Bordant maps

Definition

Bordant maps are a fundamental concept in cobordism theory, providing a way to relate different manifolds through the notion of bordism. A bordant map is a continuous function between two manifolds that allows for the construction of a 'cobordism' between them, effectively describing how one manifold can be transformed into another within a higher-dimensional space. This concept connects various topological properties and is essential for classifying manifolds based on their cobordism classes.

5 Must Know Facts For Your Next Test

1. Bordant maps allow for the construction of cobordisms, which are crucial in studying relationships between different manifolds.
2. In cobordism theory, two manifolds are considered bordant if there exists a bordant map connecting them through an intermediate manifold.
3. Bordant maps preserve important topological features, which means that they help maintain the characteristics of the manifolds involved.
4. The study of bordant maps has implications in various fields, including algebraic topology, differential geometry, and mathematical physics.
5. Bordism groups can be formed using bordant maps, which provide algebraic structures that classify manifolds up to bordism equivalence.

Review Questions

• How do bordant maps facilitate the connection between different manifolds in cobordism theory?
  • Bordant maps play a crucial role in connecting different manifolds by providing continuous functions that describe how one manifold can transform into another. They establish the concept of bordism, which allows for the construction of an intermediate manifold that links the original manifolds together. This connection is vital for understanding the relationships between manifolds and classifying them based on their topological properties.
• Discuss the significance of bordant maps in forming cobordism groups and how this relates to manifold classification.
  • Bordant maps are significant because they enable the formation of cobordism groups, which are algebraic structures that classify manifolds up to bordism equivalence. These groups group together manifolds that can be transformed into one another through bordant maps, revealing important relationships and characteristics within topology. The classification of manifolds through these groups provides insights into their topology and geometric structure.
• Evaluate the impact of bordant maps on our understanding of homotopy and its relationship to other topological properties.
  • Bordant maps significantly impact our understanding of homotopy by illustrating how continuous transformations between manifolds can yield deeper insights into their topological features. By relating cobordism and homotopy, we see that certain properties preserved under bordant maps can lead to equivalences in homotopy classes. This interplay enhances our comprehension of manifold behavior and topology, showcasing how different areas of mathematics intersect and enrich each other.

Related terms

cobordism: A relation between two manifolds indicating that they can be connected by a higher-dimensional manifold, effectively allowing for a classification of manifolds.

manifold: A topological space that locally resembles Euclidean space and is an essential object of study in topology and geometry.

homotopy: A concept in algebraic topology that describes when two continuous functions can be continuously deformed into each other, providing insight into the topological properties of spaces.