Difference between revisions of "0708-1300/Homework Assignment 6"

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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>.
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'''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>.
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'''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
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'''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>).

Revision as of 19:44, 5 December 2007

Announcements go here
In Preparation

The information below is preliminary and cannot be trusted! (v)

Contents

Reading

At your leisure, read your class notes over the break, and especially at some point right before classes resume after the break.

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!

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

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

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

Problem 4. A "link" in {\mathbb R}^3 is an ordered pair \gamma=(\gamma_1, \gamma_2), in which \gamma_1 and \gamma_2 are smooth embeddings of the circle S^1 into {\mathbb R}^3, whose images (called "the components of \gamma") 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 \gamma, define a map \Phi_\gamma:S^1\times S^1\to S^2 by \Phi_\gamma(t_1,t_2):=\frac{\gamma_2(t_2)-\gamma_1(t_1)}{||\gamma_2(t_2)-\gamma_1(t_1)||}. Finally, let \omega be the standard volume form of S^2, and define "the linking number of \gamma=(\gamma_1, \gamma_2)" to be l(\gamma)=l(\gamma_1,\gamma_2):=\int_{S^1\times S^1}\Phi_\gamma^\star\omega. Show

  1. If two links \gamma and \gamma' are isotopic, then their linking numbers are the same: l(\gamma)=l(\gamma').
  2. If \omega' is a second 2-form on S^2 for which \int_{S^2}\omega'=1 and if l'(\gamma) is defined in the same manner as l(\gamma) except replacing \omega with \omega', then l(\gamma)=l'(\gamma). (In particular this is true if \omega' is very close to a \delta-function form at the north pole of S^2).
  3. Compute (but just up to an overall sign) the linking number of the link L11a193, displayed below:
L11a193.png 0708-1300-LinkComplementExample1.png 0708-1300-LinkComplementExample2.png
The links L11a193, \gamma_3 and \gamma'_3.

Just for Fun

Prove that the two (3-component) links \gamma_3 and \gamma'_3 shown above are not isotopic, yet their complements are diffeomorphic. (See more at Classes: 2004-05: Math 1300Y - Topology: Homework Assignment 5)

Due Date

This assignment is due in class on Thursday January 10, 2007.