What are T2-Spaces and Separability in Topology?

[matt grime]

Yes, jump in. Grab Maunder's Dover reprint if you can ($11.95), or perhaps Massey though a lot of people don't get on with it.

As long as you know a tiny amount of pointset topology, enough to define a continuous as function as one pulling back open sets to open sets and say what compact is, you'll be fine, and even if not you can always look back at what you need to know. Do you want to guess how long the chapters on such analytic topology as you're studying are in maunder's book? it runs to less than 7 pages of treatment. do you know what connected is? hausdorff? compact? then you're good to go.

Why? Because any intro to algebraic topology will only really try to deal with manifolds and their properties; it turns into algebra problems about 'real life' topological spaces. Stuff like nets, ultrafilters and so on really won't come anywhere near it, nor will half the material in that book you've got. It should deal with some kinds of complexes, be they simplicial or CW, try and find out what Cech homology is.

i think others would be better placed to say, but most of the pointset topology in that book is more suited to doing analysis not algebra. so it would depend on what you wanted to do.

 

[JasonRox]

How does this book look?

http://www.math.cornell.edu/~hatcher/AT/ATpage.html

I'm reading Abstract Algebra by Herstein as well as the Topology text.

I'm skeptical about jumping in right now though. It just feels that I shouldn't rush into things. If I finish the two texts I have above, which Topology should be done by April, and Abstract Algebra by let's say August. I feel that if I wait, I will acquire lots of skill and knowledge taking this route. I want to be somewhat rounded, so that if decisions change, I'm not far far off.

Anyways, I have more questions.

First, what kind of topics (readable for me) should I look for in my first Algebraic Topology text?

Second, what kind of neat things will these topics teach me? In a language I can understand if possible.

Third, what neat things are happening today in the area of Algebraic Topology? (I believe some things are now being, or may always have been, applied to number theory.)

 

[JasonRox]

Well, I did a google search about Algebraic Topology.

I have a rough idea of the things I can do, and sounds interesting.

I guess the only question I need answered now is the first one.

 

[JasonRox]

I did a little bit more reading, and yes I must say it is very interesting.

There is still no doubt in my mind that I would gain by waiting a little bit longer. Let's say until Chapter 6 - Metric Spaces.

 

[matt grime]

I suppose the first thing to look for is the definition of the fundamental group.

This is the most intuitive and easy to understand thing there is in topology. It measures the number of ways you can embed a closed loop (ie a path) starting and ending at some fixed point in your space up to the equivalence of being able to deform paths by continuous operations.

It measures, approximately, the number of holes in your space.

The fundamental group of the circle is the integers. You can wrap around the circle n times, and up to this equivalence that specifies all of the paths.

The fundamental group of two circles that are joined at one point (so called bouquet of two circles) is the free group on two elements. For n elements it is the free group on n elements. Each has n holes.

The fundamental group of the torus is the integers. It is the abelianization of the fundamental group of the bouquet of two circles for a very interesting reason.

It is intuitive, nice, and familiar since it is the analogue of the winding number in complex analysis. The link being if you integrate 1/z round a path that loops round the origin in C you are talking about loops round a path... if two paths were continuously deformable into each other (ie not going through 0) then they'd have the same winding number.

 

[mathwonk]

i agree massey's books are good for beginners, and he ahd one years ago from a british publisher on just the fundamental group.
i am out of date too, but some fairly recent stuff in alg top seem to be the ideas that grew out of gauge theory and gromov - witten invariants, and led to the solution of the thom conjecture.
then there are links with toric, hyperbolic, and symplectic geometry. on the purely algebraic side, alg top and homotopy theory have gotten quite abstract and have links with triangulated categories and such like.

matt is better poised to discuss these topics.

 

[matt grime]

Since we want to stick with the link between maths as its done and the beginning, i suppose the best simple yet complicatedmotivating example is that:

a compact connected oriented manifold without boundary is determined up to homeomorphism by its homotopy type, that is its fundamental group. In particular all fundamental groups as Z^n, n copies of Z for some n. There is exactly one for each n.

In particular there is exactly one connected compact oriented surface with trivial fundamental group, and that is the sphere.

Open question: is this true for solids and not surfaces?

Prize: $1,000,000. (Poincare conjecture; Perelmen's proof of the geometrization may have got it though).

Even calculating some things that are higher fundamental groups (rather than maps from the circle to our space we look at the maps from an n-sphere) was worthy of a fields medal.

 

[mathwonk]

I love that example of poincare, showing how unklnown are basic questions aboiut basic concepts.

as to the surfaces and their fund groups, is that fundamental groups? or homology groups? even then they are Z^(2n), aren't they?
i thought the fundamental group of a surface of genus 2 is something like the free group on 4 generators a,b,c,d modded out by the one relation, aba-1b-1cdc-1d-1.
of course for genus one this would be Frab(a,b), mod {aba-1b-1}, i.e. Za+Zb.
to see it, represent a surface of genus g as the identification space of a polygon with 4g edges.

 

[JasonRox]

Is that it for Massey?



That would be Maunder's book.

I'll consider picking it up sometime during the term, probably during the spring.

 

[devious_]

Get Willard's .

 

[matt grime]

Ah, yes, my mistake. I always get that wrong. The fundamental group of something with genus g is the free group on 2g letters modded out by 'the simplest relation' ie that

x_1x_2x_3\ldots x_{2g}x_1^{-1}\ldots x_{2g}^{-1}=1

 

[JasonRox]

Get Willard's .

I'm looking for Algebraic Topology.

 

[mathwonk]

hatchers book is well regarded graduate level alg top book, but i myself find it a little offputting.

on the other hand many people find the classic text by spanier offputting and i like it.

both are more encyclopedic though.

actual readable books are something more like works by massey, or william fulton, or vick, or a chapter in volume 1 of spivaks diff geom for the diff point of view, or munkres, or maybe marvin greenberg, or diff forms in alg top by bott and tu.

note all these books (except vick) are reasonably priced compared to many math books today.

bott and tu is more advanced than the others but it is excellent.

 

[JasonRox]

hatchers book is well regarded graduate level alg top book, but i myself find it a little offputting.
on the other hand many people find the classic text by spanier offputting and i like it.
both are more encyclopedic though.
actual readable books are something more like works by massey, or william fulton, or vick, or a chapter in volume 1 of spivaks diff geom for the diff point of view, or munkres, or maybe marvin greenberg, or diff forms in alg top by bott and tu.
note all these books (except vick) are reasonably priced compared to many math books today.
bott and tu is more advanced than the others but it is excellent.

I'm going to go look at the University Library, and I'll see what I like.

Hopefully, they are readable.

 

[JasonRox]

Well, I know the University Library has Massey's Introduction to Topology.

That's good news, but haven't had a change to get a look at it yet.

Like in the earlier post, I hope I find it readable.