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 $X \times \closedint 0 1$.
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(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 $Y$.
Also see
- Results about cobordisms can be found here.
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 $1960$.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cobordism
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cobordism