Timeline for Reverse map of a homology equivalence.
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 30, 2012 at 21:52 | vote | accept | Peter Muller | ||
| Aug 30, 2012 at 21:51 | comment | added | Peter Muller | Merci beaucoup. | |
| Aug 30, 2012 at 16:00 | comment | added | John Klein | Yep. I was just posting it as an answer, but you beat me to the punch. | |
| Aug 30, 2012 at 15:58 | answer | added | John Klein | timeline score: 13 | |
| Aug 30, 2012 at 15:57 | comment | added | Oscar Randal-Williams | @John: Such a map would lift to the universal cover of $X$, so have degree divisible by 120. | |
| Aug 30, 2012 at 15:44 | comment | added | John Klein | My guess is that the $X =$ the Poincare homology sphere should provide a counterxample. If there were a map $S^3 \to X$ which induced a homology isomorphism, then this map is necessarily of degree one and we see that the fundamental class of $X$ is spherical. I don't see it at the moment, but this is very unlikely to be true. | |
| Aug 30, 2012 at 15:31 | comment | added | Hunter Brooks | In extreme generality no - take the circle and map it onto the non-Hausdorff space "$[-1, 1]$ with two-origins" (by projecting vertically downward, mapping the north and south poles to the separate origins). This gives an isomorphism on all homology groups, but any map the other way is homotopic to a constant since the origins collapse. | |
| Aug 30, 2012 at 15:07 | history | asked | Peter Muller | CC BY-SA 3.0 |