cobordism
thumb|A cobordism (W; M, N). In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher.
Article
21 sectionsContents
- Definition
- Manifolds
- Cobordisms
- Examples
- Terminology
- Variants
- Surgery construction
- Examples
- Morse functions
- Geometry, and the connection with Morse theory and handlebodies
- History
- Categorical aspects
- Unoriented cobordism
- Cobordism of manifolds with additional structure
- Oriented cobordism
- Cobordism as an extraordinary cohomology theory
- Other results
- See also
- Notes
- References
- External links
thumb|A cobordism (W; M, N). In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher.
The boundary of a compact (n+1)-dimensional manifold W is an n-dimensional manifold \partial W that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds.