![0708-1300-ClassPhoto.jpg](/images/thumb/d/d4/0708-1300-ClassPhoto.jpg/215px-0708-1300-ClassPhoto.jpg) Add your name / see who's in!
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Week of...
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Links
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Fall Semester
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1
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Sep 10
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About, Tue, Thu
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2
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Sep 17
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Tue, HW1, Thu
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3
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Sep 24
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Tue, Photo, Thu
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4
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Oct 1
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Questionnaire, Tue, HW2, Thu
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5
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Oct 8
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Thanksgiving, Tue, Thu
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6
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Oct 15
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Tue, HW3, Thu
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7
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Oct 22
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Tue, Thu
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8
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Oct 29
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Tue, HW4, Thu, Hilbert sphere
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9
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Nov 5
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Tue,Thu, TE1
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10
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Nov 12
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Tue, Thu
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11
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Nov 19
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Tue, Thu, HW5
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12
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Nov 26
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Tue, Thu
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13
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Dec 3
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Tue, Thu, HW6
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Spring Semester
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14
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Jan 7
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Tue, Thu, HW7
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15
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Jan 14
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Tue, Thu
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16
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Jan 21
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Tue, Thu, HW8
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17
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Jan 28
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Tue, Thu
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18
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Feb 4
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Tue
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19
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Feb 11
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TE2, Tue, HW9, Thu, Feb 17: last chance to drop class
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R
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Feb 18
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Reading week
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20
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Feb 25
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Tue, Thu, HW10
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21
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Mar 3
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Tue, Thu
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22
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Mar 10
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Tue, Thu, HW11
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23
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Mar 17
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Tue, Thu
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24
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Mar 24
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Tue, HW12, Thu
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25
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Mar 31
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Referendum,Tue, Thu
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26
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Apr 7
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Tue, Thu
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R
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Apr 14
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Office hours
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R
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Apr 21
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Office hours
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F
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Apr 28
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Office hours, Final (Fri, May 2)
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Register of Good Deeds
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Errata to Bredon's Book
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Announcements go here
Reading
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
, 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
?
- How did we get
from the least action principle?
- Do you understand how Poincare's lemma entered the derivation of Maxewell's equations?
- Do you understand the operator
? (How was it used, formally derived, and what is the intuitive picture behind it?)
- What was
and how did it relate to pullbacks and homotopy.
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!
Problem 1. If
is a compact orientable
-manifold with no boundary, show that
.
Problem 2. The standard volume form on
is the form
given by
. Show that
.
Problem 3. Show that if
satisfies
, then
is exact. Deduce that if
and
satisfy
, then
as elements of
. Deduce further that
.
Problem 4. A "link" in
is an ordered pair
, in which
and
are smooth embeddings of the circle
into
, whose images (called "the components of
") 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
, define a map
by
. Finally, let
be the standard volume form of
, and define "the linking number of
" to be
. Show
- If two links
and
are isotopic, then their linking numbers are the same:
.
- If
is a second 2-form on
for which
and if
is defined in the same manner as
except replacing
with
, then
. (In particular this is true if
is very close to a
-function form at the north pole of
).
- Compute (but just up to an overall sign) the linking number of the link L11a193, displayed below:
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The links L11a193, and .
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Just for Fun
Prove that the two (3-component) links
and
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.