@BenDover most sources seem to define euler characteristic homologically - alternating sums of counts of k-cells, or simply V-E+F for polyhedral meshes on a given surface (which one has to prove is invariant of choice of cw complex structure). you could of course define it to be 2-2g for orientable surfaces of genus g, but then you beg the obvious question of why you're even talking about "2-2g" at all (it's a rather arbitrary function of the quantity g - how do you motivate it?).
what definition of euler characteristic are you familiar with? can you give me examples of big invariants that are defined directly on the topological object instead of on representatives and then proven invariant of representative? for instance, the linking number has an integral I suppose... also as far as I know, anything homological is "indirect" in this sense of direct-definitions-on-representatives.
@anon: these indirect definitions are often easy to compute but can be theoretically quite poor. for instance, how do you get functoriality/naturality from simplicial homology? (yes, you can use simplicial approximation, but then why does the simplicial approximation you chose not matter? doable but unpleasant). so usually one passes to something that's defined more generally (and as stated uncomputable); singular homology, the fundamental group, are all defined without passing to 'reps'
it's isomorphic to the one you know for triangulable spaces. instead of using simplices that you build your space from as generators of the chain groups, you use "maps from a simplex"
this doesn't depend on anything since a map is a map is a map
this is the modern way to prove all the pleasant properties you want from homology; i don't really know if or what the non-modern way was
it seems like you had a more specific question back where i found that one from (something about embeddings of discs?) but i don't really see the actual question without hunting - did you resolve that?
was wondering if "2 isotopy classes of embeddings of D^2 in S" was a good def of orientability of a surface S (and 1 isotopy class for nonorientability) if we are restricted to working without homology or differential structure
I suppose I do want to see if you can do things with it
like, I imagine any isotopy of one embedding to a reverse embedding, if one unions the images of all the embeddings one gets a mobius band inside the surface
(some sources seem to define orientability based on whether or not it contains a mobius band)
also you can probably get the homological version too (shrink certain embeddings till you can isotope them with a given 2-cell, then figure out how to isotope it across any edge, etc.)
the way i'd think about it is in terms of the derivative of the embedding at the identity: it can't change from positive to negative if your manifold is orientable, where you have a well-defined notion of "positive determinant of derivative"
if you're doing it homologically it's the same idea, you see the induced map on H_2(D^2, D^2 - pt) -> H_2(M,M-pt). if M is orientable the latter can be consistently given an isomorphism with Z, and the definition of 'consistently given' gives that the previous map always sends the positive generator to the positive generator if you vary the choice of embedding
nonorientable means you can't make a consistent such isomorphism, which means you can find some loop, traveling along which any isomorphism has to change eventually; so slide your disc along this loop
(as a note, it's nontrivial that the space of orientation-preserving embeddings D^n -> M^n is connected. Smale proved this for smooth embeddings, i think, and it follows from the annulus theorem for continuous ones; maybe there's an easier way?)
nah, i now think it's not so bad. shrink the disc to be very small. isotope it to wherever your second disc is centered. shrink it too. now all you want to show is that the space of orientation-preserving embeddings D^n -> R^n that fix 0 is connected; in fact, it's contractible, by the homotopy $f_t(x) = f(tx)/t$. for smooth things this has a limit (the derivative of the map at the identity). i'm a little worried about continuous stuff though
yeah it doesn't necessarily have a limit if your original thing is continuous
@anon: lee mosher claims here that you absolutely need the annulus theorem for the topological case.
well, it usually doesn't. i guess if i read it i'd assume what's meant is "objects are good (compactly generated weakly hausdorff?) spaces, morphisms are homotopy classes of morphisms"
Sure. But in theory, you could try to define a homotopy category for topological spaces. Objects are topological spaces and $\operatorname{Hom}(X, Y) = [X, Y]$.
if you're working with CW complexes the isomorphisms are just basepoint-preserving homotopy equivalences; if $f$ has a homotopy inverse $g$ you can homotope $g$ to preserve the basepoint
maybe it's still true for non-CW complexes
@MichaelAlbanese: I'm not sure if the previous two statements are true anymore.
@robjohn So, similarly I can show that $\displaystyle\lim_{(x,y)\to(0,0)}xy\ln(x-y)$ also doesn't exist, by approaching on the line $x-y=0$. is this correct?
I guess I know that fear. It's that same thought that makes me hesitant to ask questions on MSE or of experts in case they're trivial and I haven't thought enough.
It's silly but knowing that doesn't get me to ask them.
I feel differently about asking questions on MSE and MO. I am hesitant to ask a question on MO that is outside of my field of study because I cannot accurately judge the difficulty of the question. On MSE, I don't have any problems asking a basic question from outside my area, even if it isn't that interesting.
I asked a question on MO not long ago about a statement in a paper I didn't understand. Turns out I was reading a 5-page notices version and there was a 50-page version that actually explained it. I felt very silly.
I'm going to talk to the guy who answered it sometime this week since we're at the same conference. I wonder if he'll remember what a doof I am.
@Cristopher Yes. However, if you want to approach along a curve on which the function is defined (say for instance that the line $x-y=0$ is excluded), you can approach along the curve $(t+e^{-1/t^4},t-e^{-1/t^4})$ as $t\to0$.
I figure I usually learn the most when I ask something stupid. For better or worse, the embarrassment burns the problem into your mind and you never forget after that
It went great! We just finished since it was just the month of June for some reason. I learned so much in such a short period of time that it's a little mindblowing.
And I feel like it went a long way in developing my mathematical maturity.
It's possible @Paul. We definitely did some work that hasn't been done before, but I don't know if we've done enough to justify a whole paper just yet. I think we'll be working on it further this coming semester.
For what it's worth, though, we're in the process of at least TeX'ing up everything we did.
Sounds good. I actually just visited "my local REU" today, they are about a month in and all the groups gave some talks about what they have done so far.
Does anyone know of a good analysis text which covers functions of bounded variation, equicontinuity, complete metric spaces, and Riemann-Stiltjes which isn't Rudin?
Btw, @TedShifrin, I sort of revisited your volume of a tricylinder problem today. I used single-variable calculus to solve the bicylinder without too much effort, and I'm going to try to generalize that to the tricylinder. Still need to figure out how to avoid calculus altogether though :P
Just to understand why I'm seeking those topics... I'm basically looking for a better version of Protter and Morrey which I learned from in my undergrad.
yeah @TedShifrin its optional to change honours advisors but then I don't know how would that affect our relationship or if it will affect me in the future
yeah I guess what I could do is just suffer that year and just do it completely by myself and have just be in the background for the just being name of an advisor because I just don't want to have any hate between me and faculty members until I graduate which would be not nice.
I mean he has never time for me and also very unprofessional there was other prof who wanted to really work with me on honours project but I choose wrongly but now if I tell him I will change him I guess I think it would affect our relationship.
@KarimMansour Yeah, that's why one person I know went off to work as an actuary after finishing ABD. She was all over that one story about Tim Hunt on FB
@MikeMiller hi mike. im struggling with this problem: conclude from the fact that there is no odd map $S^n\to S^m$ if $n>m$ that any odd map $f:S^n\to S^n$ has odd degree.
i can't figure it out for the life of me. it shouldn't be that hard
I think you're tryhing to prove that an odd map $f: S^n \to S^n$ has odd degree from scratch. Use the fact that there's no odd map $S^n \to S^m$ for $n>m$ somehow. (How? I dunno.)
I was trying to find a closed form for
$$\int_0^1 \frac{\text{Li}_2 \left(-\frac{1}{1-z}\right)-\text{Li}_2 \left(-\frac{1}{1+z}\right)}{z}dz = -2.454199511\cdots$$
where $\text{Li}_2(z)$ is the dilogarithm function. Numerically, it seems very close to $-\frac{49}{24}\zeta(3)$.
How can we prov...
Je comprend mieux pourquoi je n'aboutissais pas avec $\sum_{k=0}^\infty\frac{x^k}{(n+k)!}=\frac{e^x}{x^n}\frac{\Gamma(n+1)-\Gamma(n+1,x)}{\Gamma(n+1)}$ q_q
@LeGrandDODOM Soit A l’ensemble des entiers positifs dont l’écriture décimale ne contient pas le chiffre 0. Il faut trouver toutes les valeurs de $\alpha$ pour lesquelles la série $\sum_{n\in A}1/n^\alpha$ converge.
@anon anon, the standard definition of the Euler characteristic is the alternating sum of betti numbers (which are by definition the ranks of the homology groups with integer coefficents). Using this definition, it is obvious that it is a topological invariant. This is shown to coincide with the number of cells definition by cellular homology. The reason why one often defines these things on finite CW complexes is that these have a nice finiteness properties.. ctd
meaning that the homology groups are finitely generated. As far as I know, there is no definition of the Euler characteristic that is meaningful for every topological space. You need some finiteness conditions (e.g. that it is a compact manifold). "as far as I know, anything homological is "indirect" in this sense of direct-definitions-on-representatives". I must disagree here. Homology is defined directly for ANY space, one does not define it first for CW complexes or whatever.
I just checked out spivaks comprehensive intro to diff geometry and 1) why does the third edition have five volumes, 2) Why does the fifth volume look bad compared to the first?
same for cohomology, and all the invariants derived from these theories. also the most classical topological invariants such as compactness, connecteness, the number of path components, etc. are not defined in terms of CW complexes first.
@Gato En effet, en France en tout cas on est peu porté sur les résolutions de sommes/intégrales compliquées. Moi en lisant le livre de Furdui j'ai trouvé plein d'exercices super simpas introuvables ailleurs
I was trying to find a closed form for
$$\int_0^1 \frac{\text{Li}_2 \left(-\frac{1}{1-z}\right)-\text{Li}_2 \left(-\frac{1}{1+z}\right)}{z}dz = -2.454199511\cdots$$
where $\text{Li}_2(z)$ is the dilogarithm function. Numerically, it seems very close to $-\frac{49}{24}\zeta(3)$.
How can we prov...
I typed in tan 90 degrees in google and it gave 1.6331779e+16. How did it come to this answer? Limits? Some magic?
