Timeline for What immersed closed curves on the double-torus are non-trivial when lifted to the unit tangent bundle?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Sep 14, 2010 at 23:20 | comment | added | Pablo Lessa | Excelent. That was exactly what I was looking for. Thanks. | |
| Sep 14, 2010 at 18:29 | comment | added | Sergei Ivanov | See e.g. Hatcher, "Algebraic topology", chapter 4. The map from $\pi_2(base)$ to $\pi_1(fiber)$ is the following: an element of $\pi_2(M)$ is a map $D^2\to M$ that sends the boundary circle to the marked point $p\in M$. Lift that map to the total space (using homotopy lifting property). The boundary circle gets lifted to an element of $\pi_1$ of the fiber. This is the canonical image of the original element of $\pi_2(M)$. | |
| Sep 14, 2010 at 17:55 | vote | accept | Pablo Lessa | ||
| Sep 14, 2010 at 17:55 | comment | added | Pablo Lessa | Looks good! Thank you. Although I must admit I'm a little fuzzy on long exact sequences of homotopy groups. In particular, what is the canonical map from $\pi_2(M)$ to $\pi_1(S^1)$ in this case? Also: Any references? | |
| Sep 14, 2010 at 17:06 | history | answered | Sergei Ivanov | CC BY-SA 2.5 |