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
G
{\ displaystyle g}
is obtained by attaching disjoint handles to a 3-dimensional full sphere .
G
{\ 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
B.
3
{\ displaystyle B ^ {3}}
f
1
,
...
,
f
G
:
B.
2
×
{
0
,
1
}
→
∂
B.
3
{\ displaystyle f_ {1}, \ ldots, f_ {g}: B ^ {2} \ times \ left \ {0,1 \ right \} \ rightarrow \ partial B ^ {3}}
H
G
{\ displaystyle H_ {g}}
H
G
: =
(
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3
⋃
∪
i
=
1
G
(
B.
2
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[
0
,
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)
i
)
/
∼
{\ displaystyle H_ {g}: = (B ^ {3} \ bigcup \ cup _ {i = 1} ^ {g} (B ^ {2} \ times \ left [0,1 \ right]) _ {i }) / \ sim}
under the equivalence relation for .
x
∼
f
i
(
x
)
{\ displaystyle x \ sim f_ {i} (x)}
x
∈
(
B.
2
×
{
0
,
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}
)
i
,
i
=
1
,
...
,
G
{\ displaystyle x \ in (B ^ {2} \ times \ left \ {0,1 \ right \}) _ {i}, i = 1, \ ldots, g}
H
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 .
G
{\ displaystyle g}
G
=
0
{\ displaystyle g = 0}
G
=
2
{\ displaystyle g = 2}
: Full pretzel
G
=
3
{\ 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 .
C.
{\ displaystyle C}
S.
×
[
0
,
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{\ displaystyle S \ times \ left [0,1 \ right]}
S.
{\ displaystyle S}
S.
×
{
1
}
{\ displaystyle S \ times \ left \ {1 \ right \}}
∂
-
C.
: =
S.
×
{
0
}
{\ displaystyle \ partial _ {-} C: = S \ times \ left \ {0 \ right \}}
∂
+
C.
: =
∂
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∖
∂
-
C.
{\ displaystyle \ partial _ {+} C: = \ partial C \ setminus \ partial _ {-} C}
Handle body is obtained for , in this case it is .
S.
=
∅
{\ displaystyle S = \ emptyset}
∂
-
C.
=
∅
{\ 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
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