Homotopy group - Revision history

Vipul at 03:47, 31 December 2010

2010-12-31T03:47:54Z

← Older revision		Revision as of 03:47, 31 December 2010
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	* Two maps <math>f_1,f_2</math> are composed as follows. <math>S^n \setminus \{ p \}</math> is identified with the open northern and open southern hemisphere of a new sphere via homeomorphic identifications <math>\varphi_1</math> and <math>\varphi_2</math> from these hemispheres to <math>S^n \setminus \{ p \}</math> (these identifications are universally fixed, independent of <math>X</math>; there's a natural choice for them). The composite map is now defined as follows: as <math>f_1 \circ \varphi_1</math> on the northern hemisphere, as <math>f_2 \circ \varphi_2</math> on the southern hemisphere, and as the constant map to <math>p</math> on the equator. The basepoint is a fixed point on the equator (again, this choice is independent of <math>X</math> and is universally fixed).		* Two maps <math>f_1,f_2</math> are composed as follows. <math>S^n \setminus \{ p \}</math> is identified with the open northern and open southern hemisphere of a new sphere via homeomorphic identifications <math>\varphi_1</math> and <math>\varphi_2</math> from these hemispheres to <math>S^n \setminus \{ p \}</math> (these identifications are universally fixed, independent of <math>X</math>; there's a natural choice for them). The composite map is now defined as follows: as <math>f_1 \circ \varphi_1</math> on the northern hemisphere, as <math>f_2 \circ \varphi_2</math> on the southern hemisphere, and as the constant map to <math>p</math> on the equator. The basepoint is a fixed point on the equator (again, this choice is independent of <math>X</math> and is universally fixed).

			The multiplication defined above can be viewed as arising from the corresponding [[comultiplication]] on the <math>n</math>-sphere <math>S^n</math>, because of the contravariant nature of ''maps from''.
	===Proof that this gives a group===		===Proof that this gives a group===

Vipul at 02:56, 24 December 2010

2010-12-24T02:56:06Z

← Older revision		Revision as of 02:56, 24 December 2010
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	However, knowledge of the homotopy groups does not determine the homotopy type, or even the ''weak'' homotopy type, of the topological space.		However, knowledge of the homotopy groups does not determine the homotopy type, or even the ''weak'' homotopy type, of the topological space.

			==Facts==

			* For a [[H-space]], all homotopy groups, including the fundamental group, are [[abelian group]]s. This is a consequence of the [[Eckmann-Hilton principle]]. In fact, the group operation coincides with the operation induced by pointwise multiplication of loops (when we go down to homotopy classes). Also, <math>\pi_0</math> gets the structure of a (not necessarily abelian) group. {{further\|[[fundamental group of H-space is abelian]]}}
			* For any topological space, all higher homotopy groups, i.e., all <math>\pi_n, n \ge 2</math>, are [[abelian group]]s. This can be viewed as a consequence of the Eckmann-Hilton principle, or the fact for <math>n \ge 2</math>, <math>S^n</math> can be ''rotated'' about any equatorial axis to interchange the roles of the north and the south poles. {{further\|[[higher homotopy groups are abelian]]}}

Vipul: /* As homotopy classes of based maps */

2010-12-24T02:49:12Z

As homotopy classes of based maps

← Older revision		Revision as of 02:49, 24 December 2010
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	* Consider the based <math>n</math>-[[sphere]] <math>(S^n,p)</math> where <math>p</math> is a chosen basepoint. As a set, <math>\pi_n(X,x_0)</math> is the set of homotopy classes of all based maps from <math>(S^n,p)</math> to <math>(X,x_0)</math>, where the ''homotopy classes'' are with respect to homotopies that preserve basepoints.		* Consider the based <math>n</math>-[[sphere]] <math>(S^n,p)</math> where <math>p</math> is a chosen basepoint. As a set, <math>\pi_n(X,x_0)</math> is the set of homotopy classes of all based maps from <math>(S^n,p)</math> to <math>(X,x_0)</math>, where the ''homotopy classes'' are with respect to homotopies that preserve basepoints.
	* Two maps <math>f_1,f_2</math> are composed as follows. <math>S^n \setminus \{ p \}</math> is identified with the open northern and open southern hemisphere of a new sphere via homeomorphic identifications <math>\varphi_1</math> and <math>\varphi_2</math> from these hemispheres to <math>S^ \setminus \{ p \}</math> (these identifications are universally fixed, independent of <math>X</math>; there's a natural choice for them). The composite map is now defined as follows: as <math>f_1 \circ \varphi_1</math> on the northern hemisphere, as <math>f_2 \circ \varphi_2</math> on the southern hemisphere, and as the constant map to <math>p</math> on the equator. The basepoint is a fixed point on the equator (again, this choice is independent of <math>X</math> and is universally fixed).		* Two maps <math>f_1,f_2</math> are composed as follows. <math>S^n \setminus \{ p \}</math> is identified with the open northern and open southern hemisphere of a new sphere via homeomorphic identifications <math>\varphi_1</math> and <math>\varphi_2</math> from these hemispheres to <math>S^n \setminus \{ p \}</math> (these identifications are universally fixed, independent of <math>X</math>; there's a natural choice for them). The composite map is now defined as follows: as <math>f_1 \circ \varphi_1</math> on the northern hemisphere, as <math>f_2 \circ \varphi_2</math> on the southern hemisphere, and as the constant map to <math>p</math> on the equator. The basepoint is a fixed point on the equator (again, this choice is independent of <math>X</math> and is universally fixed).

	===Proof that this gives a group===		===Proof that this gives a group===

Vipul: /* The case n = 1 */

2010-12-24T02:48:51Z

The case n = 1

← Older revision		Revision as of 02:48, 24 December 2010
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	In this case, we get the [[fundamental group]] <math>\pi_1(X,x_0)</math>. Recall that for <math>f_1,f_2</math> based maps from the [[circle]] to <math>(X,x_0)</math>, we think of <math>f_1,f_2</math> as maps from <math>[0,1]</math> to <math>X</math> with <math>f_1(0) = f_1(1) = f_2(0) = f_2(1) = x_0</math>. The usual way of composing is to define:		In this case, we get the [[fundamental group]] <math>\pi_1(X,x_0)</math>. Recall that for <math>f_1,f_2</math> based maps from the [[circle]] to <math>(X,x_0)</math>, we think of <math>f_1,f_2</math> as maps from <math>[0,1]</math> to <math>X</math> with <math>f_1(0) = f_1(1) = f_2(0) = f_2(1) = x_0</math>. The usual way of composing is to define:

	<math>(f_1 * f_2)(t) := \lbrace \begin{array} f_1(2t), & 0 \le t \le 1/2 \\ f_2(2t - 1), & 1/2 < t \le 1 \\\end{array}</math>		<math>(f_1 * f_2)(t) := \lbrace \begin{array}{rl} f_1(2t), & 0 \le t \le 1/2 \\ f_2(2t - 1), & 1/2 < t \le 1 \\\end{array}</math>

	Here, the definition on <math>(0,1/2)</math> can be viewed as the ''northern hemisphere'' definition, the definition on <math>(1/2,1)</math> can be viewed as the ''southern hemisphere'' definition, with the ''equator'' corresponding to the two points <math>1/2</math> and <math>\! 0 \sim 1</math>, of which we choose the latter as basepoint.		Here, the definition on <math>(0,1/2)</math> can be viewed as the ''northern hemisphere'' definition, the definition on <math>(1/2,1)</math> can be viewed as the ''southern hemisphere'' definition, with the ''equator'' corresponding to the two points <math>1/2</math> and <math>\! 0 \sim 1</math>, of which we choose the latter as basepoint.

Vipul: Created page with '==Definition== ===As homotopy classes of based maps=== Suppose $(X,x_0)$ is a defining ingredient::based topological space and $n$ is a positive integ...'

2010-12-24T02:38:50Z

Created page with '==Definition== ===As homotopy classes of based maps=== Suppose <math>(X,x_0)</math> is a defining ingredient::based topological space and <math>n</math> is a positive integ...'

New page

==Definition==

===As homotopy classes of based maps===

Suppose <math>(X,x_0)</math> is a [[defining ingredient::based topological space]] and <math>n</math> is a positive integer. The '''homotopy group''' <math>\pi_n(X,x_0)</math> is defined as follows:

* Consider the based <math>n</math>-[[sphere]] <math>(S^n,p)</math> where <math>p</math> is a chosen basepoint. As a set, <math>\pi_n(X,x_0)</math> is the set of homotopy classes of all based maps from <math>(S^n,p)</math> to <math>(X,x_0)</math>, where the ''homotopy classes'' are with respect to homotopies that preserve basepoints.
* Two maps <math>f_1,f_2</math> are composed as follows. <math>S^n \setminus \{ p \}</math> is identified with the open northern and open southern hemisphere of a new sphere via homeomorphic identifications <math>\varphi_1</math> and <math>\varphi_2</math> from these hemispheres to <math>S^ \setminus \{ p \}</math> (these identifications are universally fixed, independent of <math>X</math>; there's a natural choice for them). The composite map is now defined as follows: as <math>f_1 \circ \varphi_1</math> on the northern hemisphere, as <math>f_2 \circ \varphi_2</math> on the southern hemisphere, and as the constant map to <math>p</math> on the equator. The basepoint is a fixed point on the equator (again, this choice is independent of <math>X</math> and is universally fixed).

===Proof that this gives a group===

===The case <math>n = 0</math>===

The definition of homotopy group still gives a ''set'' definition for <math>n = 0</math>. <math>S^0</math> is a two-point space, and one of these points must go to a fixed basepoint, while the other can go anywhere. Thus, the set of all based maps is the set of points in <math>X</math>, and the set of homotopy classes is the set of path components. Thus, <math>\pi_0(X,x_0)</math> is the [[set of path components]] in <math>X</math>. Note that it is independent of <math>x_0</math> because <matH>S^0</math> being discrete, the image of the basepoint does not affect where the other point goes.

However, the composition operation does not make sense for <math>n = 0</math>, because <math>S^0</math> has an ''empty'' equator. Hence, <math>\pi_0</math> is ''only a set'' and has no group structure for arbitary topological spaces. (It does have a group structure when the topological space is a [[H-space]], induced by the multiplication in the topological space).

===The case <math>n = 1</math>===

In this case, we get the [[fundamental group]] <math>\pi_1(X,x_0)</math>. Recall that for <math>f_1,f_2</math> based maps from the [[circle]] to <math>(X,x_0)</math>, we think of <math>f_1,f_2</math> as maps from <math>[0,1]</math> to <math>X</math> with <math>f_1(0) = f_1(1) = f_2(0) = f_2(1) = x_0</math>. The usual way of composing is to define:

<math>(f_1 * f_2)(t) := \lbrace \begin{array} f_1(2t), & 0 \le t \le 1/2 \\ f_2(2t - 1), & 1/2 < t \le 1 \\\end{array}</math>

Here, the definition on <math>(0,1/2)</math> can be viewed as the ''northern hemisphere'' definition, the definition on <math>(1/2,1)</math> can be viewed as the ''southern hemisphere'' definition, with the ''equator'' corresponding to the two points <math>1/2</math> and <math>\! 0 \sim 1</math>, of which we choose the latter as basepoint.

===Omission of basepoint===

For a [[path-connected space]], the homotopy groups <math>\pi_n</math> for all basepoints are isomorphic. In fact, any choice of path between two points can be used to define an isomorphism between the <math>\pi_n</math>s at these basepoints. The key fact that we need to use here is that the inclusion of a point in <math>S^n</math> is a [[cofibration]] (which is easily seen by noting that <math>S^n</math> is the boundary of <math>D^n</math>, or more generally from the fact that [[manifold implies nondegenerate]]).

In general, the homotopy group <math>\pi_n</math> may differ for different path components. For a [[homogeneous space]], or more generally for a space where all the path components are homeomorphic, the isomorphism class of <math>\pi_n</math> does not depend upon the choice of basepoint.

===Dependence on homotopy type===

The homotopy groups <math>\pi_n</math> depend only on the homotopy type of the [[based topological space]]. In fact, they depend only on the homotopy type of the path component of the basepoint in the topological space.

However, knowledge of the homotopy groups does not determine the homotopy type, or even the ''weak'' homotopy type, of the topological space.

Homotopy group - Revision history

Vipul at 03:47, 31 December 2010

Vipul at 02:56, 24 December 2010

Vipul: /* As homotopy classes of based maps */

Vipul: /* The case n = 1 */

Vipul: Created page with '==Definition== ===As homotopy classes of based maps=== Suppose (X,x_0) is a defining ingredient::based topological space and n is a positive integ...'

Vipul: Created page with '==Definition== ===As homotopy classes of based maps=== Suppose $(X,x_0)$ is a defining ingredient::based topological space and $n$ is a positive integ...'