Timeline for The 0th homology of a path-connected space
Current License: CC BY-SA 2.5
14 events
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| Oct 8, 2010 at 5:54 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Oct 7, 2010 at 14:08 | comment | added | Tyler Lawson | @Martin: Cech homology has at least three definitions: one is the inverse limit of homology groups over open covers (Aleksandrov–Cech homology), one takes values in pro-abelian groups, and one applies a derived version of the inverse limit functor to the chain complexes in the system of open covers (the version I was using). I'm not sure of a reference and I would have to check closely to see if it satisfies the other axioms. One has to define the relative Cech homology using a mapping cylinder so that the long exact sequence becomes a tautology. | |
| Oct 7, 2010 at 13:43 | answer | added | Tom Goodwillie | timeline score: 5 | |
| Oct 7, 2010 at 13:34 | comment | added | Martin Brandenburg | Hm, does Cech cohomology satisfy the axioms of a homology theory (without disjoint union)? If yes, what is a reference for this fact? | |
| Oct 7, 2010 at 13:06 | answer | added | Tyler Lawson | timeline score: 6 | |
| Oct 7, 2010 at 10:55 | comment | added | Tom Goodwillie | It is not hard to come up with a space whose Cech homology differs from its singular homology. But can you make an example where the $0$th homology is bigger (not smaller) than $0$th singular? Isn't that basically the question? | |
| Oct 7, 2010 at 10:27 | comment | added | Martin Brandenburg | @Andrew: Of course Chris' interpretation is correct. The linked question is another one. | |
| Oct 7, 2010 at 10:24 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Oct 7, 2010 at 9:05 | comment | added | Andrew Stacey | As your question isn't quite clear as stated (is Chris's interpretation correct? and you should correct it in line with Tom's comment) I can't be sure of this, but it looks as though you're asking essentially the same thing as is asked in this question: mathoverflow.net/questions/1750 | |
| Oct 7, 2010 at 6:06 | comment | added | Tim Porter | @Charles It may depend which exact version of that space (TSC) you are thinking of.Essentially the usual TSC has the same Cech invariants as if the sin 1/x bit was omitted. Cech homology is a shape invariant and the usual variants of the TSC (thought of as being in the plane) have contractible open neighbourhoods. | |
| Oct 6, 2010 at 23:55 | comment | added | Charles Rezk | What is the Cech homology of the topologist's sine curve? | |
| Oct 6, 2010 at 19:16 | comment | added | Tom Goodwillie | Maybe you shouldn't say "generalized homology theory" if you want the dimension axiom. | |
| Oct 6, 2010 at 18:13 | comment | added | Chris Schommer-Pries | So really your question is to produce such an H and a space X to evaluate it on. You'll of course need an H which is not invariant under weak homotopy equivalences, so singular homology is out. | |
| Oct 6, 2010 at 17:23 | history | asked | Martin Brandenburg | CC BY-SA 2.5 |