Timeline for Is a finite CW complex minus a point still homotopy equivalent to a finite CW complex?
Current License: CC BY-SA 3.0
12 events
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| Jul 4, 2012 at 18:56 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Jun 12, 2012 at 9:07 | comment | added | André Henriques | The Hawaiian earing space has a sequence of loops that converge to a constant loop. In the space I constructed, there is no sequence of loops converging to a constant loop, for the simple reason that the limit point has been removed. Here's another example where a similar phenomenon shows up: $\mathbb R^2\setminus \{1/2,1/3,1/4,...\}$ is h.e. to the Hawaiian earings, but $\mathbb R^2\setminus (\{0\}\cup\{1/2,1/3,1/4,...\})$ is homotopy equivalent to an infinite wedge of $S^1$s. | |
| Jun 12, 2012 at 9:02 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Jun 12, 2012 at 9:02 | comment | added | Oblomov | @Andr\'e: Actually, are you sure that the resulting space is really a a infinite wedge of circles. It seems to me that it might even not be a CW complex but rather a ''sort of Hawaian earrings`` space. | |
| Jun 12, 2012 at 9:00 | comment | added | André Henriques | Yes indeed. Sorry for the confusion. | |
| Jun 12, 2012 at 8:11 | comment | added | Oblomov | @Matin: I guess that Andr\'e thinks of $\mathrm{S}^1$ as the quotient of the interval $[-1,1]$ by its boundary. Its attaching map takes the same value at $-1$ and at $1$. | |
| Jun 12, 2012 at 7:08 | comment | added | Martin Brandenburg | @André: Could you explain the definition of the attaching map? Your function is not periodic. | |
| Jun 11, 2012 at 15:42 | comment | added | André Henriques | There's nothing yuck about this particular example. But I'll agree that the notion of CW-complex is a bit "yuck". Depending on one's perspective, the notion of CW-complex is either too general, or not general enough. The two notions that I believe to have a "good level of generality" are 1) regular CW-complex, and 2) retract of a CW-complex. | |
| Jun 11, 2012 at 15:30 | comment | added | Spice the Bird | All I have to say about this example is "yuck". Good answer though. | |
| Jun 11, 2012 at 15:14 | vote | accept | Oblomov | ||
| Jun 11, 2012 at 15:14 | vote | accept | Oblomov | ||
| Jun 11, 2012 at 15:14 | |||||
| Jun 11, 2012 at 13:38 | history | answered | André Henriques | CC BY-SA 3.0 |