Drorbn - User contributions [en]

Help:Contents

2008-03-22T17:27:09Z

Canghel: /* Wiki Editing Help */

The web site is a wiki, as in [http://www.wikipedia.org Wikipedia] - meaning that anyone can and is welcome to edit almost anything and in particular, anybody can post notes, comments, pictures, whatever.

==Some Rules==
* This wiki is a part of my ([[User:Drorbn|Dror's]]) academic web page. All postings on it must be related to one of the projects I'm involved with.
* If there's no specific reason for your edit to be anonymous, please log in and don't have it anonymous.
* Criticism is fine, but no insults or foul language, please.
* I ([[User:Drorbn|Dror]]) will allow myself to exercise editorial control, when necessary.

==Wiki Editing Help==
Pages on this wiki are can and are edited in a similar way to pages on [http://en.wikipedia.org/wiki/Main_Page Wikipedia].
For details, see [http://www.mediawiki.org/wiki/Help:Contents MediaWiki:Help:Content], [http://en.wikipedia.org/wiki/Wikipedia:How_to_edit_a_page Wikipedia:How to edit a page],
[http://meta.wikimedia.org/wiki/Help:Editing WikiMedia:Help:Editing]
and [http://meta.wikimedia.org/wiki/Help:Displaying_a_formula WikiMedia:Help:Displaying a formula].

Some stuff to help get started with contributing to the website: [[WikiHelp]].

==Some Local Help==
===Local Templates===
A full list of local templates is [{{SERVER}}/drorbn/index.php?title=Special%3AAllpages&from=&namespace=10 here].

====Mathematica-Related Templates====
Used for simulating mathematica sessions: [[Template:In]], [[Template:Out]], [[Template:InOut]], [[Template:Print]], [[Template:Message]].

====Mathematical Typesetting====
Used for typesetting and referring to equations: [[Template:Equation]], [[Template:EqRef]], [[Template:Equation*]].

====Citations and Notes====
Bottom notes: [[Template:note]], [[Template:ref]].

Side notes: [[Template:Begin Side Note]], [[Template:End Side Note]].

===Local Clip Art===
<gallery>
Image:BigCirc_symbol.png|BigCirc_symbol.png
Image:doublepoint_symbol.png|doublepoint_symbol.png
Image:overcrossing_symbol.png|overcrossing_symbol.png
Image:undercrossing_symbol.png|undercrossing_symbol.png
</gallery>

==Admin==

See [[Installation Notes]].

Help:Contents

2008-03-22T17:26:22Z

Canghel: /* Wiki Editing Help */

The web site is a wiki, as in [http://www.wikipedia.org Wikipedia] - meaning that anyone can and is welcome to edit almost anything and in particular, anybody can post notes, comments, pictures, whatever.

==Some Rules==
* This wiki is a part of my ([[User:Drorbn|Dror's]]) academic web page. All postings on it must be related to one of the projects I'm involved with.
* If there's no specific reason for your edit to be anonymous, please log in and don't have it anonymous.
* Criticism is fine, but no insults or foul language, please.
* I ([[User:Drorbn|Dror]]) will allow myself to exercise editorial control, when necessary.

==Wiki Editing Help==
Pages on this wiki are can and are edited in a similar way to pages on [http://en.wikipedia.org/wiki/Main_Page Wikipedia].
For details, see [http://www.mediawiki.org/wiki/Help:Contents MediaWiki:Help:Content], [http://en.wikipedia.org/wiki/Wikipedia:How_to_edit_a_page Wikipedia:How to edit a page],
[http://meta.wikimedia.org/wiki/Help:Editing WikiMedia:Help:Editing]
and [http://meta.wikimedia.org/wiki/Help:Displaying_a_formula WikiMedia:Help:Displaying a formula].

Some stuff I wish I'd known at the beginning: [[WikiHelp]].

==Some Local Help==
===Local Templates===
A full list of local templates is [{{SERVER}}/drorbn/index.php?title=Special%3AAllpages&from=&namespace=10 here].

====Mathematica-Related Templates====
Used for simulating mathematica sessions: [[Template:In]], [[Template:Out]], [[Template:InOut]], [[Template:Print]], [[Template:Message]].

====Mathematical Typesetting====
Used for typesetting and referring to equations: [[Template:Equation]], [[Template:EqRef]], [[Template:Equation*]].

====Citations and Notes====
Bottom notes: [[Template:note]], [[Template:ref]].

Side notes: [[Template:Begin Side Note]], [[Template:End Side Note]].

===Local Clip Art===
<gallery>
Image:BigCirc_symbol.png|BigCirc_symbol.png
Image:doublepoint_symbol.png|doublepoint_symbol.png
Image:overcrossing_symbol.png|overcrossing_symbol.png
Image:undercrossing_symbol.png|undercrossing_symbol.png
</gallery>

==Admin==

See [[Installation Notes]].

WikiHelp

2008-03-22T17:20:55Z

Canghel: /* Where/How to Scan Notes */

==How to get started using Wiki==

===Where/How to Scan Notes===

# Go to Robarts Library to the "Design Space" computers (first floor, to the left of the information desk, all the way at the back of the building). All of these computers have a scanner. It's best to go in the morning or late evening to avoid line-ups.
# Click Start-->Publishing Tools-->ScanWizard 5.
# If something doesn't work right, it might be that the scanner is not turned on.
# I used the options Original-->Photo, Scan Type-->Grey, Purpose-->Laser Print Fine, and saved the file directly in the My Documents (temporary files) directory, to erase later. This seems to make the size of the file small enough for uploading.

===How to add a link to the Navigation Panel===

# This I've spend half an hour trying to figure this out, so maybe someone could add instructions? I want to link a "Thu" page to the navigation panel to add notes, maybe it's obvious, but I don't know how to do it.

0708-1300/Quotes

2008-03-22T17:13:17Z

Canghel:

===Quotes from Class===

Oh, I forgot my spaceship!

Here's the monster, let's make it a happy one.

So boring claim 8 up to 22, <math>\pi_1</math> is a functor.

My next goal is to compute the fundamental group of <math>S^3</math> in two silly ways. The first way is silly. The second is even more silly.

If life stopped at <math>H_1</math> we wouldn't be doing this.

Two are trivial and the other is so horrible as to compensate for the triviality of the other two.

I didn't mean for this formula to look so ridiculous. (<math>T^T T</math>)

===Quotes from Studying===

First math pub night:

MS: We proved the implicit function theorem with guns!

AC: Parameterized families of guns.

(explanation)

MB: So in other words, you blew up the neighbors' house.

Mario explaining orientation: I would add a <math>v_4</math> but this room is not big enough.

It's interesting that I can prove something that I can also give a counter-example to.

0708-1300/About This Class

2008-03-22T16:48:40Z

Canghel: /* Class Photo */

{{0708-1300/Navigation}}

===Crucial Information===
{{0708-1300/Crucial Information}}

'''URL:''' {{SERVER}}/drorbn/index.php?title=0708-1300.

===Optimistic Plan===
* 8 weeks of local differential geometry: the differential, the inverse function theorem, smooth manifolds, the tangent space, immersions and submersions, regular points, transversality, Sard's theorem, the Whitney embedding theorem, smooth approximation, tubular neighborhoods, the Brouwer fixed point theorem.
* 5 weeks of differential forms: exterior algebra, forms, pullbacks, <math>d</math>, integration, Stokes' theorem, div grad curl and all, Lagrange's equation and Maxwell's equations, homotopies and Poincare's lemma, linking numbers.
* 5 weeks of fundamental groups: paths and homotopies, the fundamental group, coverings and the fundamental group of the circle, Van-Kampen's theorem, the general theory of covering spaces.
* 8 weeks of homology: simplices and boundaries, prisms and homotopies, abstract nonsense and diagram chasing, axiomatics, degrees, CW and cellular homology, subdivision and excision, the generalized Jordan curve theorem, salad bowls and Borsuk-Ulam, cohomology and de-Rham's theorem, products.

===Warning===
The class will be hard and challenging and will include a substantial component of self-study. To take it you must feel at home with point-set topology, multivariable calculus and basic group theory.

===Textbooks===
We will mainly use Glen E. Bredon's Topology and Geometry (GTM 139, ISBN 978-0-387-97926-7). Additional texts include Allen Hatcher's [http://www.math.cornell.edu/~hatcher/AT/ATpage.html ''Algebraic Topology''] (Free!), Guillemin and Pollack's ''Differential Topology'' and texts by Bott and Tu, Fulton, Massey and many others.

===Wiki===
The class web site is a wiki, as in [http://www.wikipedia.org Wikipedia] - meaning that anyone can and is welcome to edit almost anything and in particular, students can post notes, comments, pictures, whatever. Some rules, though -
* This wiki is a part of my ({{Dror}}'s) academic web page. All postings on it must be class-related (or related to one of the other projects I'm involved with).
* To edit a page on this wiki you must login; to get yourself a login name, email {{Dror}} your full name, email address and preferred login name and you will receive a password via email within a day or two.
* I ({{Dror}}) will allow myself to exercise editorial control, when necessary.
* The titles of all pages/images related to this class should begin with "0708-1300/" or with "0708-1300-", just like the title of this page.
* Some further editing help is available at [[Help:Contents]].

===Good Deeds===
Students will be able to earn up to 40 "good deeds" points throughout the year (20 for each semester) for doing services to the class as a whole. There is no pre-set system for awarding these points, but the following will definitely count:
* Drawing a beautiful picture to illustrate a point discussed in class and posting it on this site.
* Taking class notes in nice handwriting, scanning them and posting them here.
* Typing up or formatting somebody else's class notes, correcting them or expanding them in any way.
* Writing an essay expanding on anything mentioned in class and posting it here; correcting or expanding somebody else's article.
* Doing anything on our [[0708-1300/To do]] list.
* Any other service to the class as a whole.

Good deed points will count towards your final grade! If you got <math>n</math> of those, they are solidly your and the formula for the final grade below will only be applied to the remaining <math>100-n</math> points. So if you got 25 good deed points (say) and your final grade is 80, I will report your grade as <math>25+80(100-25)/100=85</math>. Yet you can get an overall 100 even without doing a single good deed.

'''Important.''' For your good deeds to count, you '''must''' add your userid to the [[0708-1300/Class Photo|Class Photo]] page (see below), or else I will not know to search for your work on the web site.

[[More on Good Deeds]].

===The Final Grade===
The "base grade" for this class will be
<math>b:=0.2HW+0.15TE1+0.15TE2+0.5FE</math>, where <math>HW</math>, <math>TE1</math>,
<math>TE2</math> and <math>FE</math> are the Home Work, Term Exam 1, Term Exam 2
and Final Exam grades respectively. The final grade will be <math>f:=n+b(100-n)/100</math>, as discussed above. A monotone increasing function might or might not be applied to <math>f</math> before it is reported to the department.

===Homework===
There will be about 12 problem sets. I encourage
you to discuss the homeworks with other students or even browse the
web, so long as you do at least some of the thinking on your own and
you write up your own solutions. The assignments will be assigned on
Thursdays and each will be due on the date of the following assignment,
in class at 1PM (see the Navigation Panel). There will be 10 points penalty for late assignments
(20 points if late by more than a week and another 10 points for every
week beyond that). Your 10 best assignments will count towards your
homework grade.

===The Term Exams===
[[0708-1300/Term Exam 1|Term Exam 1]] will take place on Thursday November 8 at 6PM. Term Exam 2 will take place in the afternoon or evening outside of class time, on the week February 11. Each will be 2 hours long.

===Class Photo===
To help me learn your names, I will take a class photo on the third week of classes. I will post the picture on the wiki and you will be ''required'' to identify yourself on the [[0708-1300/Class Photo|Class Photo]] page.

[[Quotes]]

User talk:Trefor

2008-03-22T16:46:08Z

Canghel:

Hello, could you please tell me how to add a link like "Thu" to the Navigation Panel? I made a Wiki Help page, you could add it there. Thank you!

WikiHelp

2008-03-22T16:40:00Z

Canghel: /* How to get started using Wiki */

==How to get started using Wiki==

===Where/How to Scan Notes===

# Go to Robarts Library to the "Design Space" computers (first floor, to the left of the information desk, all the way at the back of the building). All of these computers have a scanner. It's best to go in the morning or late evening to avoid line-ups.
# Click Start-->Publishing Tools-->ScanWizard 5.
# If something doesn't work right, it might be that the scanner is not turned on.
# I used the options Scan Type-->Grey, Purpose-->Laser Print Fine, and saved the file directly in the My Documents (temporary files) directory, to erase later. This seems to make the size of the file small enough for uploading.

===How to add a link to the Navigation Panel===

# This I've spend half an hour trying to figure this out, so maybe someone could add instructions? I want to link a "Thu" page to the navigation panel to add notes, maybe it's obvious, but I don't know how to do it.

WikiHelp

2008-03-22T16:36:52Z

Canghel:

==How to get started using Wiki==

===Where/How to Scan Notes===

# Go to Robarts Library to the "Design Space" computers (first floor, to the left of the information desk, all the way at the back of the building). All of these computers have a scanner. It's best to go in the morning or late evening to avoid line-ups.
# Click Start-->Publishing Tools-->ScanWizard 5.
# If something doesn't work right, it might be that the scanner is not turned on.
# I used the options Scan Type-->Grey, Purpose-->Laser Print Fine, and saved the file directly in the My Documents (temporary files) directory, to erase later. This seems to make the size of the file small enough for uploading.

===How to add a link to the Navigation Panel===

# This I've spend half an hour trying to figure out, so maybe someone could add instructions?

Help:Contents

2008-03-22T16:23:05Z

Canghel: /* Wiki Editing Help */

The web site is a wiki, as in [http://www.wikipedia.org Wikipedia] - meaning that anyone can and is welcome to edit almost anything and in particular, anybody can post notes, comments, pictures, whatever.

==Some Rules==
* This wiki is a part of my ([[User:Drorbn|Dror's]]) academic web page. All postings on it must be related to one of the projects I'm involved with.
* If there's no specific reason for your edit to be anonymous, please log in and don't have it anonymous.
* Criticism is fine, but no insults or foul language, please.
* I ([[User:Drorbn|Dror]]) will allow myself to exercise editorial control, when necessary.

==Wiki Editing Help==
Pages on this wiki are can and are edited in a similar way to pages on [http://en.wikipedia.org/wiki/Main_Page Wikipedia].
For details, see [http://www.mediawiki.org/wiki/Help:Contents MediaWiki:Help:Content], [http://en.wikipedia.org/wiki/Wikipedia:How_to_edit_a_page Wikipedia:How to edit a page],
[http://meta.wikimedia.org/wiki/Help:Editing WikiMedia:Help:Editing]
and [http://meta.wikimedia.org/wiki/Help:Displaying_a_formula WikiMedia:Help:Displaying a formula].

Some stuff I wish I'd known about wiki at the beginning: [[WikiHelp]].

==Some Local Help==
===Local Templates===
A full list of local templates is [{{SERVER}}/drorbn/index.php?title=Special%3AAllpages&from=&namespace=10 here].

====Mathematica-Related Templates====
Used for simulating mathematica sessions: [[Template:In]], [[Template:Out]], [[Template:InOut]], [[Template:Print]], [[Template:Message]].

====Mathematical Typesetting====
Used for typesetting and referring to equations: [[Template:Equation]], [[Template:EqRef]], [[Template:Equation*]].

====Citations and Notes====
Bottom notes: [[Template:note]], [[Template:ref]].

Side notes: [[Template:Begin Side Note]], [[Template:End Side Note]].

===Local Clip Art===
<gallery>
Image:BigCirc_symbol.png|BigCirc_symbol.png
Image:doublepoint_symbol.png|doublepoint_symbol.png
Image:overcrossing_symbol.png|overcrossing_symbol.png
Image:undercrossing_symbol.png|undercrossing_symbol.png
</gallery>

==Admin==

See [[Installation Notes]].

Help:Contents

2008-03-22T16:22:46Z

Canghel: /* Wiki Editing Help */

The web site is a wiki, as in [http://www.wikipedia.org Wikipedia] - meaning that anyone can and is welcome to edit almost anything and in particular, anybody can post notes, comments, pictures, whatever.

==Some Rules==
* This wiki is a part of my ([[User:Drorbn|Dror's]]) academic web page. All postings on it must be related to one of the projects I'm involved with.
* If there's no specific reason for your edit to be anonymous, please log in and don't have it anonymous.
* Criticism is fine, but no insults or foul language, please.
* I ([[User:Drorbn|Dror]]) will allow myself to exercise editorial control, when necessary.

==Wiki Editing Help==
Pages on this wiki are can and are edited in a similar way to pages on [http://en.wikipedia.org/wiki/Main_Page Wikipedia].
For details, see [http://www.mediawiki.org/wiki/Help:Contents MediaWiki:Help:Content], [http://en.wikipedia.org/wiki/Wikipedia:How_to_edit_a_page Wikipedia:How to edit a page],
[http://meta.wikimedia.org/wiki/Help:Editing WikiMedia:Help:Editing]
and [http://meta.wikimedia.org/wiki/Help:Displaying_a_formula WikiMedia:Help:Displaying a formula].

Some stuff I wish I'd known about wiki at the beginning: [[1300_WikiHelp:WikiHelp]].

==Some Local Help==
===Local Templates===
A full list of local templates is [{{SERVER}}/drorbn/index.php?title=Special%3AAllpages&from=&namespace=10 here].

====Mathematica-Related Templates====
Used for simulating mathematica sessions: [[Template:In]], [[Template:Out]], [[Template:InOut]], [[Template:Print]], [[Template:Message]].

====Mathematical Typesetting====
Used for typesetting and referring to equations: [[Template:Equation]], [[Template:EqRef]], [[Template:Equation*]].

====Citations and Notes====
Bottom notes: [[Template:note]], [[Template:ref]].

Side notes: [[Template:Begin Side Note]], [[Template:End Side Note]].

===Local Clip Art===
<gallery>
Image:BigCirc_symbol.png|BigCirc_symbol.png
Image:doublepoint_symbol.png|doublepoint_symbol.png
Image:overcrossing_symbol.png|overcrossing_symbol.png
Image:undercrossing_symbol.png|undercrossing_symbol.png
</gallery>

==Admin==

See [[Installation Notes]].

0708-1300/Class Photo

2008-02-06T05:32:09Z

Canghel: /* Who We Are... */

{{0708-1300/Navigation}}

Our class on September 27, 2007:

[[Image:0708-1300-ClassPhoto.jpg|thumb|centre|600px|Class Photo: click to enlarge]]

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1
|-
!First name
!Last name
!UserID
!Email
!In the photo
!Comments
{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn @ math.toronto.edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank}}
{{Photo Entry|last=Antolin Camarena|first=Omar|userid=oantolin|email=oantolin @ math.utoronto.ca|location=furthest person to the right|comments=I'm the TA}}
{{Photo Entry|last=Bazett|first=Trefor|userid=Trefor|email=trefor.bazett @ toronto.ca|location=tallest person a little right of center in a beige shirt|comments=}}
{{Photo Entry|last=Bjorndahl|first=Adam|userid=ABjorndahl|email=adam.bjorndahl @ utoronto.ca|location=back row, fifth from the left, under the "f(tp)dt"|comments=Looking forward to a great year!}}
{{Photo Entry|last=Choi|first=Brian|userid=Brianchoi|email=brianymc.choi @ utoronto.ca|location=In the middle of the front row, the weird looking (!) guy with brown shirt over blue and white|comments=}}
{{Photo Entry|last=Chow|first=Aaron|userid=aaron.chow|email=aaron.chow @ utoronto.ca|location=Third from right, in a black shirt.|comments=Hope we have a good year together!}}
{{Photo Entry|last=DeCorte|first=Evan|userid=Evan.decorte|email= |location=third from the left, tall guy|comments=}}
{{Photo Entry|last=Fisher|first=Jonathan|userid=jonathan.fisher|email=jonathan.fisher @ utoronto.ca|location=6th from the right, brown shirt, eyes closed|comments=}}
{{Photo Entry|last=Isgur|first=Abraham|userid=Abisgu|email=abraham.isgur @ math.toronto.edu|location=2nd person in the back row, from the right, the one with the beard and long hair|comments=}}
{{Photo Entry|last=Kinzebulatov|first=Damir|userid=Dkinz|email=dkinz @ math.toronto.edu|location=In the middle in red shirt|comments=}}
{{Photo Entry|last=Liu|first=Xiao|userid=Ninetiger|email=ninetiger.liu @ utoronto.ca|location=In the first row. A boy in orange T-shirt|comments=}}
{{Photo Entry|last=Li|first=Zhiqiang|userid=li-zhiqiang|email=lizhiqiangfly @ gmail.com|location=2nd from the left, 1st boy in the front row.|comments=}}
{{Photo Entry|last=Mann|first=Katie|userid=katiemann|email=katie.mann @ utoronto.ca|location=middle, wearing "Eulers" shirt|comments=}}
{{Photo Entry|last=Mourtada|first=Mariam|userid=Mourtada|email=mariam.mourtada @ utoronto.ca|location=I am the girl in the front middle, wearing a blue shirt and catching my hands|comments=I am not wearing glasses! }}
{{Photo Entry|last=Pym|first=Brent|userid=Bpym|email=bpym @ math.toronto.edu|location=10th from the right (cumulatively), under the <math>T_p(M)\!</math>|comments=Adding this entry was my first-ever edit of a Wiki!}}
{{Photo Entry|last=Snow|first=Megan|userid=megan|email=megansnow @ gmail.com|location=back row, slightly right of centre, wearing a blue shirt over a black one|comments=}}
{{Photo Entry|last=Vera Pacheco|first=Franklin|userid=Franklin|email=franklin.vp @ gmail.com|location=Xth from left to right|comments=To find me you must first go to [[http://www.deathball.net/notpron/]] solve the first 4 pages. Once this done you will know how to find me. Once this done go back to NOTPRON an solve the rest of the puzzle}}
{{Photo Entry|last=Watts|first=Jordan|userid=Jwatts|email=jwatts @ math.toronto.edu|location=in the back, 2nd or 3rd from the left, depending on your convention|comments=My glasses become invisible in pictures.}}
{{Photo Entry|last=Wong|first=Silian|userid=kuramay|email=kurama_y @ hotmail.com|location=One of the Asian-looking girls...with sparkling teeth(??)|comments=I'll write up some comments after their existences}}
{{Photo Entry|last=Francetic|first=Nevena|userid=Nfrancetic|email=nevena.francetic @ utoronto.ca|location=A girl standing a little bit to the left of the center in a white shirt.|comments=}}
{{Photo Entry|last=Anghel|first=Catalina|userid=canghel|email=|location=in red|comments=I decided I'd better get over my fear of websites, since I'm probably going to need the brownie points. Hey, but stuff worked this time! It was actually kind of fun...}}
|}

0708-1300/Class notes for Tuesday, January 29

2008-02-06T05:23:53Z

Canghel: /* Class Notes */

{{0708-1300/Navigation}}
==Two Covering Spaces==
{| align=center cellspacing=20
|- align=center
| [[Image:0708-1300-DoubleCover.png|300px]]
| [[Image:0708-1300-FigureEightSeifert.jpg|300px]]
|- align=left
| [http://www.wolfram.com/ Mathematica] code:
ParametricPlot3D[
{
{r Cos[2t], r Sin[2t], 4 + Cos[t]},
{r Cos[t], r Sin[t], 0},
PlotPoints -> {49, 2}
},
{t, 0, 2Pi}, {r, 1, 3}
]
| Seifert surface of the Figure Eight Knot, drawn using [http://www.win.tue.nl/~vanwijk/ Jack van Wijk]'s amazing [http://www.win.tue.nl/~vanwijk/seifertview/ SeifertView].
|}

==14 Covering Spaces==

14 further covering spaces can be found on [http://www.math.cornell.edu/~hatcher/AT/ATch1.pdf#page=38 page 58 of Hatcher's book].

==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.

[[Image:0708-1300_notes_29-01-08a.jpg|200px]]
[[Image:0708-1300_notes_29-01-08b.jpg|200px]]
[[Image:0708-1300_notes_29-01-08c.jpg|200px]]
[[Image:0708-1300_notes_29-01-08d.jpg|200px]]

0708-1300/Class notes for Tuesday, January 29

2008-02-06T05:22:28Z

Canghel: /* 14 Covering Spaces */

File:0708-1300 notes 29-01-08d.jpg

2008-02-06T05:20:43Z

Canghel:

File:0708-1300 notes 29-01-08c.jpg

2008-02-06T05:20:33Z

Canghel:

File:0708-1300 notes 29-01-08b.jpg

2008-02-06T05:20:23Z

Canghel:

File:0708-1300 notes 29-01-08a.jpg

2008-02-06T05:20:11Z

Canghel:

0708-1300/Class notes for Tuesday, January 22

2008-02-06T05:04:54Z

Canghel: /* Second Hour */

{{0708-1300/Navigation}}

==Pictures for a Van-Kampen Computation==
{{In|
n = 1 |
in = <nowiki><< KnotTheory`</nowiki>}}

<tt>Loading KnotTheory` version of January 13, 2008, 20:30:12.1353. 
Read more at http://katlas.org/wiki/KnotTheory.</tt>

{{Graphics|
n = 2 |
in = <nowiki>TubePlot[TorusKnot[8, 3]]</nowiki> |
img= 0708-1300-T83.png}}

{{In|
n = 3 |
in = <nowiki>TC[r1_, t1_,r2_,t2_ ] := {
(r1 +r2 Cos[2Pi t2])Cos[2Pi t1],
(r1 +r2 Cos[2Pi t2])Sin[2Pi t1],
r2 Sin[2Pi t2]
};</nowiki>}}

{{In|
n = 4 |
in = <nowiki>InflatedTorus[p_, q_, b_] := ParametricPlot3D[
TC[
2, p t - q s,
1 + b(p^2 + q^2)s(1 - (p^2 + q^2)s), q t + p s
],
{t, 0, 1}, {s, 0, 1/(p^2 + q^2)},
PlotPoints -> {6(p^2 + q^2) + 1, 7},
DisplayFunction -> Identity
];</nowiki>}}

{{Graphics|
n = 5 |
in = <nowiki>GraphicsArray[{{InflatedTorus[3,8,1], InflatedTorus[3,8,-1]}}]</nowiki> |
img= 0708-1300-InflatedTori.png |
width = 640px}}

==Typed Notes==

===First Hour===

'''Today's Agenda:'''

1) More Examples of Van-Kampen Theorem

2) More Diagrams

3) Proof of Van-Kampen (was not done)

We began by recalling the examples from last class. I will not repeat that here, merely making a few additional comments that came out:

'''Notation:'''

Technically, <math>A*_H B</math> is poor notion as it implies that knowledge of A, B and H is sufficient to construct <math>A*_H B</math>. In fact, we ALSO need to know the maps from H into A and B respectively in order for <math>A*_H B</math> to be defined.

'''Aside'''

Last class we simply wrote down the schematic for the two holed torus as an octagon with the identifications on the edges given last class. We now consider how one arrives at this schematic.

To create the two holed torus one begins with two tori. One then cuts out a small open disk from each torus and then glues the two boundaries together. Let us consider what this looks like when considering a torus as the normal schematic with a square in the plane with the normal identification of the sides. Removing an open disk is equivalent to removing the inside of a loop starting at one of the corners and finishing at that same corner. This is equivalent to making a pentagon with sides <math>aba^{-1}b^{-1}c</math> where c is the added edge.

Consider two such pentagons, gluing along the edge c forms precisely the octagon we had for the two holed torus last class.

[[Image:0708-1300_notes_22-01-08a.jpg|200px]]

'''Proposition'''

Letting <math>\Sigma_g</math> denote the g holed torus, then <math>\Sigma_g\neq\Sigma_{g'}</math>

(Note, I used the symbol <math>\neq</math> to as the normal \ncong command doesn't seem to work. Take its meaning in context.)

Aside: Consider a functor from the category of groups to the category of Abelian groups via

<math>G\mapsto G^{ab} = G/(ab=ba)</math>

If we have a (homo)morphism from <math>G\rightarrow H</math> then the functor takes <math>H\rightarrow H^{ab}</math> and yields a map <math>G^{ab}\rightarrow H^{ab}</math> such that everything commutes.

Hence we know that <math>\pi_1^{ab}(\Sigma_g) \cong \mathbb{Z}^{2g}\neq\mathbb{Z}^{2g'} \cong\pi_1^{ab}(\Sigma_{g'})</math>

Of course, we need to know that in fact <math>\mathbb{Z}^m\neq\mathbb{Z}^n</math> if <math>m\neq n</math>

As such, since the abelianizations are not isomorphic,neither are the original groups and the spaces themselves are not homeomorphic.

'''Example'''

<math>\pi_1(\mathbb{R}P^2)</math> is <math>\pi_1</math> of the space which can be written as a disk with two antipodal points on the boundary circle on it with the identification that the top path a (going clockwise along the boundary) is glued to the bottom path (also going clock wise). But <math>\pi_1</math> of this is just <math><a>/(a^2 = e) \cong \mathbb{Z}/2</math>

'''Claim:'''

Puncturing an n-manifold, <math>n\geq 3</math>, does not change <math>\pi_1(M)</math>. I.e., if <math>p\in M^n</math> then <math>\pi_1(M)\cong\pi_1(M-\{p\})</math>

''Proof:''

Let <math>U_1 = M-\{p\}</math>

<math>U_2</math> = a coordinate patch about p.

Then <math>U_1\cap U_2 = B^n-\{p\}\cong S^{n-1}</math>

If n=3, <math>\pi_1(S^2) = \{e\}</math> as we have computed before.

Hence, <math>\pi_1(M) = \pi_1(U_1)*_{\{\}}\{\} = \pi_1(U_1)</math>

Now, <math>\pi_1(S^3)\cong\pi_1(S^3-\{p\}) = \pi_1(B^3) = \{e\}</math>

Continuing inductively the theorem holds for all n.

'''Aside:'''

If X is connected and <math>b_1,\ b_2\in X</math> then <math>\pi_1(X,b_1) = \pi_1(X,b_2)</math>. I.e., it does not matter which base point we choose in a connected space, the fundamental group is invariant of this.

''Proof''

Consider a path <math>\eta</math> from <math>b_1</math> to <math>b_2</math>. The returning path is denoted <math>\bar{\eta}</math>

Consider a loop from <math>b_2</math> called <math>\gamma</math>.

Then get a loop from <math>b_1</math> via <math>\gamma\mapsto \bar{\eta}\gamma\eta</math>

Similarly about <math>b_2, \gamma\mapsto\eta\gamma\bar{\eta}</math>

Considering the composition we get <math>\eta\bar{\eta}\gamma\eta\bar{\eta}\sim\gamma</math>.

===Second Hour===

'''Claim'''

<math>S^3</math> is the union (with common boundary) of two solid tori <math>S^1\times D^1</math>

The natural way to add two tori with common boundary would be two glue the boundaries of two disks (making <math>S^2</math>) together for each angle going around the torus thus yielding <math>S^1\times S^2</math>. Clearly this is not the same as <math>S^3</math> as the fundamental groups differ.

Instead consider the following description. Look at a torus in the zx plane, this looks like two disks with the z axis in between them such that rotating these two disks about the z axis will yield the torus.

Lets now add in the second torus into this picture. We first draw a horizontal line between the two disks. We then "blow" up from beneath so the horizontal line is slightly curved. We imagine continuing to blow yielding larger and larger loops between the two disks until it "pops" forming the pure horizontal line consisting of the loop at infinity. Do the same for the bottom. Hence, the boundaries of the two tori drawn this way clearly are the same, and between the two cover the entire zx plane (and "point at infinity). Rotating this picture about the z axis yields all of S^1 as the union of these two sets.

[[Image:0708-1300_notes_22-01-08b.jpg|200px]]

'''Claim:'''

<math>\pi_1(S^3) = \{e\}</math>

<math>U_1</math> = the normal solid torus thickened a bit and under <math>\pi_1</math> yields <math><\alpha></math>

<math>U_2</math> = the other solid torus, also thickened a bit, under <math>\pi_1</math> yields <math><\beta></math>
<math>

U_1\cap U_2</math> is a normal torus only with slightly thick walls opposed to infinitely thin ones (homotopically the same)

So, <math>\pi_1(U_1\cap U_2)\cong\mathbb{Z}^2 \cong <a,b>/ab=ba</math>

So, <math>\pi_1(S^3) \cong\mathbb{Z}*_{\mathbb{Z}\times\mathbb{Z}}\mathbb{Z}</math>

However, we still need to describe <math>i_{1*}</math> and <math>i_{2*}</math>

Do do this let me describe a,b,<math>\alpha</math> and <math>\beta</math> explicitly.

Considering the description of the two tori given above, we let a go around the outside of one of the two disks in the plane and b go from a point on the boundary of the same disk, following the rotation about the z axis, to a point on the boundary of the other dis. <math>\alpha</math> is similar to b, but thought of as being on the boundary of the OTHER torus. <math>\beta</math> consists of the path along the z axis.

Hence,

<math>i_{1*}:</math>

<math>a\rightarrow e</math>

<math>b\rightarrow \alpha</math>

<math>i_{2*}:</math>

<math> a\rightarrow\beta</math>

<math>b\rightarrow e</math> (as it is contractible)

Hence, <math>\pi_1(S^3) = F(\alpha, \beta)/(e=\beta, \alpha = e)</math>

Hence, <math>\pi_1(S^3) = \{e\}</math>

'''Example'''

Define the "Torus knot <math>T_{p,q}</math>" where p and q are relatively prime integers. The knot <math>T_{8,3}</math> is given above. We can think of this in the following ways:

1) <math>T_{p,q}</math> is the knot that wraps around the torus p times one way and q times the other way.

2) Formally, let <math>\sigma:S^1\times S^1\mathbb{R}^3</math> be standard embedding of a torus. Let <math>\gamma:[0,1]\rightarrow S^1\times S^1</math> be <math>t\rightarrow (e^{2\pi i pt}, e^{i2\pi qt})</math>

Then <math>T_{p,q}</math> is <math>\sigma\circ\gamma</math>

3) Recall that the torus can be thought of as the image of the mapping <math>\mathbb{R}^2\rightarrow\mathbb{R}^2/\mathbb{Z}^2</math>

Consider the rectangle in the real plane: ([0,p],[0,q]) and consider the path which is the diagonal line from the corner (0,0) to the corner (p,q)

No two points on this line are the same under the mapping down to the torus. If they were, then <math>\Delta y</math> and <math>\Delta x</math> would be integers and hence <math>\Delta y/\Delta x</math> would be the slope of the line. But the slope of the line is q/p which is already in lowest common terms by assumption.

Lets compute the fundamental group of the compliment of the torus knot.

<math>\pi_1(\mathbb{R}^3-T_{p,q})\cong\pi_1(S^3-T_{p,q})</math>

<math>U_1</math>: inflated bagel, constrained by <math>T_{p,q}</math>

<math>U_2</math>: inflated bubble constrained by <math>T_{p,q}</math>

(See top of page for pictures)

The intersection <math>U_1\cap U_2</math> looks somewhat like a belt. It has some thickness to it and is wrapped around the torus, eventually forming a loop. Hence it looks like a squashed disk cross a circle. Hence, under <math>\pi_1</math> this is just <math>\mathbb{Z}\cong<\gamma></math> where <math>\gamma</math> is the path parallel to <math>T_{p,q}</math>

We thus get the maps,

<math>i_{1*}: \gamma\mapsto\alpha^p</math>

<math>i_{2*}:\gamma\mapsto\beta^q</math>

Hence, <math>\pi_1(T_{p,q}^c) = <\alpha,\beta>/\alpha^p = \beta^q</math>

Thus, <math>T_{p,q}\neq T_{p',q'}</math>

'''Diagrams:'''

Recall our diagram from last class:

<math>\begin{matrix}
&\ \ \ \ U_1&&\\
&\nearrow^{i_1}&\searrow^{j_1}&\\
U_1\cap U_2&&&U_1\cup U_2\\
&\searrow_{i_2}&\nearrow^{j_2}&\\
&\ \ \ \ U_2&&\\
\end{matrix}</math>

<math>U_1\cup U_2</math> can be ''defined'' as the object such that the above diagram commutes and should the following commute:

<math>\begin{matrix}
&\ \ \ \ U_1&&\\
&\nearrow^{i_1}&\searrow^{j_1}&\\
U_1\cap U_2&&&Y\\
&\searrow_{i_2}&\nearrow^{j_2}&\\
&\ \ \ \ U_2&&\\
\end{matrix}</math>

Then there is a unique map between <math>U_1\cup U_2</math> and Y such that the composed diagram commutes.

Indeed, the same is true for general categories.

For

<math>\begin{matrix}
&\ \ \ \ G_1&&\\
&\nearrow^{i_1}&\searrow^{j_1}&\\
H&&&P\\
&\searrow_{i_2}&\nearrow^{j_2}&\\
&\ \ \ \ G_2&&\\
\end{matrix}</math>

commuting, P is defined as an object such that if also

<math>\begin{matrix}
&\ \ \ \ G_1&&\\
&\nearrow^{i_1}&\searrow^{j_1}&\\
H&&&Q\\
&\searrow_{i_2}&\nearrow^{j_2}&\\
&\ \ \ \ G_2&&\\
\end{matrix}</math>

were to commute then there is a unique morphism from P to Q such that the composed diagram computes.

In the category of groups, this "pushforward" P is unique and is isomorphic to <math>G_1*_H G_2</math>

0708-1300/Class notes for Tuesday, January 22

2008-02-06T05:04:00Z

Canghel: /* First Hour */

{{0708-1300/Navigation}}

==Pictures for a Van-Kampen Computation==
{{In|
n = 1 |
in = <nowiki><< KnotTheory`</nowiki>}}

<tt>Loading KnotTheory` version of January 13, 2008, 20:30:12.1353. 
Read more at http://katlas.org/wiki/KnotTheory.</tt>

{{Graphics|
n = 2 |
in = <nowiki>TubePlot[TorusKnot[8, 3]]</nowiki> |
img= 0708-1300-T83.png}}

{{In|
n = 3 |
in = <nowiki>TC[r1_, t1_,r2_,t2_ ] := {
(r1 +r2 Cos[2Pi t2])Cos[2Pi t1],
(r1 +r2 Cos[2Pi t2])Sin[2Pi t1],
r2 Sin[2Pi t2]
};</nowiki>}}

{{In|
n = 4 |
in = <nowiki>InflatedTorus[p_, q_, b_] := ParametricPlot3D[
TC[
2, p t - q s,
1 + b(p^2 + q^2)s(1 - (p^2 + q^2)s), q t + p s
],
{t, 0, 1}, {s, 0, 1/(p^2 + q^2)},
PlotPoints -> {6(p^2 + q^2) + 1, 7},
DisplayFunction -> Identity
];</nowiki>}}

{{Graphics|
n = 5 |
in = <nowiki>GraphicsArray[{{InflatedTorus[3,8,1], InflatedTorus[3,8,-1]}}]</nowiki> |
img= 0708-1300-InflatedTori.png |
width = 640px}}

==Typed Notes==

===First Hour===

'''Today's Agenda:'''

1) More Examples of Van-Kampen Theorem

2) More Diagrams

3) Proof of Van-Kampen (was not done)

We began by recalling the examples from last class. I will not repeat that here, merely making a few additional comments that came out:

'''Notation:'''

Technically, <math>A*_H B</math> is poor notion as it implies that knowledge of A, B and H is sufficient to construct <math>A*_H B</math>. In fact, we ALSO need to know the maps from H into A and B respectively in order for <math>A*_H B</math> to be defined.

'''Aside'''

Last class we simply wrote down the schematic for the two holed torus as an octagon with the identifications on the edges given last class. We now consider how one arrives at this schematic.

To create the two holed torus one begins with two tori. One then cuts out a small open disk from each torus and then glues the two boundaries together. Let us consider what this looks like when considering a torus as the normal schematic with a square in the plane with the normal identification of the sides. Removing an open disk is equivalent to removing the inside of a loop starting at one of the corners and finishing at that same corner. This is equivalent to making a pentagon with sides <math>aba^{-1}b^{-1}c</math> where c is the added edge.

Consider two such pentagons, gluing along the edge c forms precisely the octagon we had for the two holed torus last class.

[[Image:0708-1300_notes_22-01-08a.jpg|200px]]

'''Proposition'''

Letting <math>\Sigma_g</math> denote the g holed torus, then <math>\Sigma_g\neq\Sigma_{g'}</math>

(Note, I used the symbol <math>\neq</math> to as the normal \ncong command doesn't seem to work. Take its meaning in context.)

Aside: Consider a functor from the category of groups to the category of Abelian groups via

<math>G\mapsto G^{ab} = G/(ab=ba)</math>

If we have a (homo)morphism from <math>G\rightarrow H</math> then the functor takes <math>H\rightarrow H^{ab}</math> and yields a map <math>G^{ab}\rightarrow H^{ab}</math> such that everything commutes.

Hence we know that <math>\pi_1^{ab}(\Sigma_g) \cong \mathbb{Z}^{2g}\neq\mathbb{Z}^{2g'} \cong\pi_1^{ab}(\Sigma_{g'})</math>

Of course, we need to know that in fact <math>\mathbb{Z}^m\neq\mathbb{Z}^n</math> if <math>m\neq n</math>

As such, since the abelianizations are not isomorphic,neither are the original groups and the spaces themselves are not homeomorphic.

'''Example'''

<math>\pi_1(\mathbb{R}P^2)</math> is <math>\pi_1</math> of the space which can be written as a disk with two antipodal points on the boundary circle on it with the identification that the top path a (going clockwise along the boundary) is glued to the bottom path (also going clock wise). But <math>\pi_1</math> of this is just <math><a>/(a^2 = e) \cong \mathbb{Z}/2</math>

'''Claim:'''

Puncturing an n-manifold, <math>n\geq 3</math>, does not change <math>\pi_1(M)</math>. I.e., if <math>p\in M^n</math> then <math>\pi_1(M)\cong\pi_1(M-\{p\})</math>

''Proof:''

Let <math>U_1 = M-\{p\}</math>

<math>U_2</math> = a coordinate patch about p.

Then <math>U_1\cap U_2 = B^n-\{p\}\cong S^{n-1}</math>

If n=3, <math>\pi_1(S^2) = \{e\}</math> as we have computed before.

Hence, <math>\pi_1(M) = \pi_1(U_1)*_{\{\}}\{\} = \pi_1(U_1)</math>

Now, <math>\pi_1(S^3)\cong\pi_1(S^3-\{p\}) = \pi_1(B^3) = \{e\}</math>

Continuing inductively the theorem holds for all n.

'''Aside:'''

If X is connected and <math>b_1,\ b_2\in X</math> then <math>\pi_1(X,b_1) = \pi_1(X,b_2)</math>. I.e., it does not matter which base point we choose in a connected space, the fundamental group is invariant of this.

''Proof''

Consider a path <math>\eta</math> from <math>b_1</math> to <math>b_2</math>. The returning path is denoted <math>\bar{\eta}</math>

Consider a loop from <math>b_2</math> called <math>\gamma</math>.

Then get a loop from <math>b_1</math> via <math>\gamma\mapsto \bar{\eta}\gamma\eta</math>

Similarly about <math>b_2, \gamma\mapsto\eta\gamma\bar{\eta}</math>

Considering the composition we get <math>\eta\bar{\eta}\gamma\eta\bar{\eta}\sim\gamma</math>.

===Second Hour===

'''Claim'''

<math>S^3</math> is the union (with common boundary) of two solid tori <math>S^1\times D^1</math>

The natural way to add two tori with common boundary would be two glue the boundaries of two disks (making <math>S^2</math>) together for each angle going around the torus thus yielding <math>S^1\times S^2</math>. Clearly this is not the same as <math>S^3</math> as the fundamental groups differ.

Instead consider the following description. Look at a torus in the zx plane, this looks like two disks with the z axis in between them such that rotating these two disks about the z axis will yield the torus.

Lets now add in the second torus into this picture. We first draw a horizontal line between the two disks. We then "blow" up from beneath so the horizontal line is slightly curved. We imagine continuing to blow yielding larger and larger loops between the two disks until it "pops" forming the pure horizontal line consisting of the loop at infinity. Do the same for the bottom. Hence, the boundaries of the two tori drawn this way clearly are the same, and between the two cover the entire zx plane (and "point at infinity). Rotating this picture about the z axis yields all of S^1 as the union of these two sets.

'''Claim:'''

<math>\pi_1(S^3) = \{e\}</math>

<math>U_1</math> = the normal solid torus thickened a bit and under <math>\pi_1</math> yields <math><\alpha></math>

<math>U_2</math> = the other solid torus, also thickened a bit, under <math>\pi_1</math> yields <math><\beta></math>
<math>

U_1\cap U_2</math> is a normal torus only with slightly thick walls opposed to infinitely thin ones (homotopically the same)

So, <math>\pi_1(U_1\cap U_2)\cong\mathbb{Z}^2 \cong <a,b>/ab=ba</math>

So, <math>\pi_1(S^3) \cong\mathbb{Z}*_{\mathbb{Z}\times\mathbb{Z}}\mathbb{Z}</math>

However, we still need to describe <math>i_{1*}</math> and <math>i_{2*}</math>

Do do this let me describe a,b,<math>\alpha</math> and <math>\beta</math> explicitly.

Considering the description of the two tori given above, we let a go around the outside of one of the two disks in the plane and b go from a point on the boundary of the same disk, following the rotation about the z axis, to a point on the boundary of the other dis. <math>\alpha</math> is similar to b, but thought of as being on the boundary of the OTHER torus. <math>\beta</math> consists of the path along the z axis.

Hence,

<math>i_{1*}:</math>

<math>a\rightarrow e</math>

<math>b\rightarrow \alpha</math>

<math>i_{2*}:</math>

<math> a\rightarrow\beta</math>

<math>b\rightarrow e</math> (as it is contractible)

Hence, <math>\pi_1(S^3) = F(\alpha, \beta)/(e=\beta, \alpha = e)</math>

Hence, <math>\pi_1(S^3) = \{e\}</math>

'''Example'''

Define the "Torus knot <math>T_{p,q}</math>" where p and q are relatively prime integers. The knot <math>T_{8,3}</math> is given above. We can think of this in the following ways:

1) <math>T_{p,q}</math> is the knot that wraps around the torus p times one way and q times the other way.

2) Formally, let <math>\sigma:S^1\times S^1\mathbb{R}^3</math> be standard embedding of a torus. Let <math>\gamma:[0,1]\rightarrow S^1\times S^1</math> be <math>t\rightarrow (e^{2\pi i pt}, e^{i2\pi qt})</math>

Then <math>T_{p,q}</math> is <math>\sigma\circ\gamma</math>

3) Recall that the torus can be thought of as the image of the mapping <math>\mathbb{R}^2\rightarrow\mathbb{R}^2/\mathbb{Z}^2</math>

Consider the rectangle in the real plane: ([0,p],[0,q]) and consider the path which is the diagonal line from the corner (0,0) to the corner (p,q)

No two points on this line are the same under the mapping down to the torus. If they were, then <math>\Delta y</math> and <math>\Delta x</math> would be integers and hence <math>\Delta y/\Delta x</math> would be the slope of the line. But the slope of the line is q/p which is already in lowest common terms by assumption.

Lets compute the fundamental group of the compliment of the torus knot.

<math>\pi_1(\mathbb{R}^3-T_{p,q})\cong\pi_1(S^3-T_{p,q})</math>

<math>U_1</math>: inflated bagel, constrained by <math>T_{p,q}</math>

<math>U_2</math>: inflated bubble constrained by <math>T_{p,q}</math>

(See top of page for pictures)

The intersection <math>U_1\cap U_2</math> looks somewhat like a belt. It has some thickness to it and is wrapped around the torus, eventually forming a loop. Hence it looks like a squashed disk cross a circle. Hence, under <math>\pi_1</math> this is just <math>\mathbb{Z}\cong<\gamma></math> where <math>\gamma</math> is the path parallel to <math>T_{p,q}</math>

We thus get the maps,

<math>i_{1*}: \gamma\mapsto\alpha^p</math>

<math>i_{2*}:\gamma\mapsto\beta^q</math>

Hence, <math>\pi_1(T_{p,q}^c) = <\alpha,\beta>/\alpha^p = \beta^q</math>

Thus, <math>T_{p,q}\neq T_{p',q'}</math>

'''Diagrams:'''

Recall our diagram from last class:

<math>\begin{matrix}
&\ \ \ \ U_1&&\\
&\nearrow^{i_1}&\searrow^{j_1}&\\
U_1\cap U_2&&&U_1\cup U_2\\
&\searrow_{i_2}&\nearrow^{j_2}&\\
&\ \ \ \ U_2&&\\
\end{matrix}</math>

<math>U_1\cup U_2</math> can be ''defined'' as the object such that the above diagram commutes and should the following commute:

<math>\begin{matrix}
&\ \ \ \ U_1&&\\
&\nearrow^{i_1}&\searrow^{j_1}&\\
U_1\cap U_2&&&Y\\
&\searrow_{i_2}&\nearrow^{j_2}&\\
&\ \ \ \ U_2&&\\
\end{matrix}</math>

Then there is a unique map between <math>U_1\cup U_2</math> and Y such that the composed diagram commutes.

Indeed, the same is true for general categories.

For

<math>\begin{matrix}
&\ \ \ \ G_1&&\\
&\nearrow^{i_1}&\searrow^{j_1}&\\
H&&&P\\
&\searrow_{i_2}&\nearrow^{j_2}&\\
&\ \ \ \ G_2&&\\
\end{matrix}</math>

commuting, P is defined as an object such that if also

<math>\begin{matrix}
&\ \ \ \ G_1&&\\
&\nearrow^{i_1}&\searrow^{j_1}&\\
H&&&Q\\
&\searrow_{i_2}&\nearrow^{j_2}&\\
&\ \ \ \ G_2&&\\
\end{matrix}</math>

were to commute then there is a unique morphism from P to Q such that the composed diagram computes.

In the category of groups, this "pushforward" P is unique and is isomorphic to <math>G_1*_H G_2</math>

File:0708-1300 notes 22-01-08b.jpg

2008-02-06T05:02:24Z

Canghel: two bagels = S^3

two bagels = S^3

File:0708-1300 notes 22-01-08a.jpg

2008-02-06T05:01:57Z

Canghel: gluing tori

gluing tori

0708-1300/Class notes for Thursday, January 17

2008-02-06T04:53:11Z

Canghel:

{{0708-1300/Navigation}}

'''Van-Kampen's Theorem'''

Let X be a point pointed topological space such that <math>X = U_1\cup U_2</math> where <math>U_1</math> and <math>U_2</math> are open and the base point b is in the (connected) intersection.

Then, <math>\pi_1() = \pi_1(U_1)*_{\pi_1(U_1\cap U_2)}\pi_1(U_2)</math>

<math>\begin{matrix}
&\ \ \ \ U_1&&\\
&\nearrow^{i_1}&\searrow^{j_1}&\\
U_1\cap U_2&&&U_1\cup U_2 = X\\
&\searrow_{i_2}&\nearrow^{j_2}&\\
&\ \ \ \ U_2&&\\
\end{matrix}</math>

where all the i's and j's are inclusions.

Lets consider the image of this under the functor <math>\pi_1</math>

<math>\begin{matrix}
&\ \ \ \ \pi_1(U_1)&&\\
&\nearrow^{i_{1*}}&\searrow^{j_{1*}}&\\
\pi_1(U_1\cap U_2)&&& \pi_1(X)\\
&\searrow_{i_{2*}}&\nearrow^{j_{2*}}&\\
&\ \ \ \ \pi(U_2)&&\\
\end{matrix}</math>

Now consider the situation as groups:

<math>\begin{matrix}
&\ \ \ \ G_1&&\\
&\nearrow_{\varphi_1}&\searrow&\\
H&&&G_1*_H G_2\\
&\searrow_{\varphi_2}&\nearrow&\\
&\ \ \ \ G_2&&\\
\end{matrix}</math>

Where <math>G_1 *_H G_2 = </math>{ words with letters alternating between being in <math>G_1</math> and <math>G_2</math>, ignoring e } / See Later

Considering just the set without the identification, we note this is a group with the operation being concatenation of words followed by reduction.

Ex: <math>a_1b_1a_2 + a_3b_2a_4 = a_1b_1ab_2a_4</math> where <math>a = a_2a_3</math>

'''Claim:'''

This is really a group.

So far, we have only defined the "free group of <math>G_1</math> and <math>G_2</math>". We now consider the identification (denoted above by 'See Later') which is

<math>\forall h\in H, \phi_1(h) = \phi_2(h</math>)

With this identification we have properly defined <math>G_1 *_H G_2</math>

Note: <math>G_1 *_H G_2</math> is equivalent to { words in <math>G_1\cap G_2\}/ (e_1 = \{\}, e_2 = \{\}, g,h\in G_i, g\cdot h = gh)</math>

'''Example 0'''

<math>\pi_1(S^n) </math> for <math> n\geq 2</math>

We can think of <math>S^n</math> as the union of two slightly overlapping open hemispheres which leaves the intersection as a band about the equator. As long as <math>n\geq 2</math> this is connected (but fails for <math>S^1</math>)

So, <math>\pi_1(S^n) = \pi_1(U_1)*_{\pi(U_1\cap U_2)}\pi_1(U_2)</math>

But, since the hemispheres themselves are contractible, <math>\pi_1(U_1) = \pi_1(U_2) = \{e\}</math>

Hence, <math>\pi_1(S^n) = \{e\}</math>

'''Example 1'''

Let us consider <math>\pi_1</math> of a a figure eight. Let <math>U_1</math> denote everything above a line slightly beneath the intersection and <math>U_2</math> everything below a line slightly above the intersection point.

Now both <math>U_1</math> and <math>U_2</math> are homotopically equivalent to a loop and so <math>\pi_1(U_1) = \pi_2(U_2) = \mathbb{Z}</math>. We can think of these being the groups generated by a loop going around once, I.e., isomorphic to <math><\alpha></math> and <math><\beta></math> respectively.

The intersection is an X, contractible to a point and so <math>\pi_1(U_1\cap U_2) = \{e\}</math>

So <math>\pi_1</math>(figure 8)<math> = <\alpha>*_{\{\}}<\beta> = F(\alpha,\beta)</math> the free group generated by <math>\alpha</math> and <math>\beta</math>

This is non abelian

'''Example 2'''

<math>\pi_1(\mathbb{T}^2)</math>

We consider <math>\mathbb{T}^2</math> in the normal way as a square with the normal identifications on the sides. We then consider two concentric squares inside this and define <math>U_1</math> as everything inside the larger square and <math>U_2</math> as everything outside the smaller square.

Clearly <math>U_1</math> is contractible, and hence <math>\pi_1(U_1) = \{e\}</math>

Now, the intersection of <math>U_1</math> and <math>U_2</math> is equivalent to an annulus and so <math>\pi_1(U_1\cap U_2) = \mathbb{Z} = <\gamma></math> where <math>\gamma</math> is just a loop in the annulus.

Now considering <math>U_2</math>, we note that each of the four outer corner points in the big square are identified, and when we identify edges we are left with something equivalent to a figure 8.

Hence <math>\pi_1(U_2) = F(\alpha, \beta)</math> as in example 1

Hence, <math>\pi_1(\mathbb{T}^2) = \{e\}*F(\alpha,\beta)/(i_{1*}(\gamma) = i_{2*}(\gamma))</math>

Now, <math>i_{1*}(\gamma) = e</math>

and

<math>i_{2*}(\gamma) = \alpha\beta\alpha^{-1}\beta^{-1}</math>

I.e., <math>\pi_1(\mathbb{T}^2) = F(\alpha,\beta)/ e = \alpha\beta\alpha^{-1}\beta^{-1}</math>

<math> = F(\alpha,\beta)/(\alpha\beta = \beta\alpha)</math>

This is just the Free Abelian group on two symbols and,

<math>= \{\alpha^n\beta^m\} = \mathbb{Z}^2</math>

Hence, <math>\pi_1(\mathbb{T}^2) = \mathbb{Z}^2</math>

'''Example 3'''

The two holed torus: <math>\Sigma_2</math>

Consider the schematic for this surface, consising of an octagon with edges labeled <math>a_1,b_1,a_1^{-1},b_1^{-1},a_2,b_2,a_2^{-1},b_2^{-1}</math>

As in the previous example, consider two concentric circles inside the octagon. Let everything inside the larger circle be <math>U_1</math> and everything outside the smaller circle be <math>U_2</math>.

Clearly <math>\pi_1(U_1) = \{e\}</math> as before.

<math>\pi_1(U_1\cap U_2) = <\gamma></math> as before.

Now, <math>U_2</math> this times when doing the identifications looks like a clover (4 loops intersecting at one point)

Completely analogously to before, we see that <math>\pi_1(U_2) = F(\alpha_1, \beta_1, \alpha_2, \beta_2)</math>

Again, <math>i_{1*}(\gamma) = e</math>

<math>i_{2*}(\gamma) = \alpha_1\beta_1\alpha_1^{-1}\alpha_2\beta_2\alpha_1^{-1}\beta_2^{-1}</math>

Therefore,

<math>\pi_{\Sigma_2} = F(\alpha_1, \beta_1, \alpha_2, \beta_2)/(e =\alpha_1\beta_1\alpha_1^{-1}\alpha_2\beta_2\alpha_1^{-1}\beta_2^{-1})</math>

The ''abelianization'' of this group is

<math>\pi_1^{ab}(\Sigma_2) = \pi_1(\Sigma_2)/ gh=hg = F.A.G (\alpha_1,\alpha_2,\beta_1,\beta^2) = \mathbb{Z}^4 \neq \mathbb{Z}^2</math>

In case someone might want diagrams for the examples above:

[[Image:0708-1300_notes_17-01-08c.jpg|200px]]

File:0708-1300 notes 17-01-08c.jpg

2008-02-06T04:49:44Z

Canghel: VanKampen examples diagrams

VanKampen examples diagrams