Timeline for What is the quantity 2(handles)+crosscaps called?
Current License: CC BY-SA 3.0
14 events
when toggle format | what | by | license | comment | |
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May 6, 2011 at 7:45 | vote | accept | Tony Huynh | ||
May 6, 2011 at 4:03 | answer | added | Bill Thurston | timeline score: 10 | |
May 6, 2011 at 1:26 | answer | added | Qiaochu Yuan | timeline score: 9 | |
May 5, 2011 at 23:43 | answer | added | Gerry Myerson | timeline score: 2 | |
May 5, 2011 at 23:04 | comment | added | Sergey Melikhov | Tony, I wouldn't care if your question was not about terminology. Reading it literally, "It is well-known that up to homeomorphism, ..., where $N_k$ is the sphere with $k$ crosscaps." I did get an impression that a crosscap is referred to as a $2$-manifold. If that is a standard terminology somewhere, fine. I was not used to it. | |
May 5, 2011 at 22:19 | comment | added | Tony Huynh | @Sergey: I don't think I ever said a crosscap is a surface. Correctly me if I'm wrong, but I thought that the term sphere with $k$ crosscaps was pretty widely used to mean the surface obtained by cutting $k$ holes out a sphere and gluing a Moebius band along the boundary of each hole? | |
May 5, 2011 at 22:09 | comment | added | Simon Rose | @Tony - The problem with calling this "generalized genus" is that a generalization of a quantity should agree with the original one where they are both defined. However, in this case your generalized genus of a torus is 2, while the genus is 1. This seems not to be ideal. | |
May 5, 2011 at 21:38 | comment | added | Sergey Melikhov | A crosscap is not a surface but a singularity of a generic map of the surface into $\Bbb R^3$. The surface is called the Moebius band. | |
May 5, 2011 at 21:21 | history | edited | Tony Huynh | CC BY-SA 3.0 |
added reference tag
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May 5, 2011 at 21:16 | comment | added | Tony Huynh | I am just wondering how badly I would be bastardizing nomenclature if I just called it genus? | |
May 5, 2011 at 21:14 | comment | added | Tony Huynh | Thanks. Yes, I thought of using that, but I didn't really want to bring in homology, and I thought there would be a standard name like say generalized genus to use. | |
May 5, 2011 at 21:09 | comment | added | Qiaochu Yuan | Isn't this just $\dim H_1$ (say with rational coefficients)? | |
May 5, 2011 at 20:57 | answer | added | Simon Rose | timeline score: 5 | |
May 5, 2011 at 20:46 | history | asked | Tony Huynh | CC BY-SA 3.0 |