Glossary of cue sports terms: Difference between revisions

Revision as of 06:53, 16 January 2014

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 $π_{k}$ assign to each path-connected topological space $X$ the group $π_{k} (X)$ of homotopy classes of continuous maps $S^{k} \to X .$

Another construction on a space $X$ is the group of all self-homeomorphisms $X \to X$ , 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 $π_{0} .$

Example

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.

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.

Template:Topology-stub

@@ Line 1: / Line 1: @@
-Jayson Berryhill is how I'm called and my spouse doesn't like it at all. For many years he's been residing in Mississippi and he doesn't plan on changing it. To climb is some thing I truly appreciate performing. He functions as a bookkeeper.<br><br>my web site: [http://www.edmposts.com/build-a-beautiful-organic-garden-using-these-ideas/ online psychic reading]
+{{distinguish|homotopy}}
+In [[algebraic topology]], an area of [[mathematics]], a '''homeotopy group''' of a [[topological space]] is a [[homotopy group]] of the group of [[homeomorphism|self-homeomorphism]]s of that space.
+==Definition==
+The [[homotopy group]] [[functor]]s <math>\pi_k</math> assign to each [[path-connected]] topological space <math>X</math> the group <math>\pi_k(X)</math> of [[homotopy class]]es of continuous maps <math>S^k\to X.</math>
+Another construction on a space <math>X</math> is the [[Homeomorphism group|group of all self-homeomorphisms]] <math>X \to X</math>, denoted <math>{\rm Homeo}(X).</math> If ''X'' is a [[locally compact]], [[locally connected]] [[Hausdorff space]] then a fundamental result of [[R. Arens]] says that <math>{\rm Homeo}(X)</math> will in fact be a [[topological group]] under the [[compact-open topology]].
+Under the above assumptions, the '''homeotopy''' groups for <math>X</math> are defined to be:
+:<math>HME_k(X)=\pi_k({\rm Homeo}(X)).</math>
+Thus <math>HME_0(X)=\pi_0({\rm Homeo}(X))=MCG^*(X)</math> is the '''extended''' [[mapping class group]] for <math>X.</math> In other words, the extended mapping class group is the set of connected components of <math>{\rm Homeo}(X)</math> as specified by the functor <math>\pi_0.</math>
+==Example==
+According to the [[Dehn-Nielsen theorem]], if <math>X</math> is a closed surface then <math>HME_0(X)={\rm Out}(\pi_1(X)),</math> 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.
+[[Category:Algebraic topology]]
+[[Category:Homeomorphisms]]
+{{topology-stub}}

Glossary of cue sports terms: Difference between revisions

Revision as of 06:53, 16 January 2014

Definition

Example

References

Navigation menu

Search