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Meta
Tag Archives: completed cohomology
ArXiv x 3
Three recent arXiv preprints this week caught my interest and seemed worth mentioning here. The first is a paper by Oscar Randal-Williams, which considers (among other things) the cohomology of congruence subgroups of \(\mathrm{SL}_N(\mathbf{Z})\) in the stable range. This is … Continue reading
Posted in Uncategorized
Tagged Andrew Blumberg, Andrew Putman, Benson Farb, Bill Thurston, completed cohomology, congruence subgroups, Dalimil Mazáč, Jordan Ellenberg, Melanie Wood, Michael Mandell, Nigel Boston, Oscar Randal-Williams, Peter Kravchuk, Simon Marshall, Sridip Pal, Thomas Church, Will Hearst, Will Sawin, William Stein
2 Comments
The stable cohomology of SL(F_p)
Back by popular demand: an actual mathematics post! Today’s problem is the following: compute the cohomology of \(\mathrm{SL}(\mathbf{F}_p)\) for a (mod-p) algebraic representation. Step 0 is to say what this problem actually is. It makes sense to talk about certain … Continue reading
Posted in Mathematics, Uncategorized
Tagged completed cohomology, completed K-theory, Eric Friedlander, Evens, K-theory, stable cohomology
4 Comments
Ventotene, Part II
I promised to return to a more mathematical summary of the conference in Ventotene, and indeed I shall do so in the next two posts. One of the themes of the conference was bounding the order of the torsion subgroup … Continue reading
Posted in Uncategorized
Tagged Akshay Venkatesh, Borel, completed cohomology, Nicolas Bergeron, Peter Scholze, Serre, torsion
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H_2(Gamma_N(p),Z)
In this post (which is a follow-up to the last post), I wanted to compute the group \( H_2(\Gamma_N(p),\mathbf{Z})\), where \( \Gamma_N(p)\) is the congruence subgroup of \( \mathrm{SL}_N(\mathbf{Z})\) for large enough \( N\) and \( p\) is prime. In … Continue reading
Stable completed homology without Quillen-Lichtenbaum
Having just made (hopefully) the final revisions on my paper on stable completed cohomology groups, I wanted to record here a few remarks which didn’t otherwise make it into the paper. The first is that, in addition to the result … Continue reading
Posted in Mathematics
Tagged Borel, completed cohomology, K-theory, Milnor, Moore, Neisendorfer, Paul Goerss, Quillen, Soule
2 Comments
Local representations occurring in cohomology
Michael Harris was in town for a few days, and we chatted about the relationship between my conjectures on completed cohomology groups with Emerton and the recent work of Scholze. The brief summary is that Scholze’s results are not naively … Continue reading
Posted in Mathematics
Tagged completed cohomology, Matthew Emerton, Michael Harris, Peter Scholze, torsion
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Virtual Congruence Betti Numbers
Suppose that \(G\) is a real semisimple group and that \(X = \Gamma \backslash G/K\) is a compact arithmetic locally symmetric space. Let us call a cohomology class tautological if it is invariant under the group \(G\). For example, if … Continue reading