Handle body

In mathematics , handle bodies are 3-dimensional structures, the edges of which are surfaces .

definition

A full sphere with 3 disjoint handles.

The handle body of the sex ${\displaystyle g}$ is obtained by attaching disjoint handles to a 3-dimensional full sphere . ${\displaystyle g}$

In formulas: Let a solid sphere, let injective continuous mappings with disjoint images, then we define the handle body as the quotient of ${\displaystyle B^{3}}$ ${\displaystyle f_{1},\ldots ,f_{g}:B^{2}\times \left\{0,1\right\}\rightarrow \partial B^{3}}$ ${\displaystyle H_{g}}$

${\displaystyle H_{g}:=(B^{3}\bigcup \cup _{i=1}^{g}(B^{2}\times \left[0,1\right])_{i})/\sim }$

under the equivalence relation for . ${\displaystyle x\sim f_{i}(x)}$ ${\displaystyle x\in (B^{2}\times \left\{0,1\right\})_{i},i=1,\ldots ,g}$

${\displaystyle H_{g}}$is an orientable 3-dimensional manifold with a border , its border is a surface of the gender . The full ball is referred to as the handle body by gender . ${\displaystyle g}$ ${\displaystyle g=0}$

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  ${\displaystyle g=1}$: Full torus

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  ${\displaystyle g=2}$: Full pretzel

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  ${\displaystyle g=3}$

Compression body

A more general term, which is mainly used in the theory of 3-manifolds with a boundary , is the term of the compression body .

A compression body is created from a product , for a closed surface , by gluing 2 handles along it . One designates and . ${\displaystyle C}$ ${\displaystyle S\times \left[0,1\right]}$ ${\displaystyle S}$ ${\displaystyle S\times \left\{1\right\}}$ ${\displaystyle \partial _{-}C:=S\times \left\{0\right\}}$ ${\displaystyle \partial _{+}C:=\partial C\setminus \partial _{-}C}$

Handle body is obtained for , in this case it is . ${\displaystyle S=\emptyset }$ ${\displaystyle \partial _{-}C=\emptyset }$

literature

• Bonahon : Geometric structures on 3-manifolds. Handbook of geometric topology, 93-164, North-Holland, Amsterdam, 2002.
• Bonahon: Cobordism of automorphisms of surfaces. Ann. Sci. École Norm. Sup. (4) 16 (1983) no. 2, 237-270. pdf
• Lackenby, Purcell: Geodesics and compression bodies pdf
• Oertel: Automorphisms of three-dimensional handlebodies. Topology 41 (2002), no. 2, 363-410. pdf