Classification of Manifolds - Maps Between Manifolds - Low Codimension

Low Codimension

Analogously to the classification of manifolds, in high codimension (meaning more than 2), embeddings are classified by surgery, while in low codimension or in relative dimension, they are rigid and geometric, and in the middle (codimension 2), one has a difficult exotic theory (knot theory).

• In codimension greater than 2, embeddings are classified by surgery theory.
• In codimension 2, particularly embeddings of 1-dimensional manifolds in 3-dimensional ones, one has knot theory.
• In codimension 1, a codimension 1 embedding separates a manifold, and these are tractable.
• In codimension 0, a codimension 0 (proper) immersion is a covering space, which are classified algebraically, and these are more naturally thoughts of as submersions.
• In relative dimension, a submersion with compact domain is a fiber bundle (just as in codimension 0 = relative dimension 0), which are classified algebraically.