# 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 $X \times \closedint 0 1$.

Work In ProgressExtract the following statement into its own page (or pages). This will require someone who knows what they are talking about. prime.mover is out of his depth.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{WIP}}` from the code. |

(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 $1960$.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**cobordism**