Finding Fundamental Group: Step-by-Step Guide

[curtdbz]

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).
Explain.


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.

 

[soarer]

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 $$\mathbb{C} - 0 \times \mathbb{C}$$ (why?), so pi_1 = $$\pi_1 (\mathbb{C} - 0) = \pi_1 (S^1) = \mathbb{Z}$$