Vipul at 23:51, 19 December 2010

2010-12-19T23:51:55Z

← Older revision		Revision as of 23:51, 19 December 2010
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	==Definition==		==Definition==

	===As a topological space===		===As a (based) topological space===
	Suppose <math>(X,x_0)</math> is a [[based topological space]], i.e., <math>X</math> is a [[topological space]] and <math>x_0</math> is a point in <math>X</math>. The '''loop space''' of <math>(X,x_0)</math>, denoted <math>\Omega(X,x_0)</math>, is defined as follows:		Suppose <math>(X,x_0)</math> is a [[based topological space]], i.e., <math>X</math> is a [[topological space]] and <math>x_0</math> is a point in <math>X</math>. The '''loop space''' of <math>(X,x_0)</math>, denoted <math>\Omega(X,x_0)</math>, is defined as follows:

	* As a set, it is the set of all [[continuous map]]s from the based [[unit circle]] <math>S^1</math> (i.e., the unit circle with a fixed basepoint) to <math>(X,x_0)</math>. In other words, all these maps send the chosen basepoint of <math>S^1</math> to <math>x_0</math>.		* As a set, it is the set of all [[continuous map]]s from the based [[unit circle]] <math>S^1</math> (i.e., the unit circle with a fixed basepoint) to <math>(X,x_0)</math>. In other words, all these maps send the chosen basepoint of <math>S^1</math> to <math>x_0</math>.
	* The topology on this set is the [[compact-open topology]] (or rather, the [[subspace topology]] from the compact-open topology on all continuous maps from <math>S^1</math> to <math>X</math>).		* The topology on this set is the [[compact-open topology]] (or rather, the [[subspace topology]] from the compact-open topology on all continuous maps from <math>S^1</math> to <math>X</math>).
			* The basepoint of this topological space is chosen as the ''constant'' loop that stays at the point <math>x_0</math>.

	==As a H-space===		==As a H-space===
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	The loop space <math>\Omega(X,x_0)</math> admits a multiplicative structure by ''concatenation and reparametrization'', where, for loops <math>f_1</math> and <math>f_2</math>, we define <math>f_1 *f_2</math> as the loop:		The loop space <math>\Omega(X,x_0)</math> admits a multiplicative structure by ''concatenation and reparametrization'', where, for loops <math>f_1</math> and <math>f_2</math>, we define <math>f_1 *f_2</math> as the loop:

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

	This multiplicative structure is continuous, making the loop space a [[topological magma]]. However, it is ''not'' a [[topological monoid]], because the multiplication is not strictly associative and does not have a strict identity element. Instead, it is a [[H-space]], in the sense that the multiplication is associative up to homotopy and there is an element that works as an identity element up to homotopy:		This multiplicative structure is continuous, making the loop space a [[topological magma]]. However, it is ''not'' a [[topological monoid]], because the multiplication is not strictly associative and does not have a strict identity element. Instead, it is a [[H-space]], in the sense that the multiplication is associative up to homotopy and there is an element that works as an identity element up to homotopy:
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	What is important is not just that there exist individual homotopies for the associativity of each triple, but that these homotopies vary continuously, so that we get a homotopy at the level of the topological space <math>\Omega(X,x_0)</math>.		What is important is not just that there exist individual homotopies for the associativity of each triple, but that these homotopies vary continuously, so that we get a homotopy at the level of the topological space <math>\Omega(X,x_0)</math>.

			==Relation with other constructs==

			===Relationship with fundamental group===

			The [[fundamental group]] of a based topological space <math>(X,x_0)</math>, denoted <math>\pi_1(X,x_0)</math>, can be identified with the set of [[path component]]s of <math>Omega(X,x_0)</math>. In other words:

			<math>\! \pi_1(X,x_0) \cong \pi_0(\Omega(X,x_0))</math>

			The identification is as follows: we know that the elements of <math>\pi_1(X,x_0)</math> are precisely the homotopy classes of loops in <math>X</math> based at <math>x_0</math>. The ''homotopy classes'', in turn, are precisely the path components of <math>\Omega(X,x_0)</math>, because a homotopy of based loops is a path in the space of based loops under the compact-open topology.

			In addition to the identification as a ''set'', we can also make the identification as a group. The left side has a group structure under concatenation. On the right side, the [[H-space]] structure of <math>\Omega(X,x_0)</math> induces a monoid structure on the space <math>\pi_0(\Omega(X,x_0))</math> of its path components. That monoid turns out to be a group, and the identification is a group isomorphism.

			===Relationship with higher homotopy groups===

			More generally, we have the following relationship:

			<math>\! \pi_k(X,x_0) \cong \pi_{k-1}(\Omega(X,x_0))</math>

			In fact, for <math>k \ge 2</math>, both sides are naturally [[abelian group]]s, and the natural identification is an isomorphism of abelian groups.

			===Iterated loop spaces===

			We can also consider the iterated loop space <math>\Omega^k(X,x_0)</math>. This is obtained by iterating the [[loop space]] construction <math>k</math> times. Note that at each stage, the new basepoint is chosen as the ''constant'' loop taking the old basepoint as its value.

Vipul: Created page with '==Definition== ===As a topological space=== Suppose $(X,x_0)$ is a based topological space, i.e., $X$ is a topological space and $x_0$ i...'

2010-12-19T23:31:41Z

Created page with '==Definition== ===As a topological space=== Suppose <math>(X,x_0)</math> is a based topological space, i.e., <math>X</math> is a topological space and <math>x_0</math> i...'

New page

==Definition==

===As a topological space===
Suppose <math>(X,x_0)</math> is a [[based topological space]], i.e., <math>X</math> is a [[topological space]] and <math>x_0</math> is a point in <math>X</math>. The '''loop space''' of <math>(X,x_0)</math>, denoted <math>\Omega(X,x_0)</math>, is defined as follows:

* As a set, it is the set of all [[continuous map]]s from the based [[unit circle]] <math>S^1</math> (i.e., the unit circle with a fixed basepoint) to <math>(X,x_0)</math>. In other words, all these maps send the chosen basepoint of <math>S^1</math> to <math>x_0</math>.
* The topology on this set is the [[compact-open topology]] (or rather, the [[subspace topology]] from the compact-open topology on all continuous maps from <math>S^1</math> to <math>X</math>).

==As a H-space===

For convenience, we treat the unit circle as the quotient of the [[closed unit interval]] <math>[0,1]</math> under the identification of <math>0</math> and <math>1</math>, and the identified <math>0 \sim 1</math> point is treated as the basepoint.

The loop space <math>\Omega(X,x_0)</math> admits a multiplicative structure by ''concatenation and reparametrization'', where, for loops <math>f_1</math> and <math>f_2</math>, we define <math>f_1 *f_2</math> as the loop:

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

This multiplicative structure is continuous, making the loop space a [[topological magma]]. However, it is ''not'' a [[topological monoid]], because the multiplication is not strictly associative and does not have a strict identity element. Instead, it is a [[H-space]], in the sense that the multiplication is associative up to homotopy and there is an element that works as an identity element up to homotopy:

{| class="sortable" border="1"
! Condition !! Proof
|-
| Associativity up to homotopy || [[homotopy between composites associated in different ways]]
|-
| Identity element up to homotopy || [[homotopy between loop and composite with constant loop]]
|}

What is important is not just that there exist individual homotopies for the associativity of each triple, but that these homotopies vary continuously, so that we get a homotopy at the level of the topological space <math>\Omega(X,x_0)</math>.

Loop space of a based topological space - Revision history

Vipul at 23:51, 19 December 2010

Vipul: Created page with '==Definition== ===As a topological space=== Suppose (X,x_0) is a based topological space, i.e., X is a topological space and x_0 i...'

Vipul: Created page with '==Definition== ===As a topological space=== Suppose $(X,x_0)$ is a based topological space, i.e., $X$ is a topological space and $x_0$ i...'