Demonark: Did you add my question to your list of questions? Prove that any retract of a smooth manifold must be a smooth submanifold. (And why is this false for only continuous manifolds?)
Today we proved that there was no smooth function $f:X\to \delta X$ whose restriction to $\delta X$ is the identity, which Neves called a retraction in passing
That's what a retraction is, Demonark. If $A\subset X$, we say $A$ is a retract if there is a (smooth in this case) function $r: X\to A$ with $r|_A = \text{id}_A$.
Like, multi happened only for 2 weeks, and we didn't often deal with partial derivatives in that time anyway, Soug was just like "Yeah so derivative is a linear transformation, determine by action on a basis", and that was basically it
First week was definition of a manifold, tangent space, derivatives, regular/critical values. This week was Sard, FTA, Whitney, classification of curves and surfaces (without proof), and starting Brouwer's fixed point theorem @Eric
Well, we only defined manifolds as subsets of $\mathbb{R}^n$, so our version of Whitney was that if you had a compact manifold with dimension $k$, you could embed it into $\mathbb{R}^{2k+1}$
ah ok, part of the reason it took so long to get anywhere when i was in it was probably that our lecturer took like 3 weeks giving alternate definitions of basic things
Oh yeah I appreciated it big time, just that I think that when Eric did it last year, they defined manifolds abstractly so it required more setup, while we kept everything simple and could reach things faster
@TedShifrin Maybe, I dunno. I was able to solve the integral over $0$ to $\pi/2$ using derivation under the integral sign. However I ran into some problems with the one above. Bah
@Eric: Such a waste when there's beautiful applications of transversality to do. I don't know why people feel like they have to teach technical grad courses to undergraduates.
@Eric, Demonark: I've used sheaves and sheaf cohomology in my research my whole life, but I would never put them in an undergraduate diff top class. In the third quarter of graduate complex analysis, yes.
@Ted I agree, the course was so overly technical that I came out not able to do anything. A lot of people came out of it with a bad taste in their mouth concerning anything differential, which is a shame because the stuff is so beautiful.
@Ted Yeah I appreciate that we're just jumping in to conceptual content, like most of what's going on is understanding a picture properly as opposed to waddling through tons of mechanical details
Well, Demonark, I sure hope he'll eventually prove the transversality theorems in chapter 2 of G&P, but I usually do applications of 'em before I do the proofs.
@BAYMAX: You should know an example of a bounded function with no limit, @BAYMAX.
@TedShifrin Yes, but I think it's not as clear. What I want to say if I somehow immerse $X$ in $\Bbb R^n$, then you can "make $X$ self-intersect transversely" which removes all singularities if $n > k + k = 2k$. So you at least need dimension $n = 2k+1$. But of course this is not very rigorous.
@Daminark in the Riemannian geo course he's taking he's taking a really computational perspective with lots of examples, it's really pretty fun, I think the geometry/topology grad sequence here is probably the most worthwhile of the grad sequences.
@TedShifrin I'm just saying, if you have an immersed $k$-dimensional compact submanifold of $\Bbb R^n$, then you can "perturb" it to be an embedding if $n > 2k$ by self-transversality.
Which is how I think of the fact that $2k+1$ is the right dimension of Whitney.
@TedShifrin This is self-transversality, not transversality in the usual sense. If you have a smooth immersion $f : M^k \to N^n$ of a manifold $M$ with the only singularities being double points (i.e., not injective only at double points), then you can homotope it by an arbitrarily small homotopy such that locally near the double points it looks like two $k$-dimensional transverse subspaces in $\Bbb R^n$.
Demonark: You have grad school ... you don't have to jam everything into undergrad!!! I never took alg top as an undergrad. I wanted to take several complex variables and complex manifolds instead.
DogAteMy: $S$ was the unit square, $T$ removed the origin.
@Ted Agreed. I don't think this can be made into a proof any easier than the usual one you are speaking of. (After all, even if you do all of this, you need to immerse $M$ into $\Bbb R^{2k+1}$ by the same projection-to-subspaces thing... so why not just do this for embedding anyway).
I have a sequence of continuous functions $f_n:\Bbb R\to\Bbb R$ that converges pointwise on a set $X\subseteq \Bbb R$, what can be said about $X$? How ugly can it be?
You're making not only the function, but its various derivatives also, transverse to a certain submanifold in the space of functions plus k derivatives.
I feel like, on a rainy day, I should pick up Hirsch or something other and read in excruciating details the local genericity theorems for maps between manifolds.
For example, suppose you look (in the case of complex projective surfaces, but you can think about real stuff for now) at asymptotic curves on a surface. Generically, they'll have inflection points. Those inflection points make a curve that runs into the parabolic curve. How do we charcaterize those points of intersection?
Embed N in R^K, take a tubular neighborhood, use genericity of embeddings for maps to R^K, then a transversality argument to show embedding is still generic when youbproject back down.
@Balarka: McCrory and I, in our first paper out of a number on this sort of thing, then counted all the various loci on a generic surface in $\Bbb P^3$ of degree $d$. (Arnol'd had done some of the same stuff in a very brief paper. But we nailed down all the genericity stuff way more than the Russians tend to do.)
