kobordant

In topology, particularly within the field of low-dimensional topology, the term *kobordant* refers to a relationship between manifolds that can be connected through a cobordism—a higher-dimensional manifold with boundary consisting of the two original manifolds. The concept is central to the study of cobordism theory, which classifies manifolds up to cobordism equivalence.

Two manifolds are said to be *kobordant* if there exists a compact manifold with boundary such that the two original manifolds form the boundary components of this larger manifold. For example, two closed manifolds of the same dimension are kobordant if they bound parallelizable regions in a higher-dimensional manifold. This relationship is an equivalence relation, meaning it partitions the set of all manifolds into distinct cobordism classes.

The study of kobordant manifolds is particularly important in the classification of manifolds in dimensions where topological invariants are not yet fully understood. In dimensions 3 and 4, cobordism theory has played a key role in resolving long-standing problems, such as the classification of 3-manifolds and the smooth structure problem for 4-manifolds. The term *kobordant* is often used in conjunction with other invariants, such as the Euler characteristic, signature, or intersection form, to refine the classification of manifolds.

In algebraic topology, cobordism theory extends beyond smooth manifolds to include topological, PL (piecewise-linear), and other types of manifolds, though the definitions may vary slightly depending on the context. The concept is also closely related to the study of bordism groups, which formalize the equivalence classes of kobordant manifolds. These groups provide a way to measure the "size" or "complexity" of manifolds in a way that respects cobordism relations.