# Jonathan: Speculation/Review (Friday, June 17) :::{.remark} What we've done: - Started with a classical problem: scissors congruence in $E^3$. - Homological ways of approaching this (e.g. \(\K\dash\)theory ) - Combinatorial \(\K\dash\)theory - Goncharov's conjectures It may be unclear why we looked so closely at filtrations on \(\K\dash\)theory spectra. ::: ## The Rank Filtration :::{.remark} The rank filtration is due to Quillen, on his paper of finite generation of $\K_i(\OO_K)$ for rings of integers. Consider $w\sdot \Vect_F^{\fd}$ and filter by rank to get \[ w\sdot \Vect_F^{\dim \leq n} \injects w\sdot \Vect_F^{\dim \leq n-1} .\] Pull weak equivalences out by taking a cofiber to get a sequence \[ \sdot \Vect_F^{\dim \leq n-1} \to \sdot \Vect_F^{\dim \leq n} \to \Sigma^2 T(F) ,\] for $T(F)$ the Tits building. On weak equivalences, one instead gets homotopy coinvariants: \[ w\sdot \Vect_F^{\dim \leq n-1} \to w\sdot \Vect_F^{\dim \leq n} \to \Sigma^2 T(F)_{h\GL} ,\] and on homology, \[ H_i(\sdot^{\leq n-1}) \to H_i(\sdot^{\leq n}) \to H_i( \Sigma^2 T(F)_{h\GL}) \cong H_{?}(\GL_n; \St(F)) \] where $\GL_n$ is considered a discrete group. The associated graded looks like a $\GL$ version of scissors congruence. ::: :::{.remark} Some applications: - Lee-Sczarszer-$\cdots$ show $\K_3(\ZZ) \cong \ZZ/48$ using this filtration, and got some information about $\K_i(\ZZ)$ for $i=4,5$. - Recently: some progress $\K_5, \K_6$ and showed $\K_8(\ZZ) = 0$, and are currently working on $\K_{12}(\ZZ)$, again just using this filtration. See Avner Ash, Gunnels, Kupers. - This inspired figuring out why scissors congruence and the rank filtration were related. Idea: $\SC \mapstofrom \bigvee_n \gr_n \K$, and we'd like information to flow both ways. ::: ## Complexes built out of SC groups :::{.remark} There is a complex: \[ \RR \to P(\EE^3) \mapsvia{D} P(S^1) \tensor P(\EE^1) \to \Omega^1_{\RR/\ZZ} .\] Complexes: - $P_*(S^{2n-1}), P_*(H^{2n-1})$: a conjecture involving weight spaces. - $P(E^{2n-1})$: a conjecture of Goncharov, related to $\Omega^{?}_{\RR/\ZZ}$. We suspect these conjectures are all wrong! Some ways to fix them: - Conjectures should involve *rank* and not weight filtrations. - \(\K\dash\)theory (as stated) is probably not the right thing: it's more likely some variant of Hermitian \(\K\dash\)theory with an analog of the rank filtration. - Why? We care about quadratic forms! And \(\K\dash\)theory doesn't see these. - For $P(E^{2n-1})$: don't know, but probably related to $\HoH$. ::: :::{.remark} Something that might be true: the tail end of $P(S^{2n-1}_k)_\QQ$ always has homology $\K^M(k)_\QQ$. So we can probably show that the tail end of the complex $P(E^{2k-3})$ has homology $\Omega^{2k-2}_{\RR/\ZZ}$. There is a map between these complexes by including one face, which induces a map on homology, and thus a map \[ \K^M_{2n-2}(k)_\QQ \mapsvia{\dlog} \Omega^{2k-3}_{k/\QQ} ,\] which agrees with the Dennis trace. Any map from $\K$ to Kähler differentials is probably supposed to be $\HoH$! ::: :::{.question} - What is the relationship between these complexes? - What is the correct formulation of Goncharov's conjecture. - Probably some variant of Hermitian \(\K\dash\)theory. - Resolve Hilbert's third problem! - That the Dehn invariant and volume are complete invariants of $\SC$, i.e. $H_{2m-1}(P(E^{2k-1})) \mapsvia{\vol} \RR$. Computing these complexes resolves Hilbert's generalized third problem! > Read Hilbert's address, where he recalls comments from Gauss complaining about using Calculus to compute volumes! Big idea: all of this machinery gives us a foothold on a 3000 year-old problem. ::: ## Weight vs Rank Issues :::{.remark} Recall that we could define \(\K\dash\)theory spaces via $\BGL(A)^+$. This is nice for the following reason: given the standard $\K_0$ of representations of $\GL_n$ considered as an algebraic group, one can define maps \[ R_\ZZ \GL_n \to [\BGL_n(A)^+, \BGL_n(A)^+] .\] This gives a source of natural operations on \(\K\dash\)theory -- in particular, there are lambda operations $\lambda^i: \K(A) \to \K(A)$ which corresponds to functors on modules $M \mapsto [\Extalg^i M]$. There are $\psi^i: \K(A) \to \K(A)$ which one lines corresponds to $L \mapsto L\tensorpowerk{i}$ and $\gamma$ operations. The weight filtration is a filtration induced by $\gamma$ operations, but the key point is that its associated graded $\K^{(i)}(A)$ are eigenspaces for the Adams operations. Thus rationally \[ \K(A)_\QQ \cong \bigoplus_i \K^{(i)}(A)_\QQ \in \Grp .\] Not clear if this is related to $\SC$! > See paper by Grayson on Adams operations. > Read a lot of his stuff!! Very useful for \(\K\dash\)theory, deep algebraic insight. ::: :::{.remark} There is another version of the unstable rank filtration -- unknown what its relation is to the previous version near the beginning of this talk. \[ F^{\leq r} K(A)_\QQ = \Im(\Prim H_*(\GL_r)_\QQ \to \Prim H_*(\GL)_\QQ) .\] ::: :::{.question} How does this relate to other rank filtrations? In particular, the original stable rank filtration? ::: :::{.conjecture} Weight vs rank: the associated gradeds of the weight and rank filtrations are the same. ::: :::{.remark} Known for number fields, and that's it! These are not the same for the polytope algebra. ::: :::{.question} How does this relate to constructions like the polytope algebra? ::: :::{.question} What is a homotopical interpretation of the weight filtration? How about for some variant of hermitian \(\K\dash\)theory where there is geometry in play? ::: :::{.question} Is there an easier analog of the weight/rank filtrations on more geometric versions of \(\K\dash\)theory, polytope algebras, etc. ::: ## Combinatorial \(\K\dash\)theory :::{.question} Can all of the Waldhausen theorems be written down for squares \(\K\dash\)theory? > Solve and Jonathan will buy you a beer. 🍺 ::: :::{.question} In \(\K\dash\)theory of rings, $\K^M$ is the totally decomposable part in the sense that \[ (F\units)\tensorpower{F}{m} \to \K^M_m(F) \to \K_m(F) .\] and you can get your hands on $\K_n^M(F)$. What about $\K(\Var\slice k)$? Note that $F\units$ corresponds to automorphism, can you detect images in $\K_n(\Var\slice k)$ by boosting elements up from $\K_1(\Var\slice k)$? ::: :::{.remark} Want a version of hermitian \(\K\dash\)theory where $\Orth_n(R)$ pops out! And more generally $\SC$. :::