Fundamental groups of subsets of S^3

[shoplifter]

Homework Statement

View S^3 as the unit sphere in C^2. Now,

1. What are the path connected components of the subset of S^3 described by the equation x^3 + y^6 = 0, where the x and y refer to the coordinates (in C)?

2. Is it true that the similar subset x^2 + y^5 = 0 is homeomorphic to the circle?

3. what is the fundamental group of S^3 - K, where K is the subset in the 2nd part of the problem?

Homework Equations

we may (and i assume we have to) use van kampen's theorem at some point.

The Attempt at a Solution

i really can't get started. can't think of any theorem (and i combed munkres's book) involving path connected components based on subsets determined by an equation -- any help or hint will be appreciated a lot. thnx.

 

[mighty2000]

Thank you for posing this interesting problem. my approach to solving this problem would be to first understand the given equations and their corresponding subsets in S^3.

1. The equation x^3 + y^6 = 0 can be rewritten as (x^3)^2 + (y^3)^2 = 0, which is equivalent to the equation z^2 + w^2 = 0, where z = x^3 and w = y^3. This is a well-known equation for a torus in C^2, which is a 2-dimensional surface with one hole. Therefore, the path connected components of the given subset in S^3 are tori.

2. The equation x^2 + y^5 = 0 can be rewritten as (x^2)^2 + (y^5)^2 = 0, which is equivalent to the equation z^2 + w^2 = 0, where z = x^2 and w = y^5. This is a well-known equation for a circle in C^2, which is a 1-dimensional curve. Therefore, the given subset is homeomorphic to a circle.

3. To find the fundamental group of S^3 - K, where K is the subset described by the equation x^2 + y^5 = 0, we can use van Kampen's theorem. Let U be a small open neighborhood of K in S^3 and V be the complement of U in S^3. Then, the fundamental group of S^3 - K is isomorphic to the fundamental group of the wedge sum of U and V, denoted by π1(U ∨ V). Since U is homeomorphic to a torus and V is homeomorphic to a sphere, we have π1(U ∨ V) ≅ π1(T^2 ∨ S^2), which is isomorphic to the free product of the fundamental groups of T^2 and S^2, i.e. π1(T^2 ∨ S^2) ≅ π1(T^2) * π1(S^2). The fundamental group of a torus is isomorphic to Z * Z, the free product of two copies of the integers. The fundamental group of a sphere is trivial. Therefore, the fundamental group of S^3 - K is isomorphic to Z * Z.

I hope this