0708-1300/Homework Assignment 6 - Revision history

Franklin: /* Just for Fun */

2007-12-13T21:22:03Z

Just for Fun

← Older revision		Revision as of 17:22, 13 December 2007
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	==Just for Fun==		==Just for Fun==
	Prove that the two (3-component) links <math>\gamma_3</math> and <math>\gamma'_3</math> shown above are not isotopic, yet their complements are diffeomorphic. (See more at {{Home Link\|classes/0405/Topology/HW5/HW.html\|Classes: 2004-05: Math 1300Y - Topology: Homework Assignment 5}})		Prove that the two (3-component) links <math>\gamma_3</math> and <math>\gamma'_3</math> shown above are not isotopic, yet their complements are diffeomorphic. (See more at {{Home Link\|classes/0405/Topology/HW5/HW.html\|Classes: 2004-05: Math 1300Y - Topology: Homework Assignment 5}})

			[[0708-1300/justforfun6\|Here]] is one idea. Don't look at it before thinking the problem for a while.

	==Due Date==		==Due Date==

Drorbn at 14:47, 6 December 2007

2007-12-06T14:47:33Z

← Older revision		Revision as of 10:47, 6 December 2007
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	==Doing==		==Doing==
	Solve the following problems and submit your solutions of problems 1, 3 and 4. This is a very challenging collection of problems; I expect most of you to do problem 2 with no difficulty (it is a repeat of an older problem), problem 1 with some effort, and I hope each of you will be able to do at least one further problem. It will be great if some of you will do all problems!		Solve the following problems and submit your solutions of problems 2, 3 and 4. This is a very challenging collection of problems; I expect most of you to do problem 1 with no difficulty (it is a repeat of an older problem), problem 2 with some effort, and I hope each of you will be able to do at least one further problem. It will be great if some of you will do all problems!

	'''Problem 1.''' If <math>M</math> is a ~~compact orientable~~ <math>n</math>~~-manifold~~ ~~with~~ no ~~boundary,~~ ~~show~~ that <math>~~H^n_~~{dR}~~(M)~~\~~neq 0~~</math>.		'''Problem 1.''' The standard volume form on <math>S^2</math> is the form <math>\omega</math> given by <math>\omega=\frac{1}{4\pi}\left(xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\right)</math>. Show that <math>\int_{S^2}\omega=1</math>.

	'''Problem 2.''' ~~The standard volume form on~~ <math>~~S^2~~</math> is ~~the~~ ~~form~~ <math>~~\omega~~</math> ~~given~~ by ~~<math>\omega=\frac{1}{4\pi}\left(xdy\wedge~~ ~~dz+ydz\wedge dx+zdx\wedge dy\right)</math>. Show~~ that <math>~~\int_~~{~~S^2~~}\~~omega=1~~</math>.		'''Problem 2.''' If <math>M</math> is a compact orientable <math>n</math>-manifold with no boundary, show that <math>H^n_{dR}(M)\neq 0</math>.

	'''Problem 3.''' Show that if <math>\omega\in\Omega^2(S^2)</math> satisfies <math>\int_{S_2}\omega=0</math>, then <math>\omega</math> is exact. Deduce that if <math>w_1\in\Omega^2(S^2)</math> and <math>w_2\in\Omega^2(S^2)</math> satisfy <math>\int_{S_2}\omega_1=\int_{S_2}\omega_2</math>, then <math>[\omega_1]=[\omega_2]</math> as elements of <math>H^2_{dR}(S^2)</math>. Deduce further that <math>\dim H^2_{dR}(S^2)=1</math>.		'''Problem 3.''' Show that if <math>\omega\in\Omega^2(S^2)</math> satisfies <math>\int_{S_2}\omega=0</math>, then <math>\omega</math> is exact. Deduce that if <math>w_1\in\Omega^2(S^2)</math> and <math>w_2\in\Omega^2(S^2)</math> satisfy <math>\int_{S_2}\omega_1=\int_{S_2}\omega_2</math>, then <math>[\omega_1]=[\omega_2]</math> as elements of <math>H^2_{dR}(S^2)</math>. Deduce further that <math>\dim H^2_{dR}(S^2)=1</math>.

Drorbn at 14:39, 6 December 2007

2007-12-06T14:39:47Z

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	{{0708-1300/Navigation}}		{{0708-1300/Navigation}}
	{{In Preparation}}
	==Reading==		==Reading==
	At your leisure, read your class notes over the break, and especially at some point right before classes resume ~~after~~ ~~the break~~.		At your leisure, read your class notes over the break, and especially at some point right before classes resume next semester. Here are a few questions you can ask yourself while reading:
			* Do you understand pullbacks of differential forms?
			* Do you think you could in practice integrate any differential form on any manifold (at least when the formulas involved are not too messy)?
			* Do you understand orientations and boundaries and how they interact?
			* Why is Stokes' theorem true? Both in terms of the local meaning of <math>d</math>, and in terms of a formal proof.
			* Do you understand the two and three dimensional cases of Stokes' theorem?
			* Do you understand the Hodge star operator <math>\star</math>?
			* How did we get <math>d\star dA=J</math> from the least action principle?
			* Do you understand how Poincare's lemma entered the derivation of Maxewell's equations?
			* Do you understand the operator <math>P</math>? (How was it used, formally derived, and what is the intuitive picture behind it?)
			* What was <math>H_{dR}</math> and how did it relate to pullbacks and homotopy.

	==Doing==		==Doing==
	Solve the following problems and submit your solutions of problems 1, 3 and 4. This is a very challenging collection of problems; I expect most of you to do problem 2 with no difficulty (it is a repeat of an older problem), problem 1 with ~~not~~ ~~too much difficulty~~, and I hope each of you will be able to do at least one further problem. It will be great if some of you will do all problems!		Solve the following problems and submit your solutions of problems 1, 3 and 4. This is a very challenging collection of problems; I expect most of you to do problem 2 with no difficulty (it is a repeat of an older problem), problem 1 with some effort, and I hope each of you will be able to do at least one further problem. It will be great if some of you will do all problems!

	'''Problem 1.''' If <math>M</math> is a compact orientable <math>n</math>-manifold with no boundary, show that <math>H^n_{dR}(M)\neq 0</math>.		'''Problem 1.''' If <math>M</math> is a compact orientable <math>n</math>-manifold with no boundary, show that <math>H^n_{dR}(M)\neq 0</math>.
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	'''Problem 2.''' The standard volume form on <math>S^2</math> is the form <math>\omega</math> given by <math>\omega=\frac{1}{4\pi}\left(xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\right)</math>. Show that <math>\int_{S^2}\omega=1</math>.		'''Problem 2.''' The standard volume form on <math>S^2</math> is the form <math>\omega</math> given by <math>\omega=\frac{1}{4\pi}\left(xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\right)</math>. Show that <math>\int_{S^2}\omega=1</math>.

	'''Problem 3.''' Show that if <math>\omega\in\Omega^2(S^2)</math> satisfies <math>\int_{S_2}\omega=0</math>, then <math>\omega</math> is exact, ~~and~~ ~~therefore,~~ if <math>w_1\in\Omega^2(S^2)</math> and <math>w_2\in\Omega^2(S^2)</math> satisfy <math>\int_{S_2}\omega_1=\int_{S_2}\omega_2</math>, then <math>[\omega_1]=[\omega_2]</math> as elements of <math>H^2_{dR}(S^2)</math>.		'''Problem 3.''' Show that if <math>\omega\in\Omega^2(S^2)</math> satisfies <math>\int_{S_2}\omega=0</math>, then <math>\omega</math> is exact. Deduce that if <math>w_1\in\Omega^2(S^2)</math> and <math>w_2\in\Omega^2(S^2)</math> satisfy <math>\int_{S_2}\omega_1=\int_{S_2}\omega_2</math>, then <math>[\omega_1]=[\omega_2]</math> as elements of <math>H^2_{dR}(S^2)</math>. Deduce further that <math>\dim H^2_{dR}(S^2)=1</math>.

	'''Problem 4.''' A "link" in <math>{\mathbb R}^3</math> is an ordered pair <math>\gamma=(\gamma_1, \gamma_2)</math>, in which <math>\gamma_1</math> and <math>\gamma_2</math> are smooth embeddings of the circle <math>S^1</math> into <math>{\mathbb R}^3</math>, whose images (called "the components of <math>\gamma</math>") are disjoint. Two such links are called "isotopic", if one can be deformed to the other via a smooth homotopy along which the components remain embeddings and remain disjoint. Given a link <math>\gamma</math>, define a map <math>\Phi_\gamma:S^1\times S^1\to S^2</math> by <math>\Phi_\gamma(t_1,t_2):=\frac{\gamma_2(t_2)-\gamma_1(t_1)}{\|\|\gamma_2(t_2)-\gamma_1(t_1)\|\|}</math>. Finally, let <math>\omega</math> be the standard volume form of <math>S^2</math>, and define "the linking number of <math>\gamma=(\gamma_1, \gamma_2)</math>" to be <math>l(\gamma)=l(\gamma_1,\gamma_2):=\int_{S^1\times S^1}\Phi_\gamma^\star\omega</math>. Show		'''Problem 4.''' A "link" in <math>{\mathbb R}^3</math> is an ordered pair <math>\gamma=(\gamma_1, \gamma_2)</math>, in which <math>\gamma_1</math> and <math>\gamma_2</math> are smooth embeddings of the circle <math>S^1</math> into <math>{\mathbb R}^3</math>, whose images (called "the components of <math>\gamma</math>") are disjoint. Two such links are called "isotopic", if one can be deformed to the other via a smooth homotopy along which the components remain embeddings and remain disjoint. Given a link <math>\gamma</math>, define a map <math>\Phi_\gamma:S^1\times S^1\to S^2</math> by <math>\Phi_\gamma(t_1,t_2):=\frac{\gamma_2(t_2)-\gamma_1(t_1)}{\|\|\gamma_2(t_2)-\gamma_1(t_1)\|\|}</math>. Finally, let <math>\omega</math> be the standard volume form of <math>S^2</math>, and define "the linking number of <math>\gamma=(\gamma_1, \gamma_2)</math>" to be <math>l(\gamma)=l(\gamma_1,\gamma_2):=\int_{S^1\times S^1}\Phi_\gamma^\star\omega</math>. Show

Drorbn at 03:33, 6 December 2007

2007-12-06T03:33:44Z

← Older revision		Revision as of 23:33, 5 December 2007
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	==Doing==		==Doing==
	Solve and submit your solutions of ~~the~~ ~~following~~ ~~problems~~. This is a very challenging collection of problems; I expect most of you to do problem 2 with no difficulty, and I hope each of you will be able to do at least one further problem. It will be great if some of you will do ~~more~~!		Solve the following problems and submit your solutions of problems 1, 3 and 4. This is a very challenging collection of problems; I expect most of you to do problem 2 with no difficulty (it is a repeat of an older problem), problem 1 with not too much difficulty, and I hope each of you will be able to do at least one further problem. It will be great if some of you will do all problems!

	'''Problem 1.''' If <math>M</math> is a compact orientable <math>n</math>-manifold with no boundary, show that <math>H^n_{dR}(M)\neq 0</math>.		'''Problem 1.''' If <math>M</math> is a compact orientable <math>n</math>-manifold with no boundary, show that <math>H^n_{dR}(M)\neq 0</math>.

Drorbn at 23:44, 5 December 2007

2007-12-05T23:44:37Z

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	Solve and submit your solutions of the following problems. This is a very challenging collection of problems; I expect most of you to do problem 2 with no difficulty, and I hope each of you will be able to do at least one further problem. It will be great if some of you will do more!		Solve and submit your solutions of the following problems. This is a very challenging collection of problems; I expect most of you to do problem 2 with no difficulty, and I hope each of you will be able to do at least one further problem. It will be great if some of you will do more!

	'''Problem 1.''' If <math>M</math> is a compact orientable n-manifold with no boundary, show that <math>H^n_{dR}(M)\neq 0</math>.		'''Problem 1.''' If <math>M</math> is a compact orientable <math>n</math>-manifold with no boundary, show that <math>H^n_{dR}(M)\neq 0</math>.

	'''Problem 2.''' The "standard volume form on <math>S^2</math>" is the form <math>\omega</math> given by <math>\omega=\frac{1}{4\pi}\left(xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\right)</math>. Show that <math>\int_{S^2}\omega=1</math>.		'''Problem 2.''' The standard volume form on <math>S^2</math> is the form <math>\omega</math> given by <math>\omega=\frac{1}{4\pi}\left(xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\right)</math>. Show that <math>\int_{S^2}\omega=1</math>.

	'''Problem 3.''' Show that if <math>\omega\in\Omega^2(S^2)</math> satisfies <math>\int_{S_2}\omega=0</math>, then <math>\omega</math> is exact, and therefore, if <math>w_1\in\Omega^2(S^2)</math> and <math>w_2\in\Omega^2(S^2)</math> satisfy <math>\int_{S_2}\omega_1=\int_{S_2}\omega_2</math>, then <math>[\omega_1]=[\omega_2]</math> as elements of <math>H^n_{dR}(S^2)</math>.		'''Problem 3.''' Show that if <math>\omega\in\Omega^2(S^2)</math> satisfies <math>\int_{S_2}\omega=0</math>, then <math>\omega</math> is exact, and therefore, if <math>w_1\in\Omega^2(S^2)</math> and <math>w_2\in\Omega^2(S^2)</math> satisfy <math>\int_{S_2}\omega_1=\int_{S_2}\omega_2</math>, then <math>[\omega_1]=[\omega_2]</math> as elements of <math>H^2_{dR}(S^2)</math>.

	'''Problem 4.''' A "link" in <math>{\mathbb R}^3</math> is an ordered pair <math>\gamma=(\gamma_1, \gamma_2)</math>, in which <math>\gamma_1</math> and <math>\gamma_2</math> are smooth embeddings of the circle <math>S^1</math> into <math>{\mathbb R}^3</math>, whose images (called "the components of <math>\gamma</math>") are disjoint. Two such links are called "isotopic", if one can be deformed to the other via a homotopy along which the components remain disjoint. Given a link <math>\gamma</math>, define a map <math>\Phi_\gamma:S^1\times S^1\to S^2</math> by <math>\Phi_\gamma(t_1,t_2):=\frac{\gamma_2(t_2)-\gamma_1(t_1)}{\|\|\gamma_2(t_2)-\gamma_1(t_1)\|\|}</math>. Finally, let <math>\omega</math> be the standard volume form of <math>S^2</math>, and define "the linking number of <math>\gamma=(\gamma_1, \gamma_2)</math>" to be <math>l(\gamma)=l(\gamma_1,\gamma_2):=\int_{S^1\times S^1}\Phi_\gamma^\star\omega</math>. Show		'''Problem 4.''' A "link" in <math>{\mathbb R}^3</math> is an ordered pair <math>\gamma=(\gamma_1, \gamma_2)</math>, in which <math>\gamma_1</math> and <math>\gamma_2</math> are smooth embeddings of the circle <math>S^1</math> into <math>{\mathbb R}^3</math>, whose images (called "the components of <math>\gamma</math>") are disjoint. Two such links are called "isotopic", if one can be deformed to the other via a smooth homotopy along which the components remain embeddings and remain disjoint. Given a link <math>\gamma</math>, define a map <math>\Phi_\gamma:S^1\times S^1\to S^2</math> by <math>\Phi_\gamma(t_1,t_2):=\frac{\gamma_2(t_2)-\gamma_1(t_1)}{\|\|\gamma_2(t_2)-\gamma_1(t_1)\|\|}</math>. Finally, let <math>\omega</math> be the standard volume form of <math>S^2</math>, and define "the linking number of <math>\gamma=(\gamma_1, \gamma_2)</math>" to be <math>l(\gamma)=l(\gamma_1,\gamma_2):=\int_{S^1\times S^1}\Phi_\gamma^\star\omega</math>. Show
	# If two links <math>\gamma</math> and <math>\gamma'</math> are isotopic, then their linking numbers are the same: <math>l(\gamma)=l(\gamma')</math>.		# If two links <math>\gamma</math> and <math>\gamma'</math> are isotopic, then their linking numbers are the same: <math>l(\gamma)=l(\gamma')</math>.
	# If <math>\omega'</math> is a second 2-form on <math>S^2</math> for which <math>\int_{S^2}\omega'=1</math> and if <math>l'(\gamma)</math> is defined in the same manner as <math>l(\gamma)</math> except replacing <math>\omega</math> with <math>\omega'</math>, then <math>l(\gamma)=l'(\gamma)</math>. (In particular this is true if <math>\omega'</math> is very close to a <math>\delta</math>-function form at the north pole of <math>S^2</math>).		# If <math>\omega'</math> is a second 2-form on <math>S^2</math> for which <math>\int_{S^2}\omega'=1</math> and if <math>l'(\gamma)</math> is defined in the same manner as <math>l(\gamma)</math> except replacing <math>\omega</math> with <math>\omega'</math>, then <math>l(\gamma)=l'(\gamma)</math>. (In particular this is true if <math>\omega'</math> is very close to a <math>\delta</math>-function form at the north pole of <math>S^2</math>).

Drorbn at 23:40, 5 December 2007

2007-12-05T23:40:26Z

← Older revision		Revision as of 19:40, 5 December 2007
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	'''Problem 1.''' If <math>M</math> is a compact orientable n-manifold with no boundary, show that <math>H^n_{dR}(M)\neq 0</math>.		'''Problem 1.''' If <math>M</math> is a compact orientable n-manifold with no boundary, show that <math>H^n_{dR}(M)\neq 0</math>.

	'''Problem 2.''' The "standard volume form on S^2" is the form <math>\omega</math> given by <math>\omega=\frac{1}{4\pi}\left(xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\right)</math>. Show that <math>\int_{S^2}\omega=1</math>.		'''Problem 2.''' The "standard volume form on <math>S^2</math>" is the form <math>\omega</math> given by <math>\omega=\frac{1}{4\pi}\left(xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\right)</math>. Show that <math>\int_{S^2}\omega=1</math>.

	'''Problem 3.''' Show that if <math>\omega\in\Omega^2(S^2)</math> satisfies <math>\int_{S_2}\omega=0</math>, then <math>\omega</math> is exact, and therefore, if <math>w_1\in\Omega^2(S^2)</math> and <math>w_2\in\Omega^2(S^2)</math> satisfy <math>\int_{S_2}\omega_1=\int_{S_2}\omega_2</math>, then <math>[\omega_1]=[\omega_2]</math> as elements of <math>H^n_{dR}(S^2)</math>.		'''Problem 3.''' Show that if <math>\omega\in\Omega^2(S^2)</math> satisfies <math>\int_{S_2}\omega=0</math>, then <math>\omega</math> is exact, and therefore, if <math>w_1\in\Omega^2(S^2)</math> and <math>w_2\in\Omega^2(S^2)</math> satisfy <math>\int_{S_2}\omega_1=\int_{S_2}\omega_2</math>, then <math>[\omega_1]=[\omega_2]</math> as elements of <math>H^n_{dR}(S^2)</math>.

Drorbn at 23:39, 5 December 2007

2007-12-05T23:39:15Z

← Older revision		Revision as of 19:39, 5 December 2007
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	==Doing==		==Doing==
	Solve and submit your solutions of the following problems:		Solve and submit your solutions of the following problems. This is a very challenging collection of problems; I expect most of you to do problem 2 with no difficulty, and I hope each of you will be able to do at least one further problem. It will be great if some of you will do more!

	'''Problem 1.''' If <math>M</math> is a compact orientable n-manifold with no boundary, show that <math>H^n_{dR}(M)\neq 0</math>.		'''Problem 1.''' If <math>M</math> is a compact orientable n-manifold with no boundary, show that <math>H^n_{dR}(M)\neq 0</math>.

Drorbn at 23:34, 5 December 2007

2007-12-05T23:34:58Z

← Older revision		Revision as of 19:34, 5 December 2007
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	\|colspan=3 align=center\|The links L11a193, <math>\gamma_3</math> and <math>\gamma'_3</math>.		\|colspan=3 align=center\|The links L11a193, <math>\gamma_3</math> and <math>\gamma'_3</math>.
	\|}		\|}

⚫	==Due Date==
⚫	This assignment is due in class on Thursday January 10, 2007.

	==Just for Fun==		==Just for Fun==
	Prove that the two (3-component) links <math>\gamma_3</math> and <math>\gamma'_3</math> shown above are not isotopic, yet their complements are diffeomorphic. (See more at {{Home Link\|classes/0405/Topology/HW5/HW.html\|Classes: 2004-05: Math 1300Y - Topology: Homework Assignment 5}})		Prove that the two (3-component) links <math>\gamma_3</math> and <math>\gamma'_3</math> shown above are not isotopic, yet their complements are diffeomorphic. (See more at {{Home Link\|classes/0405/Topology/HW5/HW.html\|Classes: 2004-05: Math 1300Y - Topology: Homework Assignment 5}})

		⚫	==Due Date==
		⚫	This assignment is due in class on Thursday January 10, 2007.

Drorbn at 23:33, 5 December 2007

2007-12-05T23:33:14Z

← Older revision		Revision as of 19:33, 5 December 2007
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	# If <math>\omega'</math> is a second 2-form on <math>S^2</math> for which <math>\int_{S^2}\omega'=1</math> and if <math>l'(\gamma)</math> is defined in the same manner as <math>l(\gamma)</math> except replacing <math>\omega</math> with <math>\omega'</math>, then <math>l(\gamma)=l'(\gamma)</math>. (In particular this is true if <math>\omega'</math> is very close to a <math>\delta</math>-function form at the north pole of <math>S^2</math>).		# If <math>\omega'</math> is a second 2-form on <math>S^2</math> for which <math>\int_{S^2}\omega'=1</math> and if <math>l'(\gamma)</math> is defined in the same manner as <math>l(\gamma)</math> except replacing <math>\omega</math> with <math>\omega'</math>, then <math>l(\gamma)=l'(\gamma)</math>. (In particular this is true if <math>\omega'</math> is very close to a <math>\delta</math>-function form at the north pole of <math>S^2</math>).
	# Compute (but just up to an overall sign) the linking number of the link {{KAT Link\|L11a193\|L11a193}}, displayed below:		# Compute (but just up to an overall sign) the linking number of the link {{KAT Link\|L11a193\|L11a193}}, displayed below:
⚫	[[Image:L11a193.png~~\|center~~]]

⚫	==Due Date==
⚫	This assignment is due in class on Thursday January 10, 2007.

⚫	==Just for Fun==
	Prove that the following two links are not isotopic, yet their complements are diffeomorphic:

	{\| align=center		{\| align=center
	\|-		\|-
		⚫	\|[[Image:L11a193.png]]
	\|[[Image:0708-1300-LinkComplementExample1.png]]		\|[[Image:0708-1300-LinkComplementExample1.png]]
	\|[[Image:0708-1300-LinkComplementExample2.png]]		\|[[Image:0708-1300-LinkComplementExample2.png]]
			\|-
			\|colspan=3 align=center\|The links L11a193, <math>\gamma_3</math> and <math>\gamma'_3</math>.
	\|}		\|}

		⚫	==Due Date==
⚫	(See more at {{Home Link\|classes/0405/Topology/HW5/HW.html\|Classes: 2004-05: Math 1300Y - Topology: Homework Assignment 5}})
		⚫	This assignment is due in class on Thursday January 10, 2007.

		⚫	==Just for Fun==
		⚫	Prove that the two (3-component) links <math>\gamma_3</math> and <math>\gamma'_3</math> shown above are not isotopic, yet their complements are diffeomorphic. (See more at {{Home Link\|classes/0405/Topology/HW5/HW.html\|Classes: 2004-05: Math 1300Y - Topology: Homework Assignment 5}})

Drorbn at 23:24, 5 December 2007

2007-12-05T23:24:57Z

← Older revision		Revision as of 19:24, 5 December 2007
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	'''Problem 1.''' If <math>M</math> is a compact orientable n-manifold with no boundary, show that <math>H^n_{dR}(M)\neq 0</math>.		'''Problem 1.''' If <math>M</math> is a compact orientable n-manifold with no boundary, show that <math>H^n_{dR}(M)\neq 0</math>.

	'''Problem 3.''' The "standard volume form on S^2" is the form <math>\omega</math> given by <math>\omega=\frac{1}{4\pi}\left(xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\right)</math>. Show that <math>\int_{S^2}\omega=1</math>.		'''Problem 2.''' The "standard volume form on S^2" is the form <math>\omega</math> given by <math>\omega=\frac{1}{4\pi}\left(xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\right)</math>. Show that <math>\int_{S^2}\omega=1</math>.

	'''Problem 3.''' Show that if <math>\omega\in\Omega^2(S^2)</math> satisfies <math>\int_{S_2}\omega=0</math>, then <math>\omega</math> is exact, and therefore, if <math>w_1\in\Omega^2(S^2)</math> and <math>w_2\in\Omega^2(S^2)</math> satisfy <math>\int_{S_2}\omega_1=\int_{S_2}\omega_2</math>, then <math>[\omega_1]=[\omega_2]</math> as elements of <math>H^n_{dR}(S^2)</math>.		'''Problem 3.''' Show that if <math>\omega\in\Omega^2(S^2)</math> satisfies <math>\int_{S_2}\omega=0</math>, then <math>\omega</math> is exact, and therefore, if <math>w_1\in\Omega^2(S^2)</math> and <math>w_2\in\Omega^2(S^2)</math> satisfy <math>\int_{S_2}\omega_1=\int_{S_2}\omega_2</math>, then <math>[\omega_1]=[\omega_2]</math> as elements of <math>H^n_{dR}(S^2)</math>.