Starting with C^3 \ (0,0,0), construct the complex projective space CP^2 with the constraint x^3 + y^3 + z^3. Is this homeomorphic to the torus? ------------------------------------- William R. Cordwell Sandia National Laboratories 505-844-0660 cordwell@sandia.gov
I'm afraid I couldn't understand this. Could we have a bit more background please? WFL On 5/25/11, Cordwell, William R <wrcordw@sandia.gov> wrote:
Starting with C^3 \ (0,0,0), construct the complex projective space CP^2 with the constraint x^3 + y^3 + z^3. Is this homeomorphic to the torus?
------------------------------------- William R. Cordwell Sandia National Laboratories 505-844-0660 cordwell@sandia.gov
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So, starting with the set CxCxC minus the origin, define the equivalence class of a point x to be all nonzero (complex) multiples of x, and "mod out" by that equivalence relation. This forms the complex projective space CP^2. I'm not sure how the topology is inherited or defined (topologists, please jump in)--maybe it's the largest one that keeps the mapping continuous? For this problem, one also wants initially to restrict to the subset of the original points in C^3 where x^3 + y^3 + z^3 = 0. The claim is that the final subset of CP^2 is homeomorphic to the torus. Bill C. -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Fred lunnon Sent: Wednesday, May 25, 2011 1:30 PM To: math-fun Subject: Re: [math-fun] topology question I'm afraid I couldn't understand this. Could we have a bit more background please? WFL On 5/25/11, Cordwell, William R <wrcordw@sandia.gov> wrote:
Starting with C^3 \ (0,0,0), construct the complex projective space CP^2 with the constraint x^3 + y^3 + z^3. Is this homeomorphic to the torus?
------------------------------------- William R. Cordwell Sandia National Laboratories 505-844-0660 cordwell@sandia.gov
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By changing coordinates it's the same as xyz=0 Sent from my iPhone On May 25, 2011, at 4:17 PM, "Cordwell, William R" <wrcordw@sandia.gov> wrote:
So, starting with the set CxCxC minus the origin, define the equivalence class of a point x to be all nonzero (complex) multiples of x, and "mod out" by that equivalence relation. This forms the complex projective space CP^2. I'm not sure how the topology is inherited or defined (topologists, please jump in)--maybe it's the largest one that keeps the mapping continuous?
For this problem, one also wants initially to restrict to the subset of the original points in C^3 where x^3 + y^3 + z^3 = 0. The claim is that the final subset of CP^2 is homeomorphic to the torus.
Bill C.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Fred lunnon Sent: Wednesday, May 25, 2011 1:30 PM To: math-fun Subject: Re: [math-fun] topology question
I'm afraid I couldn't understand this.
Could we have a bit more background please?
WFL
On 5/25/11, Cordwell, William R <wrcordw@sandia.gov> wrote:
Starting with C^3 \ (0,0,0), construct the complex projective space CP^2 with the constraint x^3 + y^3 + z^3. Is this homeomorphic to the torus?
------------------------------------- William R. Cordwell Sandia National Laboratories 505-844-0660 cordwell@sandia.gov
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participants (3)
-
Cordwell, William R -
Fred lunnon -
Victor S. Miller