@BalarkaSen turns out, I don't need to deal with that horrendous diagram at all
Dold-Thom itself is a nightmare to prove but if you don't deal with the isomorphisms being natural (which my book doesn't), the axioms just fall out like that
@Balarka so it turns out computing $SP(S^1)$ isn't looking too hot. For that reason I'll probably just black box the fact that $\mathbb{CP}^{\infty}$ is a $K(\mathbb{Z},2)$ and use that to prove dimension
I'm looking after an 'open' - free - real analysis book that I can use for self-learn. I was using wikibooks (I liked the fact that they constructed the naturals, integers and rationals first), but there's no offline version. Recommendations?
@TedShifrin I'm evading linear algebra. :P
I'm not really acquainted with the reals. Not anymore.
If I say that $M$ is an $A$-module generated by the set $\lbrace 1, \alpha, \dots, \alpha^{n-1}\rbrace$ does that mean $M = A + A\alpha + \dots + A\alpha^{n-1}$?
Suppose the function $f(t)$ has derivative $L$ at $t_0$, is it true that $\mid f(t_0+k)-f(t_0)\mid<Lk$ for $f$ is continuous differentiable? How can I prove this? What happens if f is only differentiable?
@BalarkaSen So, I know that the study of zeroes of vector fields on manifolds is a big deal. What about the study of zeros of $\mathbb{RP}$ fields on manifolds?
Ah, so these are rather complicated objects known as "classifying spaces". If you have a fiber bundle on a manifold with the transition functions taking values in a group $G$, there is a homotopy class of a map $M \to BG$ that classifies that fiber bundle.
So "$BG$" is more or less that object
I think I brought it up momentarily during our conversations about the clutching construction?
Suppose $M$ is a manifold of dimension $n \geq 1$ and $B \subseteq M$ is a regular co-ordinate ball. Show that $M \setminus B$ is a $n$-manifold with boundary, whose boundary is homeomorphic to $\mathbb{S}^{n-1}$
I'm trying to prove the above, I've already proved that $\text{Bd}(B)$ is homeomorphic to $\mathbb{S}^{n-1}$
And that $M \setminus \overline{B}$ is a $n$-manifold
@Kevin: Re your discussion with Balarka, this is a fancy generalization of the fact that if you have a compact manifold $M$ and a vector bundle $E$ of rank $k$ on $M$, there is a (continuous, smooth, whatever) map $M\to G(k,N)$ (for some large $N$) so that $E$ is the pullback of the tautological rank $k$ bundle $\tilde E\to G(k,N)$.
@TedShifrin Yep I was just about to say that, I picked $x \in \text{Bd}(B)$, and I tried to construct a neighbourhood $W$ of $x$ in $M \setminus B$ homeomorphic to an openset of $\mathbb{H}^n$
@TedShifrin At the moment I'm just trying to complete Lee's book on Topological Manifolds (because there's also a decent intro to Algebraic Topology there), and there are a bunch of cool exercises and problems I haven't seen anywhere else. And I'm also trying to complete G&P and Milnor
No, no mention of Grassmannian in the notes from that region of the course. There's a line 3 page discussion in Lee though where he talks about how you construct the charts and stuff that we used though
If i ahd to summarize the topics for our course I would say we covered:basics (smooth manifold definitions, derivations/tangent spaces, smooth functions, etc.), full-rank maps, submanifolds, whitney embedding, bundles and flows, homotopy (including approximations and stable properties), transversality (mod2 intersection and degree), tensors (exterior algebra, k-forms), orientations, integration, Lie derivatives, De Rham cohomology, M-V, and cup-product/Poincare duality
Kevin: Because we had a separate undergraduate G&P course, I totally skipped all the G&P transversality stuff and did things like flows and Frobenius (which are super important).
I also left Poincaré duality discussion for the algebraic topology course, although at some point late in the course (doing characteristic classes) I would talk about it.
My two-semester outline was something like: basics on manifolds, tangent spaces, mappings; vector bundles and tensors; differential forms and Stokes's Thm.; curves and surfaces (including minimal surfaces); flows, flowbox thm, Lie derivative, Frobenius; Riemannian connections, curvature, geodesics, exponential, Cartan-Hadamard Thm.; connections on vector bundles, Grassmannians, symmetric spaces, Gauss-Bonnet and characteristic classes.
One time I did Bochner techniques and some complex geometry (up to Kodaira embedding theorem).
@TedShifrin Apparently I have vastly overestimated what the students are expected to be able to do for the exam. Fortunately, I had someone more experienced look over it before I finalized it
yeah ... able to reserve the right to reschedule plane tickets
@Tobias: It takes many years of experience to gauge these things. I also make it a practice to write out complete solutions to the exam and I should be able to do so in approximately 1/3 the allotted time.
I have gotten a lot of practice doing that this term, as I did so for one old exam problem per week for them to practice on (with the solution for after they had spent some time solving it themselves)
@TedShifrin Yeah, good thing I have coworkers to ask this sort of thing (in this case a guy who both lectured the course last year and also happens to be undergraduate coordinator, so he knows what is expected)
Quick question if $M$ is a topological space, and we show that $\text{Int}(M)$ is a $n$-manifold without boundary and $\text{Bd}(M)$ is a $n-1$-manifold without boundary can we conclude that $M$ is a $n$-dimensional topological manifold with boundary?
Suppose that $x_n$ a convergent sequence of real numbers. I am trying to show that $\sup \{x_k \mid k \ge n \} - x_n$ converges to zero. I could use a hint.
@TedShifrin I haven't really gotten anywhere with the problem. I said let $\epsilon > 0$. Then there exists a natural number $N_\epsilon$ such that $\sup \{a_k \mid k \ge n \} - \epsilon < a_{N_\epsilon}$, but this doesn't seem very helpful.
