View S^3 as the sphere in C^2 (with coordinates, let's say, x and y). Let P and Q be the subsets of S^3 given by the equations x = 0 and y = 0 respectively. What is the fundamental group of S^3 − A − B?

P and Q are obviously disjoint circles imbedded in S^3. I am trying to show that the deformation retract of S^3 - P - Q is the torus T(a), where for some a in (0, 1), T(a) = {(x, y) in S^3 : |x|^2 = a} (let's take a = 1/2). After that, it is trivial that the group we are looking for is the group of a torus, i.e. Z^2.

So to get the retract, I see that every point of S^3 - Q has a unique nearest point in P and a unique minimizing geodesic in the usual metric. I know I have to use that, but I cannot use that geodesic to retract onto |x|^2 = |y|^2 = 1/2.

What is the retraction? Any help is appreciated. Many thanks.

P and Q are obviously disjoint circles imbedded in S^3. I am trying to show that the deformation retract of S^3 - P - Q is the torus T(a), where for some a in (0, 1), T(a) = {(x, y) in S^3 : |x|^2 = a} (let's take a = 1/2). After that, it is trivial that the group we are looking for is the group of a torus, i.e. Z^2.

So to get the retract, I see that every point of S^3 - Q has a unique nearest point in P and a unique minimizing geodesic in the usual metric. I know I have to use that, but I cannot use that geodesic to retract onto |x|^2 = |y|^2 = 1/2.

What is the retraction? Any help is appreciated. Many thanks.

### Recently Asked Questions

- please work through this so I can get a better understanding of the process

- What is the relationship between financial decision making and risk and return? Would all financial managers view risk—return trade—offs

- You are attempting to purchase a part from a specialty vendor. Your company requires a C p of at least 1.67 on a critical dimension of the part. The