Open book

In mathematics are open books (ger .: open book decompositions ) certain decompositions of manifolds that in the classification of contact structures and in the construction of foliations are useful.

definition

Be a closed oriented manifold. An open book on is a pair with: ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle (B,\pi )}$

1. ${\displaystyle B}$is an oriented -dimensional submanifold, the binding of the open book. ${\displaystyle (n-2)}$
2. ${\displaystyle \pi \colon M\setminus B\to S^{1}}$is a bundle of fibers , so that the inside of a compact -dimensional manifold - the page of the open book - is for all . ${\displaystyle \pi ^{-1}(\theta )}$ ${\displaystyle (n-1)}$ ${\displaystyle \Sigma _{\theta }}$ ${\displaystyle \partial \Sigma _{\theta }=B}$ ${\displaystyle \theta \in S^{1}}$

existence

Theorem of Alexander (1920): Every closed, oriented 3-manifold can be represented as an open book.

Theorem von Winkelnkemper (1973): A simply connected, closed manifold of dimensions can be represented as an open book precisely when its signature disappears. (The latter is especially true if it is not divisible by 4.)   ${\displaystyle n\geq 6}$ ${\displaystyle n}$

Foliage

Be an open book on a 3-manifold . Then has a foliation through fibers of and on an area around the binding one can define the Reeb foliation , this has in particular as a compact sheet. By means of turbulization , the foliage can be made tangential to this compact sheet, so one obtains a complete foliage . ${\displaystyle (B,\pi )}$ ${\displaystyle M}$ ${\displaystyle M-B}$ ${\displaystyle \pi }$ ${\displaystyle U(B)=B\times D^{2}}$ ${\displaystyle B\times \partial D^{2}}$ ${\displaystyle M\setminus U(B)}$ ${\displaystyle M}$

Contact structures

Be an open book on a 3-manifold . A contact structure is borne by, if ${\displaystyle (B,\pi )}$ ${\displaystyle M}$ ${\displaystyle \xi =\ker(\alpha )}$ ${\displaystyle (B,\pi )}$

1. ${\displaystyle d\alpha }$is a positive volume shape on each side and ${\displaystyle \Sigma _{\theta }}$
2. ${\displaystyle \alpha >0}$on the bond . ${\displaystyle B}$

Theorem of Thurston- Winkelnkemper (1975): Every open book has a contact structure.

Theorem of Giroux (2000): Every oriented contact structure is supported by an open book. Two contact structures carried by the same open book are isotopic.

literature

• Etnyre: Lectures on open book decompositions and contact structures (PDF; 426 kB)
• Martínez: Open Book decompositions and contact geometry (PDF; 223 kB)

Web links

• Manifold Atlas: Open Book