Timeline for Reference for intersection and linking in algebraic topology
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 2, 2010 at 20:00 | comment | added | Charlie Frohman | The interpretation of the cup product as an intersection depends on Poincare Duality, which only happens for manifolds, or for restricted kinds of homology for restricted kinds of complexes. Yet there is a way through to definining a product structure which has a lot of properties of the intersection pairing, like graded commutativity. There is a formulas for linking number in terms of intersection with a chain that is reminiscent of the formula for Massey products. Check out Griffiths and Morgan "Rational Homotopy theory and Differential Forms". | |
| Aug 2, 2010 at 12:02 | comment | added | Jeff Strom | This is my general understanding, too. But can you make the phrase "what is left of intersection" precise? (Or point me to a reference?) | |
| Aug 2, 2010 at 6:57 | answer | added | algori | timeline score: 3 | |
| Aug 1, 2010 at 21:40 | answer | added | Daniel Moskovich | timeline score: 1 | |
| Jul 30, 2010 at 17:41 | comment | added | Charlie Frohman | You can define the cup product in the singular cohomology of an arbitrary simplicial complex, and the information that it carries is what is left of intersection. Similarly, Massey products play the role of linking numbers. | |
| Jul 30, 2010 at 7:15 | history | edited | Charles Matthews | CC BY-SA 2.5 |
downcase
|
| Jul 29, 2010 at 22:30 | answer | added | Greg Friedman | timeline score: 3 | |
| Jul 29, 2010 at 20:10 | comment | added | Ryan Budney | You want a notion of linking that applies to arbitrary subcomplexes of arbitrary simplicial complexes? I suspect if you want this idea of "linking" to be at all a non-trivial reflection of what happens in manifolds, at least at this level of generality you're sunk. "Poincare Duality complexes" are more general than manifolds and have everything you need to get the machinery moving. But they're much more restrictive than arbitrary simplicial complexes. | |
| Jul 29, 2010 at 19:59 | history | asked | Jeff Strom | CC BY-SA 2.5 |