Talk:Homotopy group

It would be quite useful if the maps defining the long exact sequence of a fibration were intuitively defined.

Celso M Doria =================================

Hey, you... I think Homotopy groups is a more appropriate page, and this one can be a redirect, yes?

We need to fix a whole lot of links now, either way.


 * It has been homology group, for a long time. By the way, moving a page does create a redirect.


 * Charles Matthews 12:22, 27 May 2004 (UTC)


 * It's my understanding that the convention is to always use the singular unless there is some compelling reason to use the plural. For example, a compelling reason would be "orthogonal polynomials"; here you must have the plural because the definition of the term doesn't make sense for a single polynomial. In this case, although the homotopy groups are often studied all at once collectively, the object of "a homotopy group" is meaningful and not nonsensical, so I think this is best. Revolver 07:18, 22 Jun 2004 (UTC)

Equivalent spheres in a space are ... what?
From the article:
 * The many different ways to (continuously) map an n-dimensional sphere into a given space are collected into equivalence classes, called homotopy classes. Two mappings are homotopy equivalent if one can be continuously deformed into the other.

(Emphasis added.) Aren't they homotopic, rather than homotopy equivalent? (See Homotopy for definition of homotopy equivalent.)  &mdash;msh210 19:35, 23 Nov 2004 (UTC)


 * Definitely should be homotopic; I have changed it.
 * Neil Strickland (a topologist) 19:44, 23 Nov 2004 (UTC)

Right, maps are homotopic, spaces homotopy equivalent. Charles Matthews 20:14, 23 Nov 2004 (UTC)

Recursive definition
It would be nice to see also the recursive definition via loop space. I am lazy and wiki-ignorant to do it myself :) me 21:02, 7 May 2008 (UTC)

Braid Groups
There was a paper recently published that gave all the homotopy groups of S^2 (and therefore S^3 by the Hopf fibration) in terms of braid groups, but I have neither the link onhand nor the patience of read the whole way through to see if it actually counts as an explicit classification or what. It was linked recently on Baez's site, so it shouldn't be difficult to find if someone more knowledgeable wants to edit the page (which currently states that no classification of pi_n S^m, m > 1 exists). --98.212.191.4 (talk) 20:18, 14 June 2008 (UTC)

Sphere groups
The Hopf fibration should be mentioned as it is useful for calculating homotopy groups of the sphere.

Topology Expert (talk) 13:15, 4 December 2008 (UTC)

Reference to Categories
Reference to categories at the this level of exposition is an unnecessary complication and adds nothing to the exposition. One needs merely say that one can study topological spaces and their functions by associating topological spaces to groups and group homomorphisms. Is the group of integers simpler than the torus? — Preceding unsigned comment added by 86.27.193.180 (talk) 14:23, 31 December 2011 (UTC)

Continuous mappings?
If 1-homotopy is the group that is used to defined simply connected, then supposedly we are talking about the group of continuous mappings from S^1 to X, not just any mappings, right? This is probably implicit to topologists, not to casual readers like myself. DomQ (talk) 15:03, 20 June 2013 (UTC)

Topological spaces with differing homotopy groups are never equivalent (homeomorphic), but the converse is not true.
Isn't there something wrong here? I think this should be other way around as follows:

"All homeomorphism of a topological space have the same homotopy groups but the converse is not true." — Preceding unsigned comment added by 109.204.135.219 (talk) 16:57, 9 August 2013 (UTC)