Fundamental group - Revision history

Vipul: /* Proof that this gives a group structure */

2011-01-10T16:45:46Z

Proof that this gives a group structure

← Older revision		Revision as of 16:45, 10 January 2011
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	\| well defined \|\| if <math>f_1</math> and <math>g_1</math> are homotopic to each other, and <math>f_2</math> and <math>g_2</math> are homotopic to each other, then <math>f_1 * f_2</math> and <math>g_1 * g_2</math> are homotopic to each other. \|\| [[homotopy between composites of homotopic loops]]		\| well defined \|\| if <math>f_1</math> and <math>g_1</math> are homotopic to each other, and <math>f_2</math> and <math>g_2</math> are homotopic to each other, then <math>f_1 * f_2</math> and <math>g_1 * g_2</math> are homotopic to each other. \|\| [[homotopy between composites of homotopic loops]]
	\|-		\|-
	\| existence of identity element \|\| the identity element is the homotopy constant loop that stays at the base point. In other words, if we denote this loop by <math>e</math>, then for any loop <math>f</math>, the composite <math>e * f</math> is homotopic to <math>f</math>, and so is the composite <math>f * e</math>. \|\| [[homotopy between loop and composite with constant loop]]		\| existence of identity element \|\| the identity element is the homotopy class of the constant loop that stays at the base point. In other words, if we denote this loop by <math>e</math>, then for any loop <math>f</math>, the composite <math>e * f</math> is homotopic to <math>f</math>, and so is the composite <math>f * e</math>. \|\| [[homotopy between loop and composite with constant loop]]
	\|-		\|-
	\| existence of inverses \|\| the inverse of a loop <math>f</math> is the loop <math>t \mapsto f(1 - t)</math>, i.e., the same loop done backward. In other words the composite of <math>f</math> and this loop is homotopic to the constant loop. \|\| [[homotopy between constant loop and composite of loop with inverse]]		\| existence of inverses \|\| the inverse of a loop <math>f</math> is the loop <math>t \mapsto f(1 - t)</math>, i.e., the same loop done backward. In other words the composite of <math>f</math> and this loop is homotopic to the constant loop. \|\| [[homotopy between constant loop and composite of loop with inverse]]

Vipul: /* Relation with other constructs */

2011-01-10T16:43:36Z

Relation with other constructs

← Older revision		Revision as of 16:43, 10 January 2011
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	\|-		\|-
	\| All loops based at <math>x_0</math>, ''not'' up to homotopy \|\| [[loop space of a based topological space]] \|\| <math>\Omega(X,x_0)</math> \|\| <math>\pi_1(X,x_0) = \pi_0(\Omega(X,x_0))</math>		\| All loops based at <math>x_0</math>, ''not'' up to homotopy \|\| [[loop space of a based topological space]] \|\| <math>\Omega(X,x_0)</math> \|\| <math>\pi_1(X,x_0) = \pi_0(\Omega(X,x_0))</math>
	\|-
	~~\| All loops up to homotopies, (intermediaries must also be loops, but basepoint could vary) \|\| [[loop space of a topological space]] \|\| <math>\Omega(X)</math> \|\|~~
	\|-		\|-
	\| All loops based at <math>x_0</math>, up to homotopies where the intermediaries must be loops but need not be based at <math>x_0</math> \|\| [[conjugacy class set of fundamental group]] \|\| ? \|\| The set of conjugacy classes of <math>\pi_1(X,x_0)</math>. See also [[actions of the fundamental group]]		\| All loops based at <math>x_0</math>, up to homotopies where the intermediaries must be loops but need not be based at <math>x_0</math> \|\| [[conjugacy class set of fundamental group]] \|\| ? \|\| The set of conjugacy classes of <math>\pi_1(X,x_0)</math>. See also [[actions of the fundamental group]]

Vipul: /* Relation with other constructs */

2011-01-08T22:15:58Z

Relation with other constructs

← Older revision		Revision as of 22:15, 8 January 2011
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	\| All loops based at <math>x_0</math>, ''not'' up to homotopy \|\| [[loop space of a based topological space]] \|\| <math>\Omega(X,x_0)</math> \|\| <math>\pi_1(X,x_0) = \pi_0(\Omega(X,x_0))</math>		\| All loops based at <math>x_0</math>, ''not'' up to homotopy \|\| [[loop space of a based topological space]] \|\| <math>\Omega(X,x_0)</math> \|\| <math>\pi_1(X,x_0) = \pi_0(\Omega(X,x_0))</math>
	\|-		\|-
	\| All loops up to homotopies, (intermediaries must also be loops, but basepoint could vary) \|\| [[loop space of a topological space]] \|\| <math>\Omega(X~~,x_0~~)</math>		\| All loops up to homotopies, (intermediaries must also be loops, but basepoint could vary) \|\| [[loop space of a topological space]] \|\| <math>\Omega(X)</math> \|\|
	\|-		\|-
	\| All loops based at <math>x_0</math>, up to homotopies where the intermediaries must be loops but need not be based at <math>x_0</math> \|\| [[conjugacy class set of fundamental group]] \|\| ? \|\| The set of conjugacy classes of <math>\pi_1(X,x_0)</math>. See also [[actions of the fundamental group]]		\| All loops based at <math>x_0</math>, up to homotopies where the intermediaries must be loops but need not be based at <math>x_0</math> \|\| [[conjugacy class set of fundamental group]] \|\| ? \|\| The set of conjugacy classes of <math>\pi_1(X,x_0)</math>. See also [[actions of the fundamental group]]

Vipul: /* Proof that this gives a group structure */

2011-01-08T22:06:09Z

Proof that this gives a group structure

← Older revision		Revision as of 22:06, 8 January 2011
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	\| existence of identity element \|\| the identity element is the homotopy constant loop that stays at the base point. In other words, if we denote this loop by <math>e</math>, then for any loop <math>f</math>, the composite <math>e * f</math> is homotopic to <math>f</math>, and so is the composite <math>f * e</math>. \|\| [[homotopy between loop and composite with constant loop]]		\| existence of identity element \|\| the identity element is the homotopy constant loop that stays at the base point. In other words, if we denote this loop by <math>e</math>, then for any loop <math>f</math>, the composite <math>e * f</math> is homotopic to <math>f</math>, and so is the composite <math>f * e</math>. \|\| [[homotopy between loop and composite with constant loop]]
	\|-		\|-
	\| existence of inverses \|\| the ~~inverses~~ of a loop <math>f</math> is the loop <math>t \mapsto f(1 - t)</math>, i.e., the same loop done backward. In other words the composite of <math>f</math> and this loop is homotopic to the constant loop. \|\| [[homotopy between constant loop and composite of loop with inverse]]		\| existence of inverses \|\| the inverse of a loop <math>f</math> is the loop <math>t \mapsto f(1 - t)</math>, i.e., the same loop done backward. In other words the composite of <math>f</math> and this loop is homotopic to the constant loop. \|\| [[homotopy between constant loop and composite of loop with inverse]]
	\|-		\|-
	\| associativity \|\| for loops <math>f_1, f_2, f_3</math>, the composite <math>f_1 * (f_2 * f_3)</math> is homotopic to the composite <math>(f_1 * f_2) * f_3</math>. \|\| [[homotopy between composites associated in different ways]]		\| associativity \|\| for loops <math>f_1, f_2, f_3</math>, the composite <math>f_1 * (f_2 * f_3)</math> is homotopic to the composite <math>(f_1 * f_2) * f_3</math>. \|\| [[homotopy between composites associated in different ways]]

Vipul: /* Relation with other constructs */

2010-12-24T02:07:20Z

Relation with other constructs

← Older revision		Revision as of 02:07, 24 December 2010
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	\|-		\|-
	\| All loops based at <math>x_0</math>, up to ''slice and rearrange'' as well as homotopies where the intermediates must be loops but need not be based at <math>x_0</math>. In other words, ''homology classes'' \|\| [[first homology group]] (if <math>X</math> is path-connected) \|\| <math>H_1(X,x_0)</math> or just <math>H_1(X)</math> \|\| For a path-connected space, the first homology group is the abelianization of the fundamental group.		\| All loops based at <math>x_0</math>, up to ''slice and rearrange'' as well as homotopies where the intermediates must be loops but need not be based at <math>x_0</math>. In other words, ''homology classes'' \|\| [[first homology group]] (if <math>X</math> is path-connected) \|\| <math>H_1(X,x_0)</math> or just <math>H_1(X)</math> \|\| For a path-connected space, the first homology group is the abelianization of the fundamental group.
			\|-
			\| All paths up to homotopies that preserve endpoints of paths \|\| [[fundamental groupoid]] \|\| ? \|\| ?
	\|}		\|}

Vipul: /* Alternative definitions */

2010-12-20T23:29:32Z

Alternative definitions

← Older revision		Revision as of 23:29, 20 December 2010
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	===Alternative definitions===		===Alternative definitions===

			''This definition need not work for all topological spaces; however, it works for all [[compactly generated Hausdorff space]]s and possibly many others.''

	The '''fundamental group''' of a [[based topological space]] <math>(X,x_0)</math> is defined as the [[defining ingredient::space of path components]] of the [[defining ingredient::loop space of a based topological space\|loop space]] of <math>(X,x_0)</math>, i.e.:		The '''fundamental group''' of a [[based topological space]] <math>(X,x_0)</math> is defined as the [[defining ingredient::space of path components]] of the [[defining ingredient::loop space of a based topological space\|loop space]] of <math>(X,x_0)</math>, i.e.:

Vipul at 17:12, 20 December 2010

2010-12-20T17:12:09Z

← Older revision		Revision as of 17:12, 20 December 2010
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	The definition of the fundamental group as the space of path components of <math>\Omega(X,x_0)</math> gives a topology on the fundamental group. It turns out, though, that if the path component of <math>x_0</math> in <math>X</math> is a [[locally path-connected space]], then so is <math>\Omega(X,x_0)</math>, in which case the fundamental group has a discrete topology. Since the spaces we typically study (such as manifolds) are locally path-connected, the fundamental group is habitually viewed as a discrete group.		The definition of the fundamental group as the space of path components of <math>\Omega(X,x_0)</math> gives a topology on the fundamental group. It turns out, though, that if the path component of <math>x_0</math> in <math>X</math> is a [[locally path-connected space]], then so is <math>\Omega(X,x_0)</math>, in which case the fundamental group has a discrete topology. Since the spaces we typically study (such as manifolds) are locally path-connected, the fundamental group is habitually viewed as a discrete group.

			===Omission of basepoint===

			For a [[path-connected space]], the fundamental groups at two different points are isomorphic. Moreover, any path between these two points yields an isomorphism, and any two such isomorphisms have quotient equal to an [[inner automorphism]] of the fundamental group at one point. When viewing the fundamental group as an abstract group, it is thus often customary to omit the basepoint for a path-connected space.

			More generally, for a [[topological space]] where all the [[path component]]s are [[homeomorphism\|homeomorphic]] (or even more generally, where they are all [[homotopy-equivalent space]]s), the isomorphism class of the fundamental group is independent of the choice of basepoint. In particular, this is true for [[homogeneous space]]s such as [[topological group]]s.

			For a path-connected [[space with abelian fundamental group]], the fundamental groups at any two points are canonically identified and thus it makes even more sense to omit the basepoint.

	==Relation with other constructs==		==Relation with other constructs==

Vipul at 00:34, 20 December 2010

2010-12-20T00:34:37Z

@@ Line 56: / Line 56: @@
 |-
 | All loops based at <math>x_0</math>, up to ''slice and rearrange'' as well as homotopies where the intermediates must be loops but need not be based at <math>x_0</math>. In other words, ''homology classes'' || [[first homology group]] (if <math>X</math> is path-connected) || <math>H_1(X,x_0)</math> or just <math>H_1(X)</math> || For a path-connected space, the first homology group is the abelianization of the fundamental group.
 |}

Vipul: /* Relation with other constructs */

2010-12-20T00:23:53Z

Relation with other constructs

← Older revision		Revision as of 00:23, 20 December 2010
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	\|-		\|-
	\| All loops based at <math>x_0</math>, ''not'' up to homotopy \|\| [[loop space of a based topological space]] \|\| <math>\Omega(X,x_0)</math> \|\| <math>\pi_1(X,x_0) = \pi_0(\Omega(X,x_0))</math>		\| All loops based at <math>x_0</math>, ''not'' up to homotopy \|\| [[loop space of a based topological space]] \|\| <math>\Omega(X,x_0)</math> \|\| <math>\pi_1(X,x_0) = \pi_0(\Omega(X,x_0))</math>
			\|-
			\| All loops up to homotopies, (intermediaries must also be loops, but basepoint could vary) \|\| [[loop space of a topological space]] \|\| <math>\Omega(X,x_0)</math>
	\|-		\|-
	\| All loops based at <math>x_0</math>, up to homotopies where the intermediaries must be loops but need not be based at <math>x_0</math> \|\| [[conjugacy class set of fundamental group]] \|\| ? \|\| The set of conjugacy classes of <math>\pi_1(X,x_0)</math>. See also [[actions of the fundamental group]]		\| All loops based at <math>x_0</math>, up to homotopies where the intermediaries must be loops but need not be based at <math>x_0</math> \|\| [[conjugacy class set of fundamental group]] \|\| ? \|\| The set of conjugacy classes of <math>\pi_1(X,x_0)</math>. See also [[actions of the fundamental group]]
	\|-		\|-
	\| All loops up to homotopies~~, (intermediaries~~ must ~~also~~ be loops, ~~but basepoint could vary)~~ \|\| [[~~loop space of a topological space~~]] \|\| <math>~~\Omega~~(X,x_0)</math>		\| All loops based at <math>x_0</math>, up to ''slice and rearrange'' as well as homotopies where the intermediates must be loops but need not be based at <math>x_0</math>. In other words, ''homology classes'' \|\| [[first homology group]] (if <math>X</math> is path-connected) \|\| <math>H_1(X,x_0)</math> or just <math>H_1(X)</math> \|\| For a path-connected space, the first homology group is the abelianization of the fundamental group.
	\|}		\|}

Vipul at 00:18, 20 December 2010

2010-12-20T00:18:00Z

← Older revision		Revision as of 00:18, 20 December 2010
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	The definition of the fundamental group as the space of path components of <math>\Omega(X,x_0)</math> gives a topology on the fundamental group. It turns out, though, that if the path component of <math>x_0</math> in <math>X</math> is a [[locally path-connected space]], then so is <math>\Omega(X,x_0)</math>, in which case the fundamental group has a discrete topology. Since the spaces we typically study (such as manifolds) are locally path-connected, the fundamental group is habitually viewed as a discrete group.		The definition of the fundamental group as the space of path components of <math>\Omega(X,x_0)</math> gives a topology on the fundamental group. It turns out, though, that if the path component of <math>x_0</math> in <math>X</math> is a [[locally path-connected space]], then so is <math>\Omega(X,x_0)</math>, in which case the fundamental group has a discrete topology. Since the spaces we typically study (such as manifolds) are locally path-connected, the fundamental group is habitually viewed as a discrete group.

			==Relation with other constructs==

			{\| class="sortable" border="1"
			! Construct !! Name!! Symbol !! Relation with fundamental group <math>\pi_1(X,x_0)</math>
			\|-
			\| All loops based at <math>x_0</math>, ''not'' up to homotopy \|\| [[loop space of a based topological space]] \|\| <math>\Omega(X,x_0)</math> \|\| <math>\pi_1(X,x_0) = \pi_0(\Omega(X,x_0))</math>
			\|-
			\| All loops based at <math>x_0</math>, up to homotopies where the intermediaries must be loops but need not be based at <math>x_0</math> \|\| [[conjugacy class set of fundamental group]] \|\| ? \|\| The set of conjugacy classes of <math>\pi_1(X,x_0)</math>. See also [[actions of the fundamental group]]
			\|-
			\| All loops up to homotopies, (intermediaries must also be loops, but basepoint could vary) \|\| [[loop space of a topological space]] \|\| <math>\Omega(X,x_0)</math>
			\|}

	==Related properties of topological spaces==		==Related properties of topological spaces==