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• CommentRowNumber1.
• CommentAuthorTim_Porter
• CommentTimeFeb 26th 2012
• (edited Feb 26th 2012)

I have added more to higher generation by subgroups. As I said on another thread, this material feels as if there should be a nPOV / categorified version that could be quite interesting, so any thoughts would be welcome.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeFeb 26th 2012

Interesting! One obvious guess would be that $\mathcal{H}$ is $n$-generating iff $G$ regarded as an $(n-1)$-group is the colimit of the $H_i$ along their $n$-fold intersections. That doesn’t work for $n=1$, though.

• CommentRowNumber3.
• CommentAuthorTim_Porter
• CommentTimeMar 4th 2012
• (edited Mar 4th 2012)

I have added more material to this entry and have also started a related entry on Volodin spaces.

@Mike You are more or less right. In the Abels and Holz paper they use the nerve to calculate higher syzygies using the bar resolutions of everything in sight and their tool is a version of the homotopy colimit. My question is to push things outward beyond their results, taking this sort of induction principle and, say the van Kampen theorem to produce new results in the nPOV context.

• CommentRowNumber4.
• CommentAuthorTim_Porter
• CommentTimeMar 4th 2012

I have created a stub for Volodin model as well.

• CommentRowNumber5.
• CommentAuthorTim_Porter
• CommentTimeMar 7th 2012
• (edited Mar 7th 2012)

The paper by Tom Fiore et al would look to be relevant, so this may go back towards euler characteristics etc. Any thoughts anyone?