[<< wikipedia] Homeotopy
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.