Timeline for The second homotopy group of a simple CW-complex
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 15, 2011 at 14:00 | comment | added | Julien Marché | I agree with Guillaume, the attaching map from the boundary of the 2-cells to the figure-eight can be anything, so your assumption is too strong. To Ralph, $\pi_2(X)$ is always commutative and by Hurewicz's theorem, if $\pi_1(X)=1$, then the map $\pi_2(X)\to H_2(X)$ is an isomorphism, showing the result. Of course, one cannot suppose that $\pi_1(X)=1$! | |
| Dec 15, 2011 at 13:33 | comment | added | Ralph | I wonder about $H_2(X) = Ab \pi_2(X)$: Isn't the second (and higher) homotopy group always abelian ? I only know this formula in degree one, does it also hold in degree 2 ? (I don't know much about topology, so I my be wrong.) | |
| Dec 15, 2011 at 13:14 | comment | added | Guillaume Brunerie | A 2-cell can be attached to some interlacing of both loops (the fundamental group of the figure-8 is free on the two generators that are the two loops). | |
| Dec 15, 2011 at 13:07 | history | answered | Mikael Vejdemo-Johansson | CC BY-SA 3.0 |