I mean, in particular, if you think whether maps are null-homotopic is entirely determined by $\pi_1$, then you've just proved every path-connected space with trivial fundamental group is contractible, which is patently absurd.
The only statement that I know works is "If the induced map is zero on all extraordinary cohomology theories, then it's null-homotopic"... but this is completely worthless
1) Use the fact above (if a map $X \to Y$ is zero on fundamental groups, it factors through the universal cover $\tilde Y \to Y$) to calculate the homotopy groups of surfaces of positive genus. 2) Find a map between surfaces that's not null-homotopic but zero on all homotopy groups.
And then read Ch1 of Hatcher because that covering space shit is clutch
@Danu You should learn them and answer the question. The definition is in the beginning of Hatcher ch4. It is not hard to read, and you don't need background to read the definition.
@Huy The past few days I've been home sick. I'm going in today because the LA topology meeting is today and I probably shouldn't miss that. But I haven't gotten a lot of thinking done. Otherwise I have a project I should be working on.
Is there an arbitrarily large set of naturals so that the sum of each $2$ has exactly $n$ prime divisors where $n$ is fixed? What about an infinite set?
For $n=1$ this is clearly false, what about other $n$?
But... I found that there is only one solution for the equation that I mentioned above (equalizing two derivatives). For example, I equalized the derivatives of x^2 + 3 and -x^2 - 2x - 2 and I get as result that the slope is (-1/2). However, the slope is positive. How can this be possible?
What's your option for calculating this integrals? No full solution is necessary, it's optional as usual.
Calculate
$$\int_0^1 \frac{2 \zeta (3)\log ^3(1-x) \text{Li}_2(1-x) }{x}-\frac{2 \zeta (3) \log ^2(1-x) \text{Li}_3(1-x)}{x}+\frac{ \log (x) \log ^5(1-x)\text{Li}_2(1-x)}{x}+\frac{\log ^4(1...
oh wait a minute, its my first contribution in this new year, thought i wouldnt contribute a dime with this nick, nice i still have the passion and the craving to post
hey guys, so I am attempting to prove the rotation of axes formula
and I managed to get as far as setting up a system of equations (first I prove the cosine addition formula and then applied that to the rotated axis system)
but for the life of me I cant seem to solve the system of equations I set up. There are two equations and two unknowns. Here is how it looks like: wolframalpha.com/input/…
What's your option for calculating this integrals? No full solution is necessary, it's optional as usual.
Calculate
$$\int_0^1 \frac{2 \zeta (3)\log ^3(1-x) \text{Li}_2(1-x) }{x}-\frac{2 \zeta (3) \log ^2(1-x) \text{Li}_3(1-x)}{x}+\frac{ \log (x) \log ^5(1-x)\text{Li}_2(1-x)}{x}+\frac{\log ^4(1...
