Timeline for Dimension leaps
Current License: CC BY-SA 2.5
3 events
| when toggle format | what | by | license | comment | |
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| Dec 18, 2009 at 21:38 | comment | added | Ryan Budney | There are a few finiteness results of that type -- for example Seifert surfaces for knots (orientable surfaces that bound a knot in 3-space). If the knot is a special type "fibers over S1" and if the Surface is required to be "incompressible" (minimal genus) then it's known to be unique. You can of course always complicate surfaces by adding handles. | |
| Dec 18, 2009 at 17:42 | comment | added | Steven Sivek | R^3 doesn't have any special properties in that sense. The issue is that curves only have codimension 1 in R^2, so they can separate the plane into multiple components, whereas they can't separate R^3. You could probably formulate a similar problem about using surfaces with boundary to connect 1-manifolds (circles or line segments) in R^3 and get a finiteness result for that if you really wanted. | |
| Dec 18, 2009 at 16:57 | history | answered | psihodelia | CC BY-SA 2.5 |