Question On Homotopy Groups

  • Thread starter marlon
  • Start date
  • #26
3,768
10
mathwonk said:
do you see how this implies the fundamental theorem of algebra?

As i understand it, this theorem states that any polynome can be factored by using complex numbers. There is always a complex root.

The fact that the windingnumber is non-zero implies that the origin is always a part of the disc, right ? Now, this means that any function with winding number non-zero must cross the horizontal x axis and it must have roots in those points. Is this correct ?

marlon
 
  • #27
mathwonk
Science Advisor
Homework Helper
11,274
1,475
a non constant complex polynomial defines a continuous map from the complex plane to the complex plane which you want to prove must hit the origin.

by the winding number ideas above, if you can show that some circle in the plane is mapped so as to wind around the origin, then it follows that some point inside the circle must map onto the origin.

thus the proof reduces to showing that on a large enough circle, the winding number of the image under the polynom,ial, equals the winding number of the image under just the leading term, which is Z^n.

then one uses polar coordinates to actually compute this latter winding number to be 2pi d, where d is the degree of the polynomial.
 
  • #28
3,768
10
mathwonk said:
if you can show that some circle in the plane is mapped so as to wind around the origin, then it follows that some point inside the circle must map onto the origin.

Here i don't follow, Why can't a point inside the circle wind around the origin. Or is this just because, if the point inside the circle could wind around the origin, you'd get another circle which is topologically equivalent to the first circle.

Please, come again with the statement on the fundamental theorem of algebra because i think i missed it.

marlon
 
  • #29
mathwonk
Science Advisor
Homework Helper
11,274
1,475
how can a point wind around the origin? a loop can wind around a point. apoint cannot.
 
  • #30
3,768
10
Well, that is indeed quite obvious, so i don't see how the fundamental theorem of algebra is implied ?

marlon
 
  • #31
mathwonk
Science Advisor
Homework Helper
11,274
1,475
well where in my argument above did i say anything about a point winding around a point? that was your version.
 
  • #32
3,768
10
But, if you say that a circle is mapped as to wind around the origin, then way can't a point wind around the origin by this map???

marlon
 
  • #33
3,768
10
In all honesty, i really don't understand the use of all these elastic strings...

marlon

i never used them in QFT...
 
  • #34
mathwonk
Science Advisor
Homework Helper
11,274
1,475
imagine the map defined by the po;ynomial z^n. it takes the unit circle, and wraps it around the origin n times.

thus every circle of radius less than 1, also wraps n times around the origin, unless one of them maps onto the origin.

now this same reasoning applies to a large circle.


take nay polynomial at all of degree n. applying it to a large circle, it behaves much like the simpl,e term z^n, i.e. it winds the large circle n times around the origin.

now our polynomial behaves differently from z^n on the interior of this large circle, but nonetheless, if it maps zero to the origin we are done, and if not then it maps zero to some point say at (10,0) away from the origin. then every small circle maps near that point (10,0) hence with winding number zero.


since the large circle mapped with winding number n, somewhere between the large circle and the small one, some point of the large disc must have crossed over the origin.


i call this the invisible butterfly net proof. i.e. if the boundary of your butterfly net winds around the butterfly, then the buterfly must be inside your net, even if you can't see the net.

But in fact the last one, which ahs radius zero maps onto the origin.


now suppose some other polynomial also maps the unti circle in the same way as z^n does, but behaves differently in the disc. well if the center is not mapped onto the origin, then the center is mapped to some point away from the origin. that means by continuity that any small enough circle is mapped clsoe to that point, hence also well away from the origin. i.e. with winding number zero.


but then some one of the circles between radius 1 and radius zero, must have croissed over the origin, i.e. some point mapped to the origin.
 
  • #35
mathwonk
Science Advisor
Homework Helper
11,274
1,475
to answer your question, if T is a map from a circle to the plane that misses the origin, then the circle winds around the origin if the integral of dz/z around the image of the circle is not zero.

how do i define what it means for a point to wind around a point? the integral of dz/z taken over a point is never non zero.


you do not need to think of elastic strings, but it helps if you have done path integral. ??
 
  • #36
3,768
10
no i got, thanks again...mathwonk
 
  • #37
Haelfix
Science Advisor
1,959
226
  • #38
3,768
10
Thanks for the great link Haelfix

marlon
 
  • #39
mathwonk
Science Advisor
Homework Helper
11,274
1,475
If like me, you find Hatcher's book tedious, I suggest the more elementary book of Artin and Braun, or the differential topology book of Milnor, Topology from the differentiable veiwpoint, or the more detailed rewrite by Guillemin and Pollack, or the nice book on algebraic topology by William Fulton.

But Hatcher's book is free.

Another book I like a lot, is Differential forms in algebraic topology, by Bott / Tu.

My friends who are professional algebraic topologists also praise Hatcher, but I fail to see why. and I even liked Spanier.

Of course I have not read Hatcher closely, nor taught from it. maybe it grows on you. What do you like about it, Haelfix?

another excellent introduction for beginners is Andrew Wallace's book on the fundamental group, or any book by Wallace. Vick's Homology Theory is also easy to get into.

Ok I just looked at hatchers book, at least the first few paghes, and remember why I dislike it. He introduces the mapping cylinder on page 2. Thats ridiculous. No one can appreciate that before seeing a sphere or a torus. and the retraction illustrations are ugly computer generated ones.

You see now how impatient I am as well. No doubt this is just a poor choice of how to frame an introduction, and in the book proper he does better, but so far this is not my idea of pedagogy.

if you want an ideal example of good writing / teaching, see the first two pages of milnor's morse theory. with a few simple minded but attractive pictures he actually conveys almost as much morse theory as most people need, in the first two pages of the introduction.

but probably it is unfair to compare anyone's work to milnor's. even serre comes in second in my opinion. and maybe no one else comes close.

i will look further, but perhaps someone will kindly point me to a recommended well written section of hatcher to examine more closely.


Ok, it is indeed chapter zero which is so off putting, and chapter one is quite reasonable. In my view he should simply have begun with chapter one.
 
Last edited:
  • #40
Haelfix
Science Advisor
1,959
226
Well one thing I really like about Hatcher's book is the homework problems. Some of them are damn hard, and if you are able to tackle those, then you pretty much are well off and understand the topic at a high lvl. I remember spending many hours on a few of them.

I will agree some of the discussion gets a little complex, for instance he starts talking about non standard spaces (Eilenberg Maclane spaces) in the appendix of chapter 1. Thats very subtle material, and confused the hell out of me the first time I took the course.

Otoh hand his chapter on homology is very complete. I remember finding the answer to several problems I had (I had this fuzzy misunderstanding relating CW complexes and simplicial complexes), the latter I knew well but couldn't find many places where they talk about the relationship.
 
  • #41
mathwonk
Science Advisor
Homework Helper
11,274
1,475
yes as i look further, the book looks useful. he gives a lot of information in a few well chosen words, at least in places.

i do not know that much algebraic topology, and never read any book on it thoroughly.

i did learn some things from greenberg's nice book, which in turn is based on the lovely elementary book of artin and braun.

spivak's differential geometry book, volume 1, has a wonderful chapter on de rham cohomology which is quite readable.

i also worked out the basic techniques of de rham theory for myself when i was teaching calculus of differential forms in order to give applications of stokes theorem to geometry for my students. i still remember how excited i was when i realized that stokes theorem implied the unit circle was not homotopic to a point in the complement of the origin in the plane, and that this implied the fundamental theorem of algebra.

i also took a course from ron stern in which we read great notes of griffiths and morgan on homotopy theory and postnikov towers. the postnikov towers came in very handy years later for me when i was proving some results on the homotopy and cohomology of the moduli space of abelian varieties.

and i had a course from ed brown. brown was a master of homotopy theory and made the material seem very intuitive.

fortunately brown discussed such things as eilenberg maclane spaces, or K(<pi>,n) spaces with only one homotopy group. he built them up by attaching spheres to generate the group and then attaching cells to kill the relations.

afterwards he used these spaces to show cohomology is a representable functor in the homotopy category, a nice technique that was imitated later in algebraic geometry and deformation theory by mike schlessinger and others.

i have just looked at hatcher's discussion of this topic on page 448, and am disenchanted with his book in this section again. i.e. he uses more sophisticated language than necessary, no doubt seking greater efficiency. but i can hardly understand his statement, whereas brown himself made it seem like the most natural thing in the world.

i.e. i do not even know what an omega spectrum is, but i do know what brown theorem says and if i think hard enough to remember the idea, possibly also essentially how to prove it.

i like easy clear versions of things, not sophisticated, technical versions.

as brown put it, his theorem says that the cohomology functor on finite cw complexes, apparently rather techical and contrived, actually "occurs in nature!"

one topic i still feel ill at ease with is characteristic classes, e.g. chern classes, although of course the classifying space approach makes them technically simple to define: just embed your manifold and its tangent spaces in euclidean space and then this defines a clasifying amap from the manifold to a grassmannian, and pull bcak the standard cohomology classes from the grassmannian.

does hatcher explain these too? maybe in his later books on vector bundles.
 
Last edited:

Related Threads on Question On Homotopy Groups

  • Last Post
Replies
6
Views
5K
  • Last Post
Replies
8
Views
3K
Replies
4
Views
4K
  • Last Post
Replies
7
Views
3K
Replies
3
Views
2K
Replies
12
Views
3K
Replies
1
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
Top