Understanding Topology: Exploring T2-Spaces, Separability, and More

[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i am out of date too, but some fairly recent stuff in alg top seem to be the ideas that grew out of gauge theory and gromov - witten invariants, and led to the solution of the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.