Timeline for Question on Ball Quotients
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 30, 2012 at 6:57 | vote | accept | kla | ||
| Jul 30, 2012 at 5:41 | comment | added | Misha | @kla: You have group homomorphisms $Z^2\to Z\to \pi_1(X)$. Each is realized by a continuous map of the respective spaces $T^2\to S^1\to X$ by the same Whitehead's theorem, since circle has contractible universal cover. | |
| Jul 30, 2012 at 3:55 | comment | added | Misha | Since the universal cover of X is contractible, its homotopy groups vanish in dimensions $\ge 2$, hence homotopy class of a map to X is determined by map of fundamental groups (Whitehead's theorem). | |
| Jul 29, 2012 at 19:35 | comment | added | kla | @Misha Sorry, it is not clear to me where you used the fact that $X$ is a ball quotient when you relaize $\eta$ as a composition. Consider the following example: Let $X = E(1)$, an elliptic surface obtained by blowing up a pencil of cubics in $CP^2$ at $9$ base points of the pencil. Note that the class of the fiber torus is non-zero, and the loops of the fiber torus are nullhomotopic. $X$ is not a ball quotient in this case. | |
| Jul 29, 2012 at 18:10 | comment | added | kla | Thanks Misha. How you define the composition? | |
| Jul 29, 2012 at 15:48 | history | answered | Misha | CC BY-SA 3.0 |