Timeline for Why the Dold-Thom theorem?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 26, 2014 at 5:03 | comment | added | Ryan Budney | I haven't written down a proof, as I don't have time to sort through the details, but this argument seems like a natural thing to do. | |
| Oct 23, 2014 at 19:05 | comment | added | Chris Gerig | OK, if I buy this and $S_f$'s existence, then to get an element of $X$'s homology, I take the composition $S_f\hookrightarrow S^i\times X\twoheadrightarrow X$ and push-forward the image of the fundamental class $[S_f]$? Why would this be the "correct" element? | |
| Oct 23, 2014 at 3:39 | comment | added | Ryan Budney | From cell to cell $k$ can vary. But on the interior of each cell I'm saying the lift has to be unique up to the action of $\Sigma_k$. | |
| Oct 22, 2014 at 22:41 | comment | added | Chris Gerig | I'm sorry I cant' wrap my head around this yet, even for $i=1$. In the first paragraph, $k$ can't vary across the cells of $S^i$, right? And I don't see how the "subspace" of $S^i\times X$ come together to form a CW-complex. I also didn't understand the last sentence; how does the map $S_f\to S^i$ relate to "lifted maps to $X^k$"? And how do I ultimately get an element of homology of $X$? | |
| Oct 21, 2014 at 22:59 | comment | added | Ryan Budney | I suppose the philosophy of this answer is that the homology of a space allows for very stratified singular objects in the space to `detect holes'. Homotopy isn't nearly as flexible, but if you replace the space $X$ by $SP(X)$ you are replacing the space by one where homotopy allows for similarly stratified objects. | |
| Oct 21, 2014 at 22:56 | history | edited | Ryan Budney | CC BY-SA 3.0 |
made less vague, but still fairly vague.
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| Oct 14, 2014 at 22:13 | comment | added | Ryan Budney | Give me a few days. My inbox has become thick in the past few... | |
| Oct 13, 2014 at 0:40 | comment | added | Chris Gerig | I am confused. Could you elaborate in your answer on forming that new CW-complex? I don't see how to put together the union of copies of k-cells with varying k -- what are the attaching maps? I would think this affects whether the resulting "complex" could even have a fundamental class. | |
| Oct 7, 2014 at 22:52 | history | edited | Ryan Budney | CC BY-SA 3.0 |
small grammatical error
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| Oct 7, 2014 at 22:12 | history | answered | Ryan Budney | CC BY-SA 3.0 |