Timeline for Homology of $\mathrm{PGL}_2(F)$
Current License: CC BY-SA 3.0
15 events
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| Apr 26, 2018 at 8:00 | history | edited | Peter Scholze | CC BY-SA 3.0 |
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| Apr 24, 2018 at 8:33 | history | edited | Peter Scholze | CC BY-SA 3.0 |
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| Apr 23, 2018 at 11:26 | comment | added | Peter Scholze | @Matthias Wendt: Thanks a lot for the Borel-Yang reference! Let's not worry right now about the exact shape of the torsion; I'd be happy enough with the statement rationally (but then my "evidence" would be vacuous). About homotopy vs. homology: One reason is that the homotopy of the K-theory space is usually simpler than the homology of the K-theory space (just the primitives). Maybe the better answer is that I have some roundabout argument that led me to believe in the question, but it's not worth spelling this out while there is no nontrivial evidence. | |
| Apr 23, 2018 at 10:46 | comment | added | Matthias Wendt | For the function field case, Harder's results show that rational cohomology of $SL_2(F)$ is torsion. I don't know of any results bounding torsion there. The homology of $X_2(F)$ is the equivariant homology of a complex of configurations of points on $\mathbb{P}^1$. The exponential bounds for torsion (in cohomology) could follow if we knew that all torsion comes from symmetry groups of configurations. But I guess we don't know that. Finally: why should we assume that it's easier to study homotopy of $X_2(F)$ instead of its homology? | |
| Apr 23, 2018 at 10:42 | comment | added | Matthias Wendt | According to Borel and Yang, the rational cohomology of $SL_2(F)$ with $F$ a number field is the tensor product: for every real place an exterior algebra generated by an Euler class in degree 2, for every complex place an exterior algebra generated by a class in the Bloch group. So it's reasonable to assume that in this case $X_2(F)$ is rationally a product of 3-spheres. I haven't checked the details, but that would be partial support for assuming that the higher homotopy groups (above degree 3) of $X_2(F)$ are torsion. | |
| Apr 22, 2018 at 16:23 | comment | added | Peter Scholze | @S. carmeli: I don't think the cohomological dimension of $F$ should be relevant. In fact, there is no "Galois descent" (even approximate, like maybe in degrees bigger than the cohomological dimension or so (like for $K$-theory)) for $X_2(F)$. In fact, for $F=\overline{\mathbb F}_p$, the Bloch group is $0$, and all homotopy groups of $X_2(F)$ are bounded torsion. Thus, there is no hope to recover $X_2(\mathbb F_p)$ from $X_2(\overline{\mathbb F}_p)$. | |
| Apr 22, 2018 at 7:59 | comment | added | S. carmeli | Don't you want to assume a bound on the cohomological dimension of $F$? I think that his is sort of important in the case of a finite field, since etale homotopically the space X(F) over the algebraic closure is very close to an Eilenberg-Mclane space and then taking fixed points don't take high homotopies to far down, but this sort of reasoning breaks in high cohomological dimension. | |
| Apr 21, 2018 at 21:20 | comment | added | François Brunault | Related to your final remark, K. Knudson has computed the homology of PGL_2(A) where A is the coordinate ring of an elliptic curve, using an action of this group on a certain Bruhat-Tits tree, see "On the K-theory of elliptic curves". Of course the result involves the homology of PGL_2 of the base field k, so is computable if k is finite. | |
| Apr 21, 2018 at 15:40 | history | edited | Peter Scholze | CC BY-SA 3.0 |
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| Apr 21, 2018 at 15:38 | comment | added | Peter Scholze | @Matthias Wendt: "Bounded torsion" means "bounded exponent". I would expect something like "$\pi_i X_2(F)$ is killed by $i!$ for $i\neq 3$". I would imagine that the torsion is of the same cardinality as $F$. | |
| Apr 21, 2018 at 15:19 | comment | added | Peter Scholze | @Will Sawin: This is basically Quillen's classical computation. | |
| Apr 21, 2018 at 10:24 | comment | added | Matthias Wendt | I think for all that's known now we can't even rule out the possibility that $H_i(X_2(F))$ has cardinality of $F$ for any algebraically closed field $F$. Does "bounded torsion" mean bound on size or bound on exponent? | |
| Apr 20, 2018 at 21:28 | comment | added | Will Sawin | How does one do the computations in the $PGL_2(\mathbb F_q)$ case? | |
| Apr 20, 2018 at 15:14 | history | edited | YCor |
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| Apr 20, 2018 at 9:58 | history | asked | Peter Scholze | CC BY-SA 3.0 |