Homotopy between composites associated in different ways - Revision history

Vipul: /* Constructive/explicit version */

2011-01-08T22:08:29Z

Constructive/explicit version

← Older revision		Revision as of 22:08, 8 January 2011
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	If we denote the homotopy by <math>H</math>, we want <math>H(t,0) = a(t), H(t,1) = b(t)</math> and <math>H(0,s) = H(1,s) = x_0</math>. This homotopy is explicitly given by:		If we denote the homotopy by <math>H</math>, we want <math>H(t,0) = a(t), H(t,1) = b(t)</math> and <math>H(0,s) = H(1,s) = x_0</math>. This homotopy is explicitly given by:

	<math>H(t,s) = \lbrace\begin{array}{rl} f_1\left(\frac{4t}{1 + s}\right), & 0 \le t \le (1 + s)/4 \\ f_2(4t - 1 - s), & (1 + s)/4 < t \le (2 + s)/4 \\ f_3\left(\frac{4t - 2 - s}{~~4 -~~ 2 - s}\right), & (2 + s)/4 < t \le 1 \\\end{array}</math>		<math>H(t,s) = \lbrace\begin{array}{rl} f_1\left(\frac{4t}{1 + s}\right), & 0 \le t \le (1 + s)/4 \\ f_2(4t - 1 - s), & (1 + s)/4 < t \le (2 + s)/4 \\ f_3\left(\frac{4t - 2 - s}{2 - s}\right), & (2 + s)/4 < t \le 1 \\\end{array}</math>

	===Graphical version===		===Graphical version===

Vipul: /* Constructive/explicit version */

2011-01-08T22:04:06Z

Constructive/explicit version

← Older revision		Revision as of 22:04, 8 January 2011
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	If we denote the homotopy by <math>H</math>, we want <math>H(t,0) = a(t), H(t,1) = b(t)</math> and <math>H(0,s) = H(1,s) = x_0</math>. This homotopy is explicitly given by:		If we denote the homotopy by <math>H</math>, we want <math>H(t,0) = a(t), H(t,1) = b(t)</math> and <math>H(0,s) = H(1,s) = x_0</math>. This homotopy is explicitly given by:

	<math>H(t,s) = \lbrace\begin{array}{rl} f_1(4t/(1 + s)), & 0 \le t \le (1 + s)/4 \\ f_2(4t - 1 - s), & (1 + s)/4 < t \le (2 + s)/4 \\ f_3((4t - 2 - s~~)/(~~4 - 2 - s)), & (2 + s)/4 < t \le 1 \\\end{array}</math>		<math>H(t,s) = \lbrace\begin{array}{rl} f_1\left(\frac{4t}{1 + s}\right), & 0 \le t \le (1 + s)/4 \\ f_2(4t - 1 - s), & (1 + s)/4 < t \le (2 + s)/4 \\ f_3\left(\frac{4t - 2 - s}{4 - 2 - s}\right), & (2 + s)/4 < t \le 1 \\\end{array}</math>

	===Graphical version===		===Graphical version===

Vipul: /* Constructive/explicit version */

2011-01-08T22:03:06Z

Constructive/explicit version

← Older revision		Revision as of 22:03, 8 January 2011
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	If we denote the homotopy by <math>H</math>, we want <math>H(t,0) = a(t), H(t,1) = b(t)</math> and <math>H(0,s) = H(1,s) = x_0</math>. This homotopy is explicitly given by:		If we denote the homotopy by <math>H</math>, we want <math>H(t,0) = a(t), H(t,1) = b(t)</math> and <math>H(0,s) = H(1,s) = x_0</math>. This homotopy is explicitly given by:

	<math>H(t,s) = \lbrace\begin{array}{rl} f_1(2(1 + s)t), & 0 \le t \le (1 + s)/4 \\ f_2(4t - 1 - s), & (1 + s)/4 < t \le (2 + s)/4 \\ f_3(2t - ~~1 +~~ s(2t - 2)), & (2 + s)/4 < t \le 1 \\\end{array}</math>		<math>H(t,s) = \lbrace\begin{array}{rl} f_1(4t/(1 + s)), & 0 \le t \le (1 + s)/4 \\ f_2(4t - 1 - s), & (1 + s)/4 < t \le (2 + s)/4 \\ f_3((4t - 2 - s)/(4 - 2 - s)), & (2 + s)/4 < t \le 1 \\\end{array}</math>

	===Graphical version===		===Graphical version===

Vipul at 17:07, 20 December 2010

2010-12-20T17:07:42Z

← Older revision		Revision as of 17:07, 20 December 2010
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	<math>a</math> and <math>b</math> are homotopic loops, i.e., they are in the same homotopy class of loops based at <math>x_0</math>.		<math>a</math> and <math>b</math> are homotopic loops, i.e., they are in the same homotopy class of loops based at <math>x_0</math>.

			This version is essentially the ''associativity'' part of showing that the [[fact about::fundamental group]] of a based topological space is indeed a group.


	===Constructive/explicit version===		===Constructive/explicit version===
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	===Uniform version===		===Uniform version===

	This version is a little stronger than the other versions. Let <math>L = \Omega(X,x_0)</math> be the [[loop space of a based topological space\|loop space]] of <math>(X,x_0)</math>, i.e., the space of all loops in <math>X</math> based at <math>x_0</math> under the [[compact-open topology]]. Then, consider the following two maps:		This version is a little stronger than the other versions. Let <math>L = \Omega(X,x_0)</math> be the [[fact about::loop space of a based topological space\|loop space]] of <math>(X,x_0)</math>, i.e., the space of all loops in <math>X</math> based at <math>x_0</math> under the [[compact-open topology]]. Then, consider the following two maps:

	<math>\! A:L \times L \times L \to L, \qquad A(f_1,f_2,f_3) = (f_1 * f_2) * f_3</math>		<math>\! A:L \times L \times L \to L, \qquad A(f_1,f_2,f_3) = (f_1 * f_2) * f_3</math>

Vipul at 17:06, 20 December 2010

2010-12-20T17:06:16Z

← Older revision		Revision as of 17:06, 20 December 2010
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	<math>b = \lbrace\begin{array}{rl} f_1(2t), & 0 \le t \le 1/2 \\ f_2(4t - 2), & 1/2 < t \le 3/4 \\ f_3(4t - 3), & 3/4 < t \le 1 \\\end{array}</math>		<math>b = \lbrace\begin{array}{rl} f_1(2t), & 0 \le t \le 1/2 \\ f_2(4t - 2), & 1/2 < t \le 3/4 \\ f_3(4t - 3), & 3/4 < t \le 1 \\\end{array}</math>

	If we denote the homotopy by <math>H</math>, we want <math>H(t,0) = a(t), H(t,1) = b(t)</math> and <math>H(0,s) = H(1,s) = x_0</math>. This homotopy is explicitly given by {{~~fillin~~}}		If we denote the homotopy by <math>H</math>, we want <math>H(t,0) = a(t), H(t,1) = b(t)</math> and <math>H(0,s) = H(1,s) = x_0</math>. This homotopy is explicitly given by:

			<math>H(t,s) = \lbrace\begin{array}{rl} f_1(2(1 + s)t), & 0 \le t \le (1 + s)/4 \\ f_2(4t - 1 - s), & (1 + s)/4 < t \le (2 + s)/4 \\ f_3(2t - 1 + s(2t - 2)), & (2 + s)/4 < t \le 1 \\\end{array}</math>

	===Graphical version===		===Graphical version===

	[[File:Associativityhomotopy.png\|~~250px~~]]		[[File:Associativityhomotopy.png\|350px]]

			===Uniform version===

			This version is a little stronger than the other versions. Let <math>L = \Omega(X,x_0)</math> be the [[loop space of a based topological space\|loop space]] of <math>(X,x_0)</math>, i.e., the space of all loops in <math>X</math> based at <math>x_0</math> under the [[compact-open topology]]. Then, consider the following two maps:

			<math>\! A:L \times L \times L \to L, \qquad A(f_1,f_2,f_3) = (f_1 * f_2) * f_3</math>

			and:

			<math>\! B:L \times L \times L \to L, \qquad B(f_1,f_2,f_3) = f_1 * (f_2 * f_3)</math>

	~~{{fillin}}~~		Then, the maps <math>A</math> and <math>B</math> are [[homotopic maps]]. This is part of the proof that <math>\Omega(X,x_0)</math> is a [[H-space]], which is a homotopy variant of [[topological monoid]].

Vipul: /* Graphical version */

2010-12-18T19:22:11Z

Graphical version

← Older revision		Revision as of 19:22, 18 December 2010
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	===Graphical version===		===Graphical version===

			[[File:Associativityhomotopy.png\|250px]]

	{{fillin}}		{{fillin}}

Vipul at 18:57, 18 December 2010

2010-12-18T18:57:46Z

← Older revision		Revision as of 18:57, 18 December 2010
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	===Constructive/explicit version===		===Constructive/explicit version===

			We first note the explicit piecewise definitions of <math>a</math> and <math>b</math>:

			<math>a = \lbrace\begin{array}{rl} f_1(4t), & 0 \le t \le 1/4 \\ f_2(4t - 1), & 1/4 < t \le 1/2 \\ f_3(2t - 1), & 1/2 < t \le 1 \\\end{array}</math>

			and:

			<math>b = \lbrace\begin{array}{rl} f_1(2t), & 0 \le t \le 1/2 \\ f_2(4t - 2), & 1/2 < t \le 3/4 \\ f_3(4t - 3), & 3/4 < t \le 1 \\\end{array}</math>

			If we denote the homotopy by <math>H</math>, we want <math>H(t,0) = a(t), H(t,1) = b(t)</math> and <math>H(0,s) = H(1,s) = x_0</math>. This homotopy is explicitly given by {{fillin}}

			===Graphical version===

	{{fillin}}		{{fillin}}

Vipul: Created page with '==Statement== ===Existential version=== Suppose $f_1,f_2,f_3$ are loops based at a point $x_0$ in a topological space $X$ . We can consi...'

2010-12-18T18:49:39Z

Created page with '==Statement== ===Existential version=== Suppose <math>f_1,f_2,f_3</math> are loops based at a point <math>x_0</math> in a topological space <math>X</math>. We can consi...'

New page

==Statement==

===Existential version===

Suppose <math>f_1,f_2,f_3</math> are [[loop]]s based at a point <math>x_0</math> in a [[topological space]] <math>X</math>. We can consider two differently associated products of these three loops:

<math>\! a = (f_1 * f_2) * f_3, b = f_1 * (f_2 * f_3)</math>

<math>a</math> and <math>b</math> are homotopic loops, i.e., they are in the same homotopy class of loops based at <math>x_0</math>.

===Constructive/explicit version===

{{fillin}}

Homotopy between composites associated in different ways - Revision history

Vipul: /* Constructive/explicit version */

Vipul: /* Constructive/explicit version */

Vipul: /* Constructive/explicit version */

Vipul at 17:07, 20 December 2010

Vipul at 17:06, 20 December 2010

Vipul: /* Graphical version */

Vipul at 18:57, 18 December 2010

Vipul: Created page with '==Statement== ===Existential version=== Suppose f_1,f_2,f_3 are loops based at a point x_0 in a topological space X. We can consi...'

Vipul: Created page with '==Statement== ===Existential version=== Suppose $f_1,f_2,f_3$ are loops based at a point $x_0$ in a topological space $X$ . We can consi...'