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Fundamental Group

  1. Jan 30, 2010 #1
    I'm studying for an exam which is a couple months away and I found an old exam which asks the following:

    Find the fundamental group of:
    a) The closed subset in R3 given by the equation x - y^2 -z^2 = in the standard coordinates.
    b) The closed subset in R3 given by the equation x - y^2 -z^2 + 1 = in the standard coordinates.
    c) The one point compactifcation of the disjoint union of two open discs in R^2
    d) The space of upper triangular matrices of the size 2 by 2 over C (complex) with derminant equal to 1 considered as a closed subspace in the vector space of all matrices of the size 2 by 2 over C (complex).

    Now,my book (as well as Wikipedia) both define the fundamental group similarly. That is: fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

    What I want to know is how to solve the above? In my book, there are no examples of such things. I want to know if there is a step by step process for each, or if each case is completely different. Thank you for your time.
  2. jcsd
  3. Jan 30, 2010 #2
    It's better for you to pick up any book on algebraic topology (hatcher for instance) and read it.

    (a), (b) both spaces should be contractible if i'm not mistaken. So pi_1 = 0
    (c) The space is just S^2 v S^2, again pi_1 = 0. (To be more rigorous you may need some Seifert VanKampen here)
    (d) The space is homeomorphic to [tex]\mathbb{C} - 0 \times \mathbb{C}[/tex] (why?), so pi_1 = [tex]\pi_1 (\mathbb{C} - 0) = \pi_1 (S^1) = \mathbb{Z}[/tex]
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