In algebraic topology, an area of mathematics, a homeotopy group of a topological space is a homotopy group of the group of self-homeomorphisms of that space.
== Definition ==
The homotopy group functors
π
k
{\displaystyle \pi _{k}}
assign to each path-connected topological space
X
{\displaystyle X}
the group
π
k
(
X
)
{\displaystyle \pi _{k}(X)}
of homotopy classes of continuous maps
S
k
→
X
.
{\displaystyle S^{k}\to X.}
Another construction on a space
X
{\displaystyle X}
is the group of all self-homeomorphisms
X
→
X
{\displaystyle X\to X}
, denoted
H
o
m
e
o
(
X
)
.
{\displaystyle {\rm {Homeo}}(X).}
If X is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that
H
o
m
e
o
(
X
)
{\displaystyle {\rm {Homeo}}(X)}
will in fact be a topological group under the compact-open topology.
Under the above assumptions, the homeotopy groups for
X
{\displaystyle X}
are defined to be:
H
M
E
k
(
X
)
=
π
k
(
H
o
m
e
o
(
X
)
)
.
{\displaystyle HME_{k}(X)=\pi _{k}({\rm {Homeo}}(X)).}
Thus
H
M
E
0
(
X
)
=
π
0
(
H
o
m
e
o
(
X
)
)
=
M
C
G
∗
(
X
)
{\displaystyle HME_{0}(X)=\pi _{0}({\rm {Homeo}}(X))=MCG^{*}(X)}
is the mapping class group for
X
.
{\displaystyle X.}
In other words, the mapping class group is the set of connected components of
H
o
m
e
o
(
X
)
{\displaystyle {\rm {Homeo}}(X)}
as specified by the functor
π
0
.
{\displaystyle \pi _{0}.}
== Example ==
According to the Dehn-Nielsen theorem, if
X
{\displaystyle X}
is a closed surface then
H
M
E
0
(
X
)
=
O
u
t
(
π
1
(
X
)
)
,
{\displaystyle HME_{0}(X)={\rm {Out}}(\pi _{1}(X)),}
the outer automorphism group of its fundamental group.
== References ==
G.S. McCarty. Homeotopy groups. Trans. A.M.S. 106(1963)293-304.
R. Arens, Topologies for homeomorphism groups, Amer. J. Math. 68 (1946), 593–610.