![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
English Spelling
Many interesting rules about 0708-1300/English Spelling
Class Notes
The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.
General class comments
1) The class photo is up, please add yourself
2) A questionnaire was passed out in class
3) Homework one is due on thursday
First Hour
Today's Theme: Locally a function looks like its differential
Pushforward/Pullback
Let
be a smooth map.
We consider various objects, defined with respect to X or Y, and see in which direction it makes sense to consider corresponding objects on the other space. In general
will denote the push forward, and
will denote the pullback.
1) points pushforward
2) Paths
, ie a bunch of points, pushforward,
3) Sets
pullback via
Note that if one tried to pushforward sets A in X, the set operations compliment and intersection would not commute appropriately with the map
4) A measures
pushforward via
5)In some sense, we consider functions, "dual" to points and thus should go in the opposite direction of points, namely
6) Tangent vectors, defined in the sense of equivalence classes of paths, [
] pushforward as we would expect since each path pushes forward.
CHECK: This definition is well defined, that is, independent of the representative choice of
7) We can consider operators on functions to be in a sense dual to the functions and hence should go in the opposite direction. Hence, tangent vectors, defined in the sense of derivations, pushforward via
CHECK: This definition satisfies linearity and Liebnitz property.
Theorem 1
The two definitions for the pushforward of a tangent vector coincide.
Proof:
Given a
we can construct
as above. However from both
and
we can also construct
and
because we have previously shown our two definitions for the tangent vector are equivalent. We can then pushforward
to get
. The theorem is reduced to the claim that:
for functions
Now,
Q.E.D
Functorality
let
Consider some "object" s defined with respect to X and some "object u" defined with respect to Z. Something has the property of functorality if
and
Claim: All the classes we considered previously have the functorality property; in particular, the pushforward of tangent vectors does.
Let us consider
on
given a
We can arrange for charts
on a subset of M into
(with coordinates denoted
)and
on a subset of N into
(with coordinates denoted
)such that
and
Define
Now, for a
we can write
So,
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Now, we want to write
and so,
where the 1 is at the kth location. In other words,
So,
, i.e.,
is the differential of
at p
We can check the functorality,
, then
This is just the chain rule.
Second Hour
Defintion 1
An immersion is a (smooth) map
such that
of tangent vectors is 1:1. More precisely,
is 1:1
Example 1
Consider the canonical immersion, for m<n given by
with n-m zeros.
Example 2
This is the map from
to
that looks like a loop-de-loop on a roller coaster (but squashed into the plane of course!) The map
itself is NOT 1:1 (consider the crossover point) however
IS 1:1, hence an immersion.
Example 3
Consider the map from
to
that looks like a check mark. While this map itself is 1:1,
is NOT 1:1 (at the cusp in the check mark) and hence is not an immersion.
Example 4
Can there be objects, such as the graph of |x| that are NOT an immersion, but are constructed from a smooth function?
Consider the function
for x>0 and 0 otherwise.
Then the map
is a smooth mapping with the graph of |x| as its image, but is NOT an immersion.
Example 5
The torus, as a subset of
is an immersion
Now, consider the 1:1 linear map
where V,W are vector spaces that takes
From linear algebra we know that we can choose a basis such that T is represented by a matrix with 1's along the first m diagonal locations and zeros elsewhere.
Theorem 2
Locally, every immersion looks like the inclusion
.
More precisely, if
and
is 1:1 then there exist charts
acting on
and
acting on
such that for
such that the following diagram commutes:
that is,
Definition 2
M is a submanifold of N if there exists a mapping
such that
is a 1:1 immersion.
Example 6
Our previous example of the graph of a "loop-de-loop", while an immersion, the function is not 1:1 and hence the graph is not a sub manifold.
Example 6
The torus is a submanifold as the natural immersion into
is 1:1
Definition 3
The map
is an embedding if the subset topology on
coincides with the topology induced from the original topology of M.
Example 7
Consider the map from
whose graph looks like the open interval whose two ends have been wrapped around until they just touch (would intersect at one point if they were closed) the points 1/3 and 2/3rds of the way along the interval respectively.
The map is both 1:1 and an immersion. However, any neighborhood about the endpoints of the interval will ALSO include points near the 1/3rd and 2/3rd spots on the line, i.e., the topology is different and hence this is not an embedding.
Corollary 1 to Theorem 2
The functional structure on an embedded manifold induced by the functional structure on the containing manifold is equal to its original functional structure.
Indeed, for all smooth
and
there exists a neighborhood V of
and a smooth
such that
Proof of Corollary 1
Loosely (and a sketch is most useful to see this!) we consider the embedded submanifold M in N and consider its image, under the appropriate charts, to a subset of
. We then consider some function defined on M, and hence on the subset in
which we can extend canonically as a constant function in the "vertical" directions. Now simply pullback into N to get the extended member of the functional structure on N.
Proof of Theorem 2
We start with the normal situation of
with M,N manifolds with atlases containing
and
respectively. We also expect that for
. I will first draw the diagram and will subsequently justify the relevant parts. The proof reduces to showing a certain part of the diagram commutes appropriately.
It is very important to note that the
and
are NOT the charts we are looking for , they are merely one of the ones that happen to act about the point p.
In the diagram above,
. So,
and
. Note the
, being merely the normal composition with the appropriate charts, does not fundamentally add anything. What makes this theorem work is the function
We consider the map
given
. We note that
corresponds with the idea of "lifting" a flattened image back to its original height.
Claims:
1)
is invertible near zero. Indeed, computing
which is invertible as a matrix and hence
is invertible as a function near zero.
2) Take an
. There are two routes to get to
and upon computing both ways yields the same result. Hence, the diagram commutes.
Hence, our immersion looks (locally) like the standard immersion between real spaces given by
and the charts are the compositions going between
to
and
to
Q.E.D