PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (204)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Medium Entry average rating: No information on entry rating
homeotopy (Definition)

Let $ X$ be a topological Hausdorff space. Let $ {\rm Homeo}(X)$ be the group of homeomorphisms $ X\to X$, which can be also turn into a topological space by means of the compact-open topology. And let $ \pi_k$ be the k-th homotopy group functor.

Then the k-th homeotopy is defined as:

$\displaystyle {\cal{H}}_k(X)=\pi_k({\rm Homeo}(X))$
that is, the group of homotopy classes of maps $ S^k\to {\rm Homeo}(X)$. Which is different from $ \pi_k(X)$, the group of homotopy classes of maps $ S^k\to X$.

One important result for any low dimensional topologist is that for a surface $ F$

$\displaystyle {\cal{H}}_0(F)={\rm Out}(\pi_1(F))$
which is the $ F$'s extended mapping class group.

Reference

G.S. McCarty, Homeotopy groups, Trans. A.M.S. 106(1963)293-304.



"homeotopy" is owned by juanman. [ full author list (2) ]
(view preamble)

View style:

See Also: isotopy, group, homeomorphism, group, isotopy, homeomorphism

Other names:  mapping class group
Keywords:  homotopy group functor
Log in to rate this entry.
(view current ratings)

Cross-references: surface, maps, classes, homotopy, functor, homotopy group, compact-open topology, topological space, homeomorphisms, group, Hausdorff space
There are 2 references to this entry.

This is version 14 of homeotopy, born on 2006-02-20, modified 2006-12-24.
Object id is 7642, canonical name is Homeotopy.
Accessed 1529 times total.

Classification:
AMS MSC: 20F38 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Other groups related to topology or analysis)
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)