Homeotopy group: Difference between revisions

Latest revision as of 07:14, 1 June 2016

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 $MCG^{*}(X)$ .

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 Suppose <math>X</math> is a [[locally connected space|locally connected]] [[locally compact Hausdorff space]] and <math>k</math> is a positive integer. Denote by <math>\operatorname{Homeo}(X)</math> the [[self-homeomorphism group]] of <math>X</math>, given the structure of a topological space via the [[compact-open topology]]. <math>\operatorname{Homeo}(X)</math> becomes a [[T0 topological group]] under this topology (see [[self-homeomorphism group of locally connected locally compact Hausdorff space is a T0 topological group under the compact-open topology|here]]).
-The <math>k^{th}</math> '''homeotopy group''' of <math>X</math>, denoted <math>HME_k(X)</math>, is defined as the <math>k^{th}</math> [[homotopy group]] of <math>\operatorname{Homeo}(X)</math>. Note that since <math>X</math> is a topological ''group'', even the case <math>k = 0</math> gives a ''group'', and the case <math>k \ge 1</math> gives an [[abelian group]]. Explicitly:
+The <math>k^{th}</math> '''homeotopy group''' of <math>X</math>, denoted <math>HME_k(X)</math>, is defined as the <math>k^{th}</math> [[homotopy group]] of <math>\operatorname{Homeo}(X)</math>. Note that since <math>\operatorname{Homeo}(X)</math> is a topological ''group'', even the case <math>k = 0</math> gives a ''group'', and the case <math>k \ge 1</math> gives an [[abelian group]]. Explicitly:
 <math>HME_k(X) = \pi_k(\operatorname{Homeo}(X);\mbox{identity map})</math>
 The special case <math>k = 0</math> gives the group <math>HME_0(X)</math>. This group is also called the [[extended mapping class group]] of <math>X</math> and is denoted <math>MCG^*(X)</math>.