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March 29th, 2011, 11:14 AM   #1
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Algebraic Topology

Can anybody help me with these questions,

1. (A) Let X = {(x,y,0) in R^3 : x^2 + y^2 = 1}. Compute the fundamental group of R^3 - X.

(B) Let T a subset of R^3 be a trefoil knot. Compute the fundamental group of R^3 - T.

2. Show that there is a two sheeted covering map T -> K where T is the torus and K is the Klein bottle.
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March 29th, 2011, 12:48 PM   #2
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Re: Algebraic Topology


For the first two problems, 1A) and 1B), you will want to produce a convenient deformation retraction of the space in question to a space for which van Kampen's Theorem applies easily. In 1A), this map is easy to "see", but in the second case, it is a little bit harder.

For 1A), the deformation retract will be the wedge sum of a circle and a 2-sphere, and van Kampen then tells you that the fundamental group is simply ?.

For 1B), you will as stated have to work harder to get a good deformation retract. The fundamental group will be the quotient of the free group on two generators x and y by the subgroup generated by x²y?³. One thing to realize about this knot that is helpful is that it lives on the torus in a nice way---it wraps around twice in one direction while going around three times in the other (here I mean the two "directions" on the torus, i.e. thought of as a product of two circles).

The last question is very similar in vein to the ones you posted in your last algebraic topology post. I would go back and take a look at the solutions to those problems for inspiration.

I hope this helps a little bit at least.

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