# Fundamental Group

1. Jan 30, 2010

### 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.

2. Jan 30, 2010

### 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 = $$\pi_1 (\mathbb{C} - 0) = \pi_1 (S^1) = \mathbb{Z}$$