Homeotopy group

Definition

Suppose $X$ is a locally connected locally compact Hausdorff space and $k$ is a positive integer. Denote by $\operatorname {Homeo} (X)$ the self-homeomorphism group of $X$, given the structure of a topological space via the compact-open topology. $\operatorname {Homeo} (X)$ becomes a T0 topological group under this topology (see here).

The $k^{th}$ homeotopy group of $X$, denoted $HME_{k}(X)$, is defined as the $k^{th}$ homotopy group of $\operatorname {Homeo} (X)$. Note that since $\operatorname {Homeo} (X)$ is a topological group, even the case $k=0$ gives a group, and the case $k\geq 1$ gives an abelian group. Explicitly:

$HME_{k}(X)=\pi _{k}(\operatorname {Homeo} (X);{\mbox{identity map}})$

The special case $k=0$ gives the group $HME_{0}(X)$. This group is also called the extended mapping class group of $X$ and is denoted $MCG^{*}(X)$.