Timeline for Are "most" spaces aspherical?
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| Apr 14, 2019 at 5:20 | comment | added | cgodfrey | @KevinCasto I'm not familiar with Gromov's work on random groups but you're definitely right. All I'm saying is that infinite $\pi_1$ is a necessary condition for a non-contractible finite complex to be aspherical. There are certainly way more such complexes with infinite $\pi_1$, e.g. take the product of any finite CW complex with $S^1$. | |
| Apr 13, 2019 at 6:51 | comment | added | Kevin Casto | Surely it's much easier/weaker that "most" spaces have infinite $\pi_1$ than that "most" spaces are aspherical? (e.g., Gromov's work on random groups) | |
| Apr 13, 2019 at 5:57 | comment | added | cgodfrey | Interesting. Oh, also I guess another class of aspherical manifolds would be tori. | |
| Apr 13, 2019 at 5:39 | comment | added | Tim Campion | I absolutely agree. In fact, I'm so used to thinking along these lines that I found myself scratching my head the other day when I heard the phrase "finite aspherical space" in a talk (and then slapping my forehead when I remembered "duh, think about hyperbolic manifolds"!). So my sense is that the spaces which homotopy theorists tend to think of as "typical" are in some sense a very biased sample, just as Luck's survey indicates that the manifolds we think of as "typical" are actually very special. | |
| Apr 13, 2019 at 3:34 | history | answered | cgodfrey | CC BY-SA 4.0 |