Open manifold

In mathematics , an open manifold is a manifold without a boundary , the connected components of which are all non- compact . The opposite concept of an open manifold is that of the closed manifold .

Examples of open manifolds

• the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$
• any open subset of the ${\displaystyle \mathbb {R} ^{n}}$
• the dotted torus
• the Whitehead manifold

Tame ends

An end of an open manifold is tame, if a series finally dominated environments with ${\displaystyle U_{i}}$

${\displaystyle \bigcap _{i}{\overline {U_{i}}}=\emptyset }$ and ${\displaystyle \pi _{1}U_{i}=\pi _{1}E\ \forall i}$

owns. An open manifold is the interior of a compact, framed manifold if the ends are tame and the seven-man obstruction for all ends

${\displaystyle \left[E\right]\in \varprojlim _{i\in I}{\widetilde {K_{0}}}(\mathbb {Z} \left[\pi _{1}U_{i}\right])}$

vanishes in the projective limit of the reduced algebraic K-theory of group rings , so . ${\displaystyle \left[E\right]=0}$

literature

A. Ranicki , B. Hughes : Ends of complexes , Cambridge Tracts in Mathematics 123, Cambridge University Press (1996).