Definition:Cobordism

Definitions

Let $X^n$ and $Y^n$ be manifolds without boundary of dimension $n$.

A cobordism $W^{n + 1}$ between $X$ and $Y$ is an $\paren {n + 1}$-dimensional topological manifold such that:

$\partial W = X \cup Y$

where $\partial W$ denotes the boundary of $W$.

Oriented Cobordism

Let $X^n$ and $Y^n$ be orientable manifolds without boundary of dimension $n$.

An oriented cobordism $W^{n + 1}$ is an $\paren {n + 1}$-dimensional topological manifold such that:

$\partial W = X \cup \overline Y$

where:

$\partial W$ denotes the boundary of $W$
$\overline Y$ denotes $Y$ taken with reverse orientation.

h-Cobordism

Let $X^n$ and $Y^n$ be manifolds without boundary of dimension $n$.

Let $W$ be a cobordism between $X$ and $Y$ such that $W$ is homotopy-equivalent to $X \times \closedint 0 1$.

(formally, $\exists \phi: W \to X$ such that $\phi$ is a retract, which for $X$ and $Y$ simply connected is equivalent to $H_* \struct {W, M; \Z} = 0$)

Then $W$ is said to be an $h$-cobordism.

Also denoted as

If the dimension of $X$ and $Y$ are clear, it is commonplace to omit the indices and state that $W$ is a cobordism between $X$ and $Y$.

Historical Note

The concept of cobordism was devised by René Frédéric Thom in $1954$, who established necessary and sufficient conditions for two manifolds to be cobordant.

This work was extended by John Willard Milnor in $1960$ and Charles Terence Clegg Wall, also in $1960$.