0708-1300/Homework Assignment 6

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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:

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

Problem 3. 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^n_{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 homotopy along which the components 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

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:

0708-1300-LinkComplementExample1.png 0708-1300-LinkComplementExample2.png

(See more at Classes: 2004-05: Math 1300Y - Topology: Homework Assignment 5)