@robjohn: Here's a cool exercise for you. It's obvious that the two end pieces are cones. Can you give an abstract argument that the inner segment must be a piece of a hyperboloid with one sheet?

That's true, but not sure if by compactness. You cellular approximate.

Ah, is the direct (?) limit of compact things automatically compact? I'm being dumb.

Certainly not, R^2 is the direct limit (under inclusion) of the balls centered at 0, of increasing diameter.

Ah, so the point is that we should be able to lift to $\Lambda^n\Bbb R^N$ ... which of course doesn't work for non-oriented planes.

Is $S^\infty$ compact?

Oh, of course not.

OK, so my intuition about using compactness was probably right, then.

Yes, I too think a compactness argument would work.

A loop in $S^\infty$ has to live in some $S^k$. Or else you'd get a sequence of points with no convergent subsequence ...

Good thing Mike isn't here to tell me I'm being stoooopid.

:)

Speaking of, I think I just proved that $w_1(E) = 0$ iff the bundle $E$ is orientable.

That's correct.

@MikeMiller That shows $w_1(S) \neq 0$ for nonorientable fellows.

$w_2$ is harder to understand, unfortunately, but important.

That's what I am going to understand next, for surfaces at least.

But I have a feeling I am over-complicating it. Shrug.

For surfaces it's easy. Higher dimensions, not so ...

I already gave that ($w_n$ for rank $n$ bundle) away to you a week ago.

But I'm shutting up before Mike bitches at me again.

@Balarka: More interesting question — working with $\Bbb Z_2$ coefficients, of course, to what is $w_1$ Poincaré dual?

Ah, yes, I remember. Maybe I should understand why.

(I did immediately remember, but internet wants me to pretend I took a hard time to remember, which is fine if not annoying)

LOL, you're not on trial. But did you see my Poincaré dual query? That's not uninteresting in general.

@user3925758 got very quiet.

Lol sorry, I've been reading the Spivak's Calculus book

Oh, very cool. :)

I'm reading through the derivatives and integrals problems; I like how they aren't just busywork like my homework is

They're actually problems that discuss the nature of the topics discussed

If you want recommendations on interesting problems, send me an email (my email addy is in my profile). I will admit up front that I wrote some of 'em :P

I'm glad you found the book. It's fantabulous.

Lol, that sounds awesome! Thanks, I'll ping you up in a minute

I'd love to stay and chat, but I've got to leave for work in a few minutes and I'm not even out of my pajamas yet. I'll check back in here around 6 hours from now. Sorry if that's past some of your bedtimes(I know timezones can be a drag!)

LOL ... Have fun, @user3925758. Don't go to work in your pyjamas.

@TedShifrin Hm, I seem to have gotten a proof-sketch that $w_n$ is Poincare-dual to the Euler (homology) class as a by-product of thinking about that problem.

Which problem?

And of course you mean everything mod 2.

Poincare dual of $w_1$.

right, of course.

Ohhhh ... cool.

I've forgotten how Milnor/Stasheff even define the $w_i$. And I don't have the book any more.

Ah, Mike is lurking and he hasn't even called us stoooopid yet. :P

LOL, no longer lurking.

:P

@TedShifrin Axiomatically.

Oh, right, and then they construct 'em using Steenrod squares or something yucky.

I guess so. :(

When do they talk about linear independence of sections or obstructions to extending?

$X$ be an $n$-manifold. Euler class gives a map $[X, G_n] \to H_0(X; \Bbb Z/2)$, by pulling back the universal bundle, then getting zero of a generic section, and taking it's homology class. $w_n$ gives a map $[X, G_n] \to H^n(X; \Bbb Z/2)$, by pulling back the universal bundle and taking it's $w_n$. There two maps are actually natural transformations of the relevant functors.

Make their values to have the same place by taking Poincare dual to get $w_n$ map to $H_0(X;\Bbb Z/2)$. Now these two are natural transforms $[-, G_n] \to H_0(-; \Bbb Z/2)$ which agree on a point. Hence they agree everywhere by Yoneda lemma.

That's my proof-sketch.

I think I can safely call it a proof, long as it's not wrong. :)

The only regret being it's general nonsense, not a geometric proof.

I don't remember what the Yoneda lemma is ... and you still have an extraneous apostrophe. :D

Ugh, right.

No, you changed the wrong one.

There were two that were wrong.

Damn, right.

I can't edit anymore. Oh well.

I'm going to be a pest about this.

Apostrophe's.

It would be nicer to deduce it more directly from the axioms.

smack ... no ' for plural.

I was kidding, of course :)

or for possessive of it.

I remember loving Steenrod's old book on fiber bundles. You should look at it sometime.

@TedShifrin It says every natural transformation $F \to \hom(-, A)$ (where $F$ is a functor from an arbitrary nice category to Set, and $A$ is an element of that arbitrary nice category) is determined by the elements of $F(A)$.

In the sense there is a 1-1 correspondence between such natural transformations and elements of $F(A)$.

I suppose I knew that when I was a grad student, but now it doesn't even sound believable.

heh, it's pretty surprising.

It made me change my views that category theory is garbage and useless. (and many other facts later convinced me the opposite)

So I guess the crucial thing is to argue natural transformation.

@TedShifrin My advisor supposedly loves that book but he always dodges my algebraic topology questions...

It is a beautiful old-fashioned book, @0celo. Before the abstract Bourbaki influence hit.

@TedShifrin Interesting, thanks. I saw it in my "advisor" 's office.

I should have kept that one :(

@TedShifrin Is Kob-No influenced by Bourbaki?

I find it incredibly hard to read, but very rewarding when I do

He told me to flip it if I wanted once. I was afraid I wouldn't understand anything so I never did

Well, they were writing it in the 60s ... so probably. K-N is best to read when you know what they're doing.

No, @Balarka, you'd understand a lot of it at this point. Beautiful stuff.

And Steenrod really talks about obstruction theory concretely.

@TedShifrin Euler/SW class are both natural so I guess that isn't a problem.

Right.

So did you have any thoughts on what $w_1$ means geometrically/topologically?

@TedShifrin When you know what they're doing?

Or ... go to sleep.

@0celo: Yeah, if you already know the stuff, then you can read it. Otherwise, it's very terse and they don't have exercises.

@TedShifrin I don't know anything at all about obstruction theory, unfortunately. I think Hatcher rants a bit by motivating char classes by obstruction theory in the beginning of the chapter. I should read it.

Obstruction theory is very beautiful stuff, Balarka.

Anyhow, good night!

I'll think about the $w_1$ thing and get back to you tomorrow.

Sure. Night :)

@TedShifrin Yes, I know (the second part)

Thanks a lot for the discussion(s), and g'night.

Jasper and I have argued about this a lot. I devalue books with no exercises (unless they're seriously at a research level, not first- or second-year graduate). To me exercises are probably the most important part of a good text.

I worked hard writing exercises when I taught graduate courses.

@Balarka I'm sparingly available for today and tomorrow. If you want me to answer something, email me with lots of details.

(And undergrad, too, but there are more sources for those.)

He's going to sleep, Mike. He's making progress.

In case you don't want to read, I asked him what the P.D. of $w_1$ is.

@TedShifrin How do you know if a book is research level?

Maybe a stupid question, but are there research level books in GSM/GTM?

GTM is mostly first- and second-year graduate. Maybe more research stuff in GSM.

To be honest, I haven't looked at listings in years.

@TedShifrin Hamilton's Ricci Flow has dozens of exercises, I think it's "research level"

I consider stuff research level if you're expected to know a ton of graduate material in order to read it. Generally, it presents stuff that's being done in the field in reasonably modern times.

My goal is to solve all of the chapter 1 exercises

I haven't read Hamilton's book, but I've heard him lecture back when he was proving all that stuff in the 80s. He was great.

@TedShifrin Confusing, sorry. The book is called Hamilton's Ricci Flow

I prefer exercises at all levels, but I accept the fact that at the advanced levels most mathematicians can't be bothered.

It's by Bennet Chow et al.

Ohhh ... Chow at UCSD. I've never met him.

@TedShifrin The later exercises seem to be "verify this horrible equation/estimate, see [XXX] if you need help"

but in the first chapter there's stuff on holonomy, Weitzenböck formulas, etc.

Yeah, ok. Sort of the way research papers are written, not putting in every detail of horrid computations.

comparison theorems

But Cheeger/Ebin does all that material and has exercises, I think. I'm not sure.

CE doesn't have exercises

@MikeMiller Caught the ping before leaving. I proved that $w_1(E) = 0$ iff bundle is orientable, and that $w_n(E) = PD[e(E)]$ where $e(E)$ is the Euler class mod 2. Together, these show $w(S)$ for a nonorienable surface is $w(S) = 1 + \alpha + (-1)^n \beta$ where $\alpha$ is a nontrivial element in $H^1(S)$ and $\beta$ is fundamental class, $n$ being number of cross caps.

There are all sorts of books at a more sophisticated level that I love that I have no exercises. I'm bitching mostly about introductory graduate level.

And CE doesn't cover holonomy or Weitzenböck

@TedShifrin Is KN intro grad level?

It's not meant to be a textbook

Should have mentioned $E$ is rank $n$. But now I am truly leaving.

It is meant to be a textbook. Yes, intro grad level.

NIGHT, @Balarka.

for realz :P

Did he calculate for surfaces?

@TedShifrin But not a very good one?

@Balarka: Euler class lives in cohomology, not homology.

Ugh no dagnabbits, by $(-1)^n$ I really mean $0$ if $n$ is even, $1$ if $n$ is odd.

go to bed

@TedShifrin Bishop-Crittenden have tons of exercises, which is nice

A great little book I'm trying to track down for you is called something like Bochner Techniques in Differential Geometry, by H.H. Wu at Berkeley.

And they do it in a way that the main text is separate from the exercises

@TedShifrin yes I've heard of it

It's worth looking at. Nice treatment of Weitzenböck, etc.

Is it out of print?

I followed it for a week or two of one of my graduate geometry courses.

Your library should have it.

@Balarka Calculate $w_1^2$.

@TedShifrin doesn't look like it

Rats. UGA certainly had it

I need to figure out my reading list

@MikeMiller You mean for the nonorientable surface? That's $\beta$ when it's not 0, not? (the generator of $H^2$)

@BalarkaSen Weren't you supposed to be in bed?

I came back because I said $-1$ when I meant $0$ :P

@TedShifrin On the one hand, I want to learn representation theory

That had to be corrected.

You can't learn everything at once, @0celo. I want to learn representation theory now, too. I know very little.

My prof said to take a look at Wolf's spaces of constant curvature to kill two birds with one stone (representation theory and some classical riemannian geometry)

Or Helgason

But I find Helgason very hard to read

@TedShifrin yeah I know

There's not that much representation theory in Wolf's book, that I recall. But I might be wrong.

I should be doing quantum mechanics right now

Well, stop chatting and go do it.

@TedShifrin he has a chapter on representations of finite groups

@MikeMiller I parsed it badly. I meant I think $w_1^2 = w_2$. I'll cook up a proof tomorrow.

and the book seems interesting

@TedShifrin Have you read Helgason?

It's so...daunting. Quite a large book

I used Helgason as a reference (along with K-N) when I taught homogeneous spaces and symmetric spaces in my graduate diff geo back 20+ years ago.

I actually love his proof of Cartan-Hadamard. I always presented that in lecture. He uses moving frames for it.

Cartan-Hadamard: simply connected, complete, nonpositive curvature implies diffeomorphic to $\Bbb R^n$?

yup

Looking it up in Helgason

Of course he doesn't call it by that name

@TedShifrin I see

I recall do Carmo using Jacobi field estimates and Milnor using the loop space and proving it in one line :P

I love moving frames proofs for anything I can use them for. :)

There's probably a neat proof using heat flow or something in Li or related

@TedShifrin So, something you might understand: why do some people define manifolds to be separable

You want to get paracompactness somehow.

Why is separability better than paracompactness or second countability or what have you

@TedShifrin Right, but why separability as a defining property?

It's less scary to metric space students. It's all equivalent once you have locally Euclidean. Don't worry about it.

I know it's equivalent, I wrote out the proof that all the manifolds definitions are equivalent the other day

But isn't second countability the least scary?

If you come from a real analysis background rather than point-set topology, no.

@TedShifrin I guess we've mentioned separability in analysis but not second countability

But my first exposure to "manifold" was a physics book telling me it's second countable, so that's my first thought

I said 2nd countable, Hausdorff, typically when I taught.

it is 2nd countable, Hausdorff.

@TedShifrin right

I think you need them for partition of unity.

Different books use different (equivalent) formulations.

That's what 0celo is debating.

@Adeek You need paracompactness for that

which is equivalent

I just don't care ... move on.

yes

I guess if you use second countable and Hausdorff as formulation then every manifold is actually homeomorphic to a subset of $R^{n}$.

@Adeek Huh?

What is $n$ there?

where n is the dimension of the manifold.

Better reconsider that.

@Adeek How is $S^1$ homeomorphic to a subset of $\Bbb R^1$?

I think maybe it is a subset of some $R^l$

Yes, $2n$

I am not sure I remember reading about it before in the book that I am using

Whitney requires smoothness, of course, 0celo.

In certain cases you can do better

I see

@TedShifrin we're talking about homeomorphism here

Yeah, how do you prove topological embedding in $\Bbb R^{2n}$?

Can you only get 2n+1 for topological?

I actually have no idea. We need a topological manifold with no smooth structure, and I don't think about such things.

I think munkres actually proves it but I am only familiar with munkres until chapter 4.

yes he does

I am looking there right now

He talks about covering dimension, true.

hmm

he doesn't give a specific number

just some $N$

I see

@Adeek $2n+1$ probably works :P

He does claim for compact $m$ manifolds that $2m+1$ works. Dunno about noncompact.

hmm

I wonder if there is a bound

I mean a good one

@MikeMiller would know

The issue is of course the $C^0$ manifolds that can't be smoothed

Well, as Munkres points out, there are 1-dimensional graphs that are not planar, so you can't in general do better than $2m+1$.

The Whitney trick for smooth things definitely uses smoothness.

@TedShifrin Oh, interesting you mention that

those counterexamples are misleading

they just show that in 2, 3, 4, whatever you can't do better

There's a conjecture that smooth Whitney works in $2n+1-\alpha(n)$, where $\alpha(n)$ is the number of 1s in the binary expansion of $n$.

I don't know the status of that conjecture.

I don't either

But for Whitney immersion it is true.

Actually it's $2n-\alpha(n)$ for immersions.

@Adeek So did you read that horrible proof on $\sigma$-compact spaces from earlier?

no but I saved it to read for later

have marking assignments to do for calculus and abstract algebra

I will read it after I am done

cool

yeah one person just handed drawings including unicorns Pen!ses etc etc

pen!s drawings *

why

pen!s?

I can appreciate a good pen I guess

@Adeek what's his grade in the class?

0

he got 2 in his first assignment I guess

is this a calculus student or an algebra student?

I assume algebra is abstract algebra and not high school remedial algebra.

he said abstract @TedShifrin

Indeed.

that student is a calculus student

yeah it is introduction to ring theory and modules

and group theory

Good for you to grade and learn it better.

yeah @TedShifrin I am happy about that

I am not happy about calculus though

so boring

@Adeek at least you're not doing math 131 college algebra like some people I know

who is doing it ?

Yeah I would not like that !_!

@TedShifrin I noticed here that many profs come from germany

Germany must be strong in math

@Adeek I mean grading it.

grading/teaching

yeah college algebra sucks. I wouldn't like doing it

No stronger than England or France or Switzerland, Karim. But yes.

It is stronger than England,france,and switzerland ?

You need to stop asking such questions.

lol

I am bored

@TedShifrin ?

Well, don't bore me while you're at it.

haha sorry

I answered the question, anyhow, and then you asked it again.

Done.

I have no idea if you can topologixally embed in 2n. The stated conjecture for embeddings is surely not conjectured because it's false.

The conjecture is for immersions where it's true.

(Everything after the first sentence was about the alpha(n) stuff.)