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2013-07-18T01:39:51Z

Created page with "==Definition== Suppose <math>X</math> is a locally connected locally compact Hausdorff space. The '''extended mapping class group''' of <math>..."

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==Definition==

Suppose <math>X</math> is a [[locally connected space|locally connected]] [[locally compact Hausdorff space]]. The '''extended mapping class group''' of <math>X</math>, denoted <math>MCG^*(X)</math>, is defined as the zeroth [[defining ingredient::homeotopy group]] of <math>X</math>.

Explicitly, it is defined as follows: let <math>\operatorname{Homeo}(X)</math> denote the [[defining ingredient::self-homeomorphism group]] of <math>X</math>, viewed as a topological space with the [[compact-open topology]]. With this topology, it becomes a [[T0 topological group]] (see [[Self-homeomorphism group of locally connected locally compact Hausdorff space is a T0 topological group under the compact-open topology|here]]). We define:

<math>MCG^*(X) = \pi_0(\operatorname{Homeo}(X),\mbox{identity map}) = \operatorname{Homeo}(X)/(\mbox{Path component of identity in } \operatorname{Homeo}(X))</math>

Extended mapping class group - Revision history

Vipul: Created page with "==Definition== Suppose X is a locally connected locally compact Hausdorff space. The '''extended mapping class group''' of ..."

Vipul: Created page with "==Definition== Suppose $X$ is a locally connected locally compact Hausdorff space. The '''extended mapping class group''' of $..."$