an organized method for computing complex cohomological invariants associated with algebraic varieties and manifolds. The spectral sequence organizes information from simpler associated objects into more complex ones, ultimately revealing deeper connections between topology and algebra. This process not only simplifies computations but also enables researchers to explore profound implications in algebraic K-theory and its applications in various branches of mathematics.

Related terms

Cohomology Ring: The cohomology ring is formed by taking the direct sum of the cohomology groups and defining a product on them, allowing for algebraic manipulation of topological information.

Sheaf Cohomology:

Sheaf cohomology is a generalization of ordinary cohomology that involves sheaves, allowing for finer control over local data on topological spaces.

Homology:

Homology is another invariant of topological spaces that, unlike cohomology, typically focuses on constructing chains and studying their properties, offering complementary insights.