Induced maps between cohomology groups

Definition

Induced maps between cohomology groups are homomorphisms that arise when there is a continuous map between topological spaces, leading to a relationship between their respective cohomology groups. These maps reflect how the structure of cohomology groups transforms under continuous functions, capturing essential topological features and preserving algebraic properties of the spaces involved.

5 Must Know Facts For Your Next Test

1. Induced maps are created through continuous functions between spaces, like $f: X \to Y$, leading to a map $f^*: H^n(Y) \to H^n(X)$ on their cohomology groups.
2. They play a key role in establishing relationships between the cohomology of pairs of spaces and their subspaces, especially when analyzing relative cohomology.
3. Induced maps can be used to derive long exact sequences in cohomology, providing a systematic way to relate the cohomology of different spaces.
4. If a map is a homeomorphism, then its induced maps on cohomology are isomorphisms, indicating that the topological features of the spaces are preserved.
5. Induced maps respect cup products in cohomology, meaning that they maintain the algebraic structure inherent in the product operation across different spaces.

Review Questions

• How do induced maps illustrate the relationship between continuous functions and their effects on cohomology groups?
  • Induced maps showcase how continuous functions transform the cohomological structure of topological spaces. When a continuous function $f: X \to Y$ is applied, it generates a map $f^*$ that relates the cohomology groups $H^n(Y)$ and $H^n(X)$. This means that the properties of $Y$ can be 'pulled back' to understand $X$, thereby highlighting the interplay between topology and algebra.
• Discuss the significance of induced maps in establishing long exact sequences in cohomology theory.
  • Induced maps are crucial for creating long exact sequences in cohomology, which provide insights into how different cohomological dimensions relate to each other. These sequences typically arise from pairs of spaces or inclusions of subspaces, allowing us to see connections between their respective cohomologies. The long exact sequence captures important information about how homomorphisms induced by continuous functions link these dimensions and reveal critical structural aspects.
• Evaluate how induced maps preserve algebraic structures within the context of cup products in cohomology.
  • Induced maps have a significant role in preserving the algebraic structure defined by cup products in cohomology. When considering two spaces and their induced maps, if you have cohomology classes $a \in H^p(X)$ and $b \in H^q(Y)$, then under an induced map $f^*$, the cup product $(a \smile b) ext{ in } H^{p+q}(X)$ translates accordingly. This preservation ensures that the algebraic operations defined in one space reflect similarly when viewed through the lens of another space, solidifying the connection between topology and algebra.

Related terms

Cohomology:

A mathematical tool used to study topological spaces by assigning algebraic structures, like groups, to these spaces that capture their shape and structure.

Homomorphism: A structure-preserving map between two algebraic structures, such as groups or vector spaces, that respects the operations defined on those structures.

Long exact sequence: A sequence of abelian groups and homomorphisms that arises in homology or cohomology, which provides powerful tools for understanding relationships between different cohomological dimensions.