Timeline for Examples of "Unusual" Classifications
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 17, 2014 at 0:58 | comment | added | David Roberts♦ | Do you mean compact surfaces? I can think of some pretty nasty non-compact examples (infinite genus with infinite punctures) | |
| Mar 25, 2011 at 14:51 | comment | added | Simon Rose | @José - I think it depends on how you view the classification. Purely topologically, I would argue that the two families are: $A_g$ = the connected sum of $g$ copies of $T^2$, and $B_k$ = the connected sum of $k$ copies of $RP^2$. Ignoring the geometry, this is perfectly valid. The only dodgy part of it, perhaps, is that $B_0 = A_0$, but this sort of thing also occurs in the classification of Lie Algebras as well, so... | |
| Mar 25, 2011 at 2:48 | comment | added | Paul | How about classification of irreducible closed 2-manifolds: just $S^2, T^2, and RP^2$. All others are (connected) sums of these. So maybe this is an example of (B). @Jose why do you view g=1 as "sporadic"? (hyperbolic metric, presumably?) topologically they aren't much more special than their higher genus siblings. | |
| Mar 25, 2011 at 1:18 | comment | added | José Figueroa-O'Farrill | @Simon: I'm not really sure that this is an example of (A). I think one could argue that that orientable surfaces with genera $g>1$ are the ones making up the generic infinite family, whereas $g=0$ and $g=1$ are the two "sporadic" examples. It is just that in this case, the sporadic examples came first. | |
| Mar 24, 2011 at 18:53 | comment | added | ARupinski | Wow, I guess I totally overlooked such an obvious example from undergrad/grad topology classes. | |
| Mar 24, 2011 at 18:53 | comment | added | Simon Rose | I should add that this is an example of (A). | |
| Mar 24, 2011 at 18:48 | history | answered | Simon Rose | CC BY-SA 2.5 |