Equivariant Cohomology

Definition

Equivariant cohomology is a branch of mathematics that extends the notion of cohomology to spaces with group actions, allowing for the study of topological spaces that have a symmetry represented by a group. This concept captures both the topology of the space and the way the group acts on it, leading to insights in various fields, including algebraic topology and representation theory. It provides tools for analyzing fixed points and equivariant maps, linking closely with other powerful theorems in topology.

5 Must Know Facts For Your Next Test

1. Equivariant cohomology generalizes ordinary cohomology by incorporating information about the action of a group on a space, often denoted as $$H^*(X; G)$$ for a space $$X$$ acted on by a group $$G$$.
2. The use of equivariant cohomology is crucial when applying Lefschetz fixed-point theorem to analyze fixed points of continuous maps in spaces with symmetries.
3. This theory leads to new invariants that can differentiate between spaces that are homotopically equivalent but have different group actions.
4. Equivariant cohomology can be computed using spectral sequences or derived functors, making it a robust tool for algebraic topology.
5. Applications of equivariant cohomology include studying toric varieties and understanding the topology of symmetric spaces.

Review Questions

• How does equivariant cohomology enhance our understanding of fixed points in the context of group actions?
  • Equivariant cohomology provides a framework to study fixed points under group actions by incorporating symmetry into the analysis. This allows us to apply powerful results like the Lefschetz fixed-point theorem more effectively, revealing how many points remain unchanged under specific mappings. The interplay between the topology of the space and its symmetries gives deeper insights into the structure and behavior of such spaces.
• Discuss how Borel construction is utilized in defining equivariant cohomology and its significance.
  • Borel construction is essential for defining equivariant cohomology because it creates a new space from an original space with a group action, allowing us to study topological properties that respect this action. By forming the quotient of the product space by the group action, we capture both the original topological structure and the symmetry involved. This construction facilitates computations and insights into equivariant invariants that are crucial for understanding fixed-point phenomena.
• Evaluate the impact of equivariant cohomology on modern algebraic topology and its applications in other fields.
  • Equivariant cohomology has significantly impacted modern algebraic topology by providing tools to analyze spaces with symmetries and explore their properties more thoroughly. Its applications extend beyond pure mathematics into fields like robotics, physics, and data analysis, where symmetry plays a crucial role. By enabling researchers to understand how groups act on spaces, equivariant cohomology aids in solving complex problems involving both topology and algebra, ultimately enriching our grasp of these interconnected areas.

Related terms

Fixed Point Theorem: A theorem that states conditions under which a function will have at least one fixed point, often used in the context of continuous mappings.

Borel Construction: A method for constructing a new space from a given topological space with a group action, allowing for the analysis of equivariant properties.

Group Action: An operation by a group on a set that describes how elements of the group can manipulate or change the elements in the set, influencing its topological structure.