Timeline for vanishing of $\pi_2$ and $H_2$
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 23, 2011 at 23:29 | vote | accept | Isaac Goldbring | ||
| Feb 2, 2011 at 21:03 | comment | added | Greg Kuperberg | It is not true that there is a simpler proof, because the question basically is the Hurewicz theorem. That is, the full version of the Hurewicz theorem looks stronger, but this special case already requires basically all of the ideas. | |
| Feb 2, 2011 at 0:12 | answer | added | Charles Rezk | timeline score: 11 | |
| Feb 2, 2011 at 0:01 | answer | added | Jeff Strom | timeline score: 1 | |
| Feb 1, 2011 at 23:59 | comment | added | Somnath Basu | You would first want to show that any element $\alpha$ of $H_2(X;\mathbb{Z})$ can be realized by some map of a closed, oriented surface into $X$. If you're using simplicial homology then $\alpha$ is a bunch of triangles which glue together in $X$ and has no boundary. You need to resolve this object at the vertices where it may not be like a surface. Now use the classification of surfaces and kill off the generators ($\pi_1(X)=0$) of the fundamental group of this surface. Homologically, this means that $\alpha$ is the sum of spheres mapping into $X$ and then use $\pi_2(X)=0$. | |
| Feb 1, 2011 at 23:53 | history | asked | Isaac Goldbring | CC BY-SA 2.5 |