1. Fundmental Group

How would I go about calculating the fundamental group of R^3 minus the x-y plane? I'm guessing the Van Kampen Theorem will be used somewhere, not really sure, any ideas? Thanks.

2. Originally Posted by skamoni
How would I go about calculating the fundamental group of R^3 minus the x-y plane? I'm guessing the Van Kampen Theorem will be used somewhere, not really sure, any ideas? Thanks.
What is the deformation retract of R^3 minus the x-y plane? It is just two points whose fundamental group is a trivial group.
I don't think Van Kampen theorem is needed here.

3. Thanks.

How about the union of a sphere (in R3) and a disc (in the x-y plane), am I right in thinking that since both are simply connected, the union will be, and so the fundamental group trivial?

4. Originally Posted by skamoni
Thanks.

How about the union of a sphere (in R3) and a disc (in the x-y plane), am I right in thinking that since both are simply connected, the union will be, and so the fundamental group trivial?
Whether your union is a union having an intersection or one-pointed union (wedge sum) or disjoint union, the fundamental group of your space is trivial, implying that every loop in your space is homotopic to the constant map.
For example, if your union is one-pointed union (wedge sum) and you apply the Van Kampen theorem, the fundamental group of your space is still trivial since the free product of two trivial groups is trivial.