Timeline for Obstruction Cocycles
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 12, 2010 at 23:15 | comment | added | Paul | PS. keep in mind that even showing that $\partial^2=0$ is not so easy if you use $\partial e=\sum_i[w_i:e]w_i$ as your definition of the differential, and the question about why the obstruction cochain is a cocycle is basically a variation on the same question. On the other hand, the with defn I outlined above $\partial^2=0$ is obvious. | |
| Sep 12, 2010 at 23:09 | comment | added | Paul | Juan: try this. Consider the map of pairs $(D^{q+1}, S^{q})\to (X^{q+1}, X^{q})$ given by the characteristic map for $e$ and consider the map of long exact sequences of these pairs. The way you define the obstruction cocycle in your question pulls back to $Ze=H_{q+1}(D^{q+1},S^q)\to H_q(S^q)\pi_q(S^q)\to \pi_q(F)$. This shows that the restriction of Steenrod's obstruction cochain to $e$ is the cochain you defined in your first paragraph. Now the argument I outlined shows that Steenrod's cochain is a cocycle. | |
| Sep 12, 2010 at 21:58 | comment | added | Juan OS | You are correct about the definition of the cell complex, however my problem is that Steenrod appears to be stating that the composite of the maps in the sequence I wrote is the value of the obs. cochain, and I feel in doing so he is stating something like "the value of the map in the (topological) boundary is the same as its value in the (homological) boundary", I'll try to reread the proof from the first definition you gave of the cell complex. Thanks. | |
| Sep 12, 2010 at 21:38 | history | answered | Paul | CC BY-SA 2.5 |