Exact sequence of chain complexes

Definition

An exact sequence of chain complexes is a sequence of chain complexes and chain maps such that the image of one chain map equals the kernel of the subsequent chain map at each degree. This concept plays a crucial role in understanding how cohomology theories interact, especially when considering constructions like the excision theorem, which relates to the ability to calculate cohomology in reduced or modified spaces.

5 Must Know Facts For Your Next Test

1. An exact sequence of chain complexes provides a powerful framework for translating algebraic relationships into topological ones, revealing deeper properties of spaces.
2. In an exact sequence, for each degree n, the image of the nth chain map must equal the kernel of the (n+1)th chain map, ensuring continuity between the chains.
3. Exact sequences can be used to construct long exact sequences in cohomology, which are crucial for various applications, including computing relative cohomology.
4. The excision theorem relies on exact sequences to show that certain subspaces do not affect the overall cohomological properties of a space.
5. Understanding exact sequences of chain complexes is vital for proving many fundamental results in algebraic topology, including various forms of Mayer-Vietoris sequences.

Review Questions

• How does an exact sequence of chain complexes ensure continuity between different degrees in a chain complex?
  • An exact sequence of chain complexes ensures continuity between different degrees by enforcing that the image of each chain map corresponds exactly to the kernel of the following chain map. This means that any cycle that can be formed from a previous degree must be mapped correctly into the next degree without any gaps or overlaps. This property allows for a smooth transition between degrees, which is essential for maintaining the integrity and structure within cohomology theories.
• Discuss how exact sequences contribute to the proof and application of the excision theorem in algebraic topology.
  • Exact sequences contribute to the proof and application of the excision theorem by allowing mathematicians to analyze the cohomological properties of spaces when certain subspaces are removed. By using an exact sequence, one can demonstrate that removing a sufficiently nice subspace does not alter the overall cohomology groups, thereby simplifying computations. This is particularly useful in situations where direct calculations may be complicated or infeasible.
• Evaluate the significance of long exact sequences derived from short exact sequences in understanding relationships between homology and cohomology.
  • Long exact sequences derived from short exact sequences are significant because they create a framework for linking different homology and cohomology groups together through their relationships. By evaluating these long exact sequences, one can extract valuable information about how changes in one area affect others, providing insights into invariants of topological spaces. This interconnectedness highlights how algebraic tools can translate into geometric intuition, ultimately leading to deeper comprehension within algebraic topology.

Related terms

Chain Complex: A chain complex is a sequence of abelian groups or modules connected by homomorphisms such that the composition of any two consecutive homomorphisms is zero.

Cohomology: Cohomology is a mathematical tool used to study topological spaces via algebraic invariants, providing insights into their structure and properties through chains, cycles, and boundaries.

Homology: Homology is a concept closely related to cohomology that measures the 'holes' in a topological space using sequences of abelian groups associated with chains, cycles, and boundaries.