Definition:Cobordism

From ProofWiki
Jump to navigation Jump to search

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 $(n+1)$-dimensional 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 $(n+1)$-dimensional Definition: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 $X \times \left[{0 \,.\,.\, 1}\right]$.

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


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 $Y$.