subsets of unit sphere in C^2

**shoplifter** · May 4th 2010, 12:44 PM

View $S^3$ as the unit sphere in $\mathbb{C}^2$ . 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 $\mathbb{C}$ )?

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?

thnx

**Drexel28** · May 4th 2010, 03:14 PM

Originally Posted by shoplifter

View $S^3$ as the unit sphere in $\mathbb{C}^2$ . 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 $\mathbb{C}$ )?

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?

thnx

What do you think? You can't expect us to just give you answers, we need to seem some effort on your part!

**Jose27** · May 4th 2010, 04:10 PM

Originally Posted by shoplifter

View $S^3$ as the unit sphere in $\mathbb{C}^2$ . 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 $\mathbb{C}$ )?

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?

thnx

Are you working with the ususal topology on $\mathbb{C} ^2$ ? The Zariski topology?

**shoplifter** · May 4th 2010, 04:14 PM

i apologize - i only wanted to get started, i've thought about this for a week and haven't progressed at all. we're looking at $S^3$ as the unit sphere in $\mathbb{C}^2$ , so this is the standard topology

**Drexel28** · May 5th 2010, 01:52 PM

Originally Posted by shoplifter

i apologize - i only wanted to get started, i've thought about this for a week and haven't progressed at all. we're looking at $S^3$ as the unit sphere in $\mathbb{C}^2$ , so this is the standard topology

Let's start with one that seems easiest, whether the curve defined by $x^2+y^5=0$ is homeomorphic to $\mathbb{S}^1$ . I have literally put no real work into this, and so please do not think I am dropping hints because I know something, but my intuition (God knows how useful that is) says they are not. $\mathbb{S}^1$ is a $1$ -manifold, is that subspace?

**shoplifter** · May 5th 2010, 02:22 PM

actually my prof said they are homeomorphic. should i be looking at ways to solve the simultaneous equations (because i have two, right? the one for S^3, and the one for the subset itself)? because i tried, and it doesn't help at all

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