Bordism

Definition

Bordism is a concept in algebraic topology that studies the relationships between manifolds, particularly focusing on how manifolds can be considered equivalent if they can be connected by a 'bordism'—essentially a cobordism of higher-dimensional spaces. This idea allows mathematicians to classify manifolds based on their dimensionality and the nature of their boundaries, creating a powerful framework for understanding the topology of shapes in a cohesive manner.

5 Must Know Facts For Your Next Test

1. Bordism can be classified into different types, such as smooth bordism and topological bordism, depending on the properties of the manifolds being considered.
2. The bordism relation is an equivalence relation, meaning it satisfies reflexivity, symmetry, and transitivity among manifolds.
3. The study of bordism is closely related to other areas in algebraic topology, such as homology and cohomology theories.
4. Bordism classes can be represented using invariants, which allow for the comparison and classification of different manifolds based on their topological properties.
5. One important application of bordism theory is in the classification of manifolds in terms of their intersection forms, providing insights into more complex topological features.

Review Questions

• How does bordism relate to the concept of cobordism, and what implications does this have for the classification of manifolds?
  • Bordism is directly related to cobordism as it considers two manifolds to be equivalent if they can be connected through a higher-dimensional manifold, known as a cobordism. This relation allows mathematicians to classify manifolds based on their boundaries and the structures that connect them. By examining how manifolds interact through bordisms, we gain insights into their topological properties and relationships.
• What are some key differences between smooth bordism and topological bordism, and why are these distinctions significant in topology?
  • Smooth bordism deals with manifolds that have differentiable structures, meaning the maps between them are smooth, while topological bordism focuses on continuous mappings without requiring differentiability. These distinctions are significant because they affect how we understand the properties and classifications of manifolds within different contexts. For instance, certain invariants or features may only apply to smooth manifolds, impacting how we analyze and categorize them in algebraic topology.
• In what ways does bordism theory contribute to our understanding of homotopy groups, and what broader implications does this have in algebraic topology?
  • Bordism theory provides a framework for relating various manifolds through equivalence relations that can be analyzed in terms of homotopy groups. By studying bordism classes, we can derive insights into how these groups behave under transformations and identify relationships between different types of manifolds. This understanding deepens our grasp of the structure of topological spaces, influencing areas such as manifold theory and geometric topology while also opening pathways for new discoveries in mathematical research.

Related terms

Cobordism: A relationship between two manifolds where they are the boundaries of a higher-dimensional manifold.

Manifold: A topological space that locally resembles Euclidean space, allowing for the study of more complex shapes and structures.

Homotopy: A concept that studies the properties of topological spaces that are preserved under continuous deformations, linking closely to bordism in how spaces can be transformed.