0708-1300/Homework Assignment 9 - Revision history

Evan.decorte: /* Just for Fun */

2008-04-30T23:48:41Z

Just for Fun

← Older revision		Revision as of 19:48, 30 April 2008
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	* What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space? [[0708-1300/Covering_Product\|Here]] is an idea but don't look at it until you have think on the problem for a while.		* What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space? [[0708-1300/Covering_Product\|Here]] is an idea but don't look at it until you have think on the problem for a while.
	* This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?		* This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?
	* Here is a short introduction to regular covering spaces and an application of them to the last problem on this homework: [[~~Image~~:0708-1300-Regular-Covering-Spaces.pdf]].		* Here is a short introduction to regular covering spaces and an application of them to the last problem on this homework: [[Media:0708-1300-Regular-Covering-Spaces.pdf\|Regular Covering Spaces]] (PDF).

Evan.decorte: /* Just for Fun */

2008-04-30T23:35:40Z

Just for Fun

← Older revision		Revision as of 19:35, 30 April 2008
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	* What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space? [[0708-1300/Covering_Product\|Here]] is an idea but don't look at it until you have think on the problem for a while.		* What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space? [[0708-1300/Covering_Product\|Here]] is an idea but don't look at it until you have think on the problem for a while.
	* This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?		* This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?
			* Here is a short introduction to regular covering spaces and an application of them to the last problem on this homework: [[Image:0708-1300-Regular-Covering-Spaces.pdf]].

Evan.decorte: /* Just for Fun */

2008-04-30T23:23:49Z

Just for Fun

← Older revision		Revision as of 19:23, 30 April 2008
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	* What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space? [[0708-1300/Covering_Product\|Here]] is an idea but don't look at it until you have think on the problem for a while.		* What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space? [[0708-1300/Covering_Product\|Here]] is an idea but don't look at it until you have think on the problem for a while.
	* This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?		* This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?
	* Here is a short introduction to regular covering spaces, along with an application of them to the last problem on this homework. [[0708-1300/Regular_Covering_Spaces.pdf]]

Evan.decorte: /* Just for Fun */

2008-04-30T23:13:19Z

Just for Fun

← Older revision		Revision as of 19:13, 30 April 2008
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	* What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space? [[0708-1300/Covering_Product\|Here]] is an idea but don't look at it until you have think on the problem for a while.		* What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space? [[0708-1300/Covering_Product\|Here]] is an idea but don't look at it until you have think on the problem for a while.
	* This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?		* This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?
	* Here is a short introduction to regular covering spaces, along with an application of them to the last problem on this homework. [[0708-1300/~~Regular Covering Spaces~~.pdf]]		* Here is a short introduction to regular covering spaces, along with an application of them to the last problem on this homework. [[0708-1300/Regular_Covering_Spaces.pdf]]

Evan.decorte: /* Just for Fun */

2008-04-30T23:12:25Z

Just for Fun

← Older revision		Revision as of 19:12, 30 April 2008
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	* What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space? [[0708-1300/Covering_Product\|Here]] is an idea but don't look at it until you have think on the problem for a while.		* What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space? [[0708-1300/Covering_Product\|Here]] is an idea but don't look at it until you have think on the problem for a while.
	* This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?		* This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?
			* Here is a short introduction to regular covering spaces, along with an application of them to the last problem on this homework. [[0708-1300/Regular Covering Spaces.pdf]]

Franklin: /* Just for Fun */

2008-02-18T21:09:12Z

Just for Fun

← Older revision		Revision as of 17:09, 18 February 2008
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	==Just for Fun==		==Just for Fun==
	* What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space?		* What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space? [[0708-1300/Covering_Product\|Here]] is an idea but don't look at it until you have think on the problem for a while.
	* This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?		* This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?

Drorbn at 18:32, 18 February 2008

2008-02-18T18:32:54Z

← Older revision		Revision as of 14:32, 18 February 2008
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	==Just for Fun==		==Just for Fun==
	What happens if in problem ~~<math>~~1~~</math>~~ we consider infinitely many covering spaces.		* What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space?
			* This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?
	This is, is the product of an infinite family of covering spaces a covering space?

Franklin at 16:20, 18 February 2008

2008-02-18T16:20:25Z

← Older revision		Revision as of 12:20, 18 February 2008
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	==Due Date==		==Due Date==
	This assignment is due in class on Thursday February 28, 2008.		This assignment is due in class on Thursday February 28, 2008.

			==Just for Fun==
			What happens if in problem <math>1</math> we consider infinitely many covering spaces.
			This is, is the product of an infinite family of covering spaces a covering space?

Drorbn: /* Doing */

2008-02-14T15:40:14Z

Doing

← Older revision		Revision as of 11:40, 14 February 2008
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	==Doing==		==Doing==
	(Problems 1,2,4,5 below are taken with slight modifications from Hatcher's book, pages 79-90).		(Problems 1,2,4,5 below are taken with slight modifications from Hatcher's book, pages 79-80).

	# Show that if <math>p_1\colon X_1\to B_1</math> and <math>p_2\colon X_2 \to B_2</math> are covering spaces, then so is their product <math>p_1\times p_2\colon X_1\times X_2\to B_1\times B_2</math>.		# Show that if <math>p_1\colon X_1\to B_1</math> and <math>p_2\colon X_2 \to B_2</math> are covering spaces, then so is their product <math>p_1\times p_2\colon X_1\times X_2\to B_1\times B_2</math>.

Drorbn at 15:32, 14 February 2008

2008-02-14T15:32:11Z

← Older revision		Revision as of 11:32, 14 February 2008
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	# Construct (i.e., describe in explicit terms) a simply-connected covering space of the space <math>X\subset\mathbb{R}^3</math> that is the union of a sphere and a diameter. Do the same when <math>X</math> is the union of a sphere and a circle intersecting it in two points.		# Construct (i.e., describe in explicit terms) a simply-connected covering space of the space <math>X\subset\mathbb{R}^3</math> that is the union of a sphere and a diameter. Do the same when <math>X</math> is the union of a sphere and a circle intersecting it in two points.
	# Do the same to the space <math>Y</math> of the term test: <math>Y=\{z\in{\mathbb C}\colon\|z\|\leq 1\}/(z\sim e^{2\pi i/3}z\mbox{ whenever }\|z\|=1)</math>.		# Do the same to the space <math>Y</math> of the term test: <math>Y=\{z\in{\mathbb C}\colon\|z\|\leq 1\}/(z\sim e^{2\pi i/3}z\mbox{ whenever }\|z\|=1)</math>.
	# Find all the connected 2-sheeted and 3-sheeted covering spaces of <math>S^1\vee S^1</math>, up to isomorphism of covering spaces without base points.		# Find all the connected 2-sheeted and 3-sheeted covering spaces of the "figure eight space" <math>S^1\vee S^1</math> (two circles joined at a point), up to isomorphism of covering spaces without base points.
	# Let <math>a</math> and <math>b</math> be the generators of <math>\pi_1(S^1\vee S^1)</math> corresponding to the two <math>S^1</math> summands. Draw a picture of the covering space of <math>S^1\vee S^1</math> corresponding to the normal subgroup generated by <math>a^2</math>, <math>b^2</math>, and <math>(ab)^4</math>, and prove that this covering space is indeed the correct one.		# Let <math>a</math> and <math>b</math> be the generators of <math>\pi_1(S^1\vee S^1)</math> corresponding to the two <math>S^1</math> summands. Draw a picture of the covering space of <math>S^1\vee S^1</math> corresponding to the normal subgroup generated by <math>a^2</math>, <math>b^2</math>, and <math>(ab)^4</math>, and prove that this covering space is indeed the correct one.