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.
The homotopy group functors
π
k
assign to each path-connected topological space
X
the group
π
k
(
X
)
of homotopy classes of continuous maps
S
k
→
X
.
Another construction on a space
X
is the group of all self-homeomorphisms
X
→
X
, denoted
H
o
m
e
o
(
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
)
will in fact be a topological group under the compact-open topology.
Under the above assumptions, the homeotopy groups for
X
are defined to be:
H
M
E
k
(
X
)
=
π
k
(
H
o
m
e
o
(
X
)
)
.
Thus
H
M
E
0
(
X
)
=
π
0
(
H
o
m
e
o
(
X
)
)
=
M
C
G
∗
(
X
)
is the extended mapping class group for
X
.
In other words, the extended mapping class group is the set of connected components of
H
o
m
e
o
(
X
)
as specified by the functor
π
0
.
According to the Dehn-Nielsen theorem, if
X
is a closed surface then
H
M
E
0
(
X
)
=
O
u
t
(
π
1
(
X
)
)
,
the outer automorphism group of its fundamental group.
