Birman sequence

In the low-dimensional topology , the Birman sequence is a fundamental tool in the study of mapping class groups . It is named after the American mathematician Joan Birman .

Figure class groups

→ Main article : Figure class group

For a closed, orientable area of gender with points one defines ${\displaystyle F}$ ${\displaystyle g}$ ${\displaystyle r}$ ${\displaystyle x_{1},\ldots ,x_{r}\in F}$

${\displaystyle \Gamma _{g,r}:=Mod(F,\left\{x_{1},\ldots ,x_{r}\right\})}$

as the group of homotopy classes of homeomorphisms with , whereby the homotopias should also fix the points . In particular, one obtains for the "classic" mapping class group . ${\displaystyle \phi \colon F\to F}$ ${\displaystyle \phi (x_{1})=x_{1},\ldots ,\phi (x_{r})=x_{r}}$ ${\displaystyle x_{1},\ldots ,x_{r}}$ ${\displaystyle r=0}$ ${\displaystyle \Gamma _{g}:=Mod(F)}$

The Birman sequence is mainly used for induction proofs of properties by means of induction after . But it can also be used in the opposite direction. For example, the Madsen-Weiss theorem allows the stable homology of to be calculated and the Birman sequence can then be used to establish a reference to the homology of . ${\displaystyle \Gamma _{g,r}}$ ${\displaystyle r}$ ${\displaystyle \Gamma _{g,1}}$ ${\displaystyle \Gamma _{g}}$

Birman sequence

Let it be a compact , orientable surface of the gender and be points on it . Then you have an exact sequence ${\displaystyle F}$ ${\displaystyle g}$ ${\displaystyle x_{1},\ldots ,x_{r}}$ ${\displaystyle F}$

${\displaystyle 1\to \pi _{1}C(F,r)\to \Gamma _{g,r}\to \Gamma _{g}\to 1,}$

where denotes the configuration space of points on , i.e. the quotient of under the action of the symmetrical group . ${\displaystyle C(F,r)}$ ${\displaystyle r}$ ${\displaystyle F}$ ${\displaystyle \left\{(c_{1},\ldots ,c_{r})\in F^{r}\colon c_{i}\not =c_{j}\ \forall \ i\not =j\right\}}$ ${\displaystyle S_{r}}$

Often only the special case , i.e. the exact sequence, is used ${\displaystyle r=1}$

${\displaystyle 1\to \pi _{1}F\to \Gamma _{g,1}\to \Gamma _{g}\to 1}$

referred to as the Birman sequence.

The figures and in general are defined by the "Point-Pushing Map". ${\displaystyle \pi _{1}F\to \Gamma _{g,1}}$ ${\displaystyle \pi _{1}C(F,r)\to \Gamma _{g,r}}$

3-manifolds

There is also a Birman sequence for hyperbolic 3-manifolds , but not for Seifert fibers .

literature

• Joan Birman : Braids, left, and mapping class groups. Annals of Mathematics Studies, No. 82. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974.
• Benson Farb , Dan Margalit : A primer on mapping class groups. Princeton Mathematical Series, 49th Princeton University Press, Princeton, NJ, 2012. ISBN 978-0-691-14794-9

Individual evidence

1. ↑ Chapter 4.2 in Color Margalite, op. Cit.
2. ↑ For the exact construction of the "Point-Pushing Map" see chapter 4.2.2 in Farb-Margalit, op. Cit.
3. ↑ Jessica Banks : The Birman exact sequence for 3-manifolds. ArXiv