Recent content by mich0144

how do you go about constructing covering spaces I know the definition of a covering and the usual ones for a circle and torus are easy to see but for example constructing a covering space of a sphere + a diameter how would you tackle something like this.

why is S^n/S^m homotopic to S^n-m-1. the book just made this remark how do you see this geometrically. how do you compute fundamental groups of matrices like O(3) and SO(3) or SL(2) and whatnot.

ok so I figured out some tricks you basically just mod it by the relation generated by identifying the edges of the polygon representation and set it equal to 1, this seems to work for most case.

so if I were using seifert van kampen on a torus as an illustration. decompose it as U = torus with a hole, and V as a patch larger than the whole. now fundamental group of U is the figure eight which is Z*Z i think if I unroll it into a square and fundamental group of patch is trivial...

yea that's a good idea i know the 2manifolds classifications I think, is there a good book or any kind of document with lots of examples for using seifert van kampen I'm reading hatcher but it's a little too terse and my algebra background isn't too good

I haven't gotten to homology yet but thanks

So I've read through beginning alg topology really fast and there are a lot of theorems and methods for computing fundamental groups but what are the most useful tools? When asked to compute the fundamental group what should one do? try to find a deformation retract and compute the fund group...

Yea I think they are infact the same group like you said demonstrated by the isomorphism but the actions act differently, I'm not sure but maybe converting right to left action has something to do with abelianization so. I need to review my algebra this problem can't be so trivial.

Thanks in your definition as well as in hatcher they show the deck transformation to be a left group action and lift to be a right this is what's confusing me, why is this so. I thought it was purely notational can't you write either in the other way.

yea I think that's the implied one I just got to calculate the actual fundamental group when i I get around to it thanks for the help.

So you're saying the two actions are always the same if we take p to be the universal map, what characteristic of universal map are you using to conclude this and what's a counterex. for a nonuniversal map. the question (27 in hatcher) specifically asks about s1 V s1 and s1 x s1 and then ask...

I don't understand how does 0 give (1,1), (φ(x,y))^0 gives (x*(2^0), y/(2^0)) = (x,y) In other words this is when phi is applied 0 times so nothing happens to (x,y) of course. I think you thought it was (x*2)^k. I should have added parenthesis to begin with.

x as in the x coordinate from (x,y) in R² so k in Z corresponds to taking x,y to x*2^k, y/2^k

as in each integer a is associated to (φ(x,y))^a. This works as well it seems and it's easy to see this is properly discontinuous, you can just pick your neighborhood U to be an interval less than 2x on the right endpoint and all the power translates will take it out of the neighborhood.

I agree quasar I'v never see a problem where they don't specify the generated action no matter how obvious it is, for ex another action where integers are associated to powers of φ(x, y) also fits the definition of a homomorphism this seems pretty natural as well, so I have no idea.