Cobordism

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 an ${\displaystyle (n+1)}$-dimensional manifold ${\displaystyle W}$ is an ${\displaystyle n}$-dimensional manifold ${\displaystyle \partial W}$ 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.

A cobordism between manifolds ${\displaystyle M}$ and ${\displaystyle N}$ is a compact manifold ${\displaystyle W}$ whose boundary is the disjoint union of ${\displaystyle M}$ and ${\displaystyle N}$, ${\displaystyle \partial W=M\sqcup N}$.

Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right. Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to diffeomorphism or homeomorphism in dimensions ≥ 4 – because the word problem for groups cannot be solved – but it is possible to classify manifolds up to cobordism. Cobordisms are central objects of study in geometric topology and algebraic topology. In geometric topology, cobordisms are intimately connected with Morse theory, and h-cobordisms are fundamental in the study of high-dimensional manifolds, namely surgery theory. In algebraic topology, cobordism theories are fundamental extraordinary cohomology theories, and categories of cobordisms are the domains of topological quantum field theories.

Roughly speaking, an ${\displaystyle n}$-dimensional manifold ${\displaystyle M}$ is a topological space locally (i.e., near each point) homeomorphic to an open subset of Euclidean space ${\displaystyle \mathbb {R} ^{n}}$. A manifold with boundary is similar, except that a point of ${\displaystyle M}$ is allowed to have a neighborhood that is homeomorphic to an open subset of the half-space

Those points without a neighborhood homeomorphic to an open subset of Euclidean space are the boundary points of ${\displaystyle M}$; the boundary of ${\displaystyle M}$ is denoted by ${\displaystyle \partial M}$. Finally, a closed manifold is, by definition, a compact manifold without boundary (${\displaystyle \partial M=\emptyset }$).

An ${\displaystyle (n+1)}$-dimensional cobordism is a quintuple ${\displaystyle (W;M,N,i,j)}$ consisting of an ${\displaystyle (n+1)}$-dimensional compact differentiable manifold with boundary, ${\displaystyle W}$; closed ${\displaystyle n}$-manifolds ${\displaystyle M}$, ${\displaystyle N}$; and embeddings ${\displaystyle i\colon M\hookrightarrow \partial W}$, ${\displaystyle j\colon N\hookrightarrow \partial W}$ with disjoint images such that

The terminology is usually abbreviated to ${\displaystyle (W;M,N)}$. ${\displaystyle M}$ and ${\displaystyle N}$ are called cobordant if such a cobordism exists. All manifolds cobordant to a fixed given manifold ${\displaystyle M}$ form the cobordism class of ${\displaystyle M}$.