# Elise: Construction the Goncharov map (Friday, June 17) :qa :::{.remark} Idea: iterate the Dehn invariant as follows \[ P(X^n) \mapsvia{D^i} P(X^i)\tensor P(\SS^{n-i-1}) .\] ::: :::{.example title="?"} Even $D^i$ vanish, and e.g. the following diagram commutes: \begin{tikzcd} {P(X^5)} && {P(X^3)\tensor P(S^1)} \\ \\ {P(X_1)\tensor P(S^3)} && {P(X^1)\tensor P(S^1) \tensor P(S^1)} \arrow["{\id\tensor D_1}", from=3-1, to=3-3] \arrow["{D_1}", from=1-1, to=3-1] \arrow["{D_1\tensor \id}", from=1-3, to=3-3] \arrow["{D_3}", from=1-1, to=1-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJQKFheNSkiXSxbMiwwLCJQKFheMylcXHRlbnNvciBQKFNeMSkiXSxbMCwyLCJQKFhfMSlcXHRlbnNvciBQKFNeMykiXSxbMiwyLCJQKFheMSlcXHRlbnNvciBQKFNeMSkgXFx0ZW5zb3IgUChTXjEpIl0sWzIsMywiXFxpZFxcdGVuc29yIERfMSJdLFswLDIsIkRfMSJdLFsxLDMsIkRfMVxcdGVuc29yIFxcaWQiXSxbMCwxLCJEXzMiXV0=) ::: :::{.remark} Problem: we're taking iterated homology, which is difficult to reason about. Solution: construct a simplicial set that has homology $P(X, 1)$ and define the Dehn invariant directly on this. The goal is to delay taking coinvariants $H_0(G, \wait)$ as long as possible. ::: :::{.conjecture} Goncharov: there exists a homomorphism \[ H_m(P_*(X^{2n-1})) \mapsvia{\phi_n} \qty{\gr_n^\gamma \K_{n+m} (\CC)_\QQ\tensor \QQ^{?}}^{\pm} .\] ::: :::{.remark} Spoiler: there is a map \[ H_{n+m}(I(X); \ZZ\invert{2}^t) \mapsvia{\theta_n} H_m(P_*(X)) .\] If $X=S^n$ then the LHS is $\Orth_n$. ::: :::{.definition title="?"} Let $X = S^n$ or $H^n$ given by $q= \sum x_i^2$ and $q = -x_0^2 + \sum x_i^2$ respectively. A subspace $U$ of $X$ is the linear subspace $U'$ such that $\ro{q}{U'}$ is nondegenerate and has maximal negative signature. ::: :::{.example title="?"} If $X = H^n$, then $U \cong H^i$ and $U\perp \cong S^{n-i?}$. For $X = S^n$, $U\cong S^i$ and $U\perp \cong S^?$. ::: :::{.definition title="RT buildings"} The **RT-building** $\cx{F}^X$ is the simplicial set where $[i]$ maps to chains of nonempty subspaces $U_0 \injects \cdots U_i = X$, where the face maps are deleting $U_j$ and the degeneracies are repeating them. ::: :::{.fact} $I(X)\actson \cx{F}^X$ by translating each open set in a chain. ::: :::{.theorem title="?"} There is an isomorphism \[ P(X, 1) &\iso H_{n+1}(S^0 \smashprod \cx{F}^X) \\ \ts{x_0,\cdots, x_n} &\mapsto \sum_{\sigma\in \Sigma_{n+1}} \sign(\sigma) [\sp X_{\sigma(0)} \injects \sp(X_{\sigma(0)}, X_{\sigma(1)}), \cdots] .\] Thus $P(X, G) \cong H_0(G, ?)$. ::: :::{.theorem title="?"} $H_n(\cx{F}^X) \neq 0$ only in $n=\dim X$. ::: :::{.remark} Next goal: construct the Dehn invariant on $\cx{F}^X$. Idea: the angle can be captured by projecting onto orthogonal complements. ![](figures/2022-06-17_15-29-32.png) ::: :::{.definition title="?"} For pointed simplicial sets $X, Y$, the reduced join $X\tilde\ast Y$ is defined by \[ (X\tilde\ast Y)_n = \bigvee_{i+j=m-1} X_i \smashprod Y^j \homotopic S^1 \smashprod X \smashprod Y .\] ::: :::{.definition title="?"} Let $U \leq X$ be a proper nonempty subspace and define \[ D_u: \cx{F}^X &\to \cx{F}^X \tilde\ast \cx{F}^X \\ \vector U \da (U_0 \subseteq \cdots \subseteq U_n) &\mapsto \begin{cases} \vector U \smashprod \pr_{U\perp} \vector U & j = \max\ts{i\st U_i = U} \\ \pt & \text{ if there does not exist a $j$ such that $U_j = U$}. \end{cases} \] ::: :::{.definition title="Derived Dehn invariant"} The dimension $i$ derived Dehn invariant $D_i$ is the lift: \begin{tikzcd} && {\bigvee_{U\subseteq X, \dim U = i} \cx{F}^{U} \tilde\ast \cx{F}^{U\perp}} \\ \\ {\cx{F}^X} && {\prod_{U\subseteq X, \dim U = i} \cx{F}^{U} \tilde\ast \cx{F}^{U\perp}} \arrow[from=3-1, to=3-3] \arrow["{D_i}", dashed, from=3-1, to=1-3] \arrow[from=1-3, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJcXGN4e0Z9XlgiXSxbMiwyLCJcXHByb2Rfe1VcXHN1YnNldGVxIFgsIFxcZGltIFUgPSBpfSBcXGN4e0Z9XntVfSBcXHRpbGRlXFxhc3QgXFxjeHtGfV57VVxccGVycH0iXSxbMiwwLCJcXGJpZ3ZlZV97VVxcc3Vic2V0ZXEgWCwgXFxkaW0gVSA9IGl9IFxcY3h7Rn1ee1V9IFxcdGlsZGVcXGFzdCBcXGN4e0Z9XntVXFxwZXJwfSJdLFswLDFdLFswLDIsIkRfaSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsyLDFdXQ==) This is $I(X)$ equivariant, and $H_0(I(X); H_{n+1}(S^0\smashprod D_i))$ is the classical Dehn invariant. ::: :::{.lemma title="?"} For $V\subseteq U$, \begin{tikzcd} {\cx{F}^{X}} && {\bigvee_{U\subseteq X, \dim U = j} \cx{F}^{U}\tilde\ast \cx{F}^{U\perp}} \\ \\ {\bigvee_{U\subseteq X, \dim U = j} \cx{F}^{V}\tilde\ast \cx{F}^{V\perp}} && {\bigvee_{U\subseteq X, \dim U = j} \cx{F}^{U}\tilde\ast \cx{F}^{U\perp \cap V} \tilde\ast\cx{F}^{V\perp}} \arrow["{D_i}", from=1-1, to=1-3] \arrow["{D_i, \id}", from=3-1, to=3-3] \arrow["{D_j}", from=1-1, to=3-1] \arrow["{\id, D_{j-i-1}}", from=1-3, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) ::: :::{.remark} Goal: get a spectral sequence converges to the total homotopy cofiber of something, which we'll analyze using the fact that we can swap the order of homotopy fibers and coinvariants, since they are both colimits. ::: :::{.remark} What is this spectral sequence? For a functor $F: I_n\to \Top_*$ where $I_n$ is the cube diagram of dimension $n$, \[ E^1_{p, q} = \bigoplus _{A = (b, a_1, \cdots, a_{n-p-1})} \tilde H_1(F(A)) \abuts \tilde H_{p+1}(\cof^{th} F) .\] Here $I_n$ is the cube diagram: \begin{tikzcd} && {(3)} && {(2, 1)} \\ {I_3:} \\ && {( 1, 2)} && {(1,1,1)} \arrow[from=1-3, to=3-3] \arrow[from=3-3, to=3-5] \arrow[from=1-3, to=1-5] \arrow[from=1-5, to=3-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMiwwLCIoMykiXSxbNCwwLCIoMiwgMSkiXSxbMiwyLCIoIDEsIDIpIl0sWzQsMiwiKDEsMSwxKSJdLFswLDEsIklfMzoiXSxbMCwyXSxbMiwzXSxbMCwxXSxbMSwzXV0=) ::: :::{.definition title="?"} \[ D^0: I_n &\to \Top_* \\ (b, a_1, \cdots, a_k) &\mapsto \qty{ \bigvee_{d\im V_j = a_j-1, \dim W = b} \cx{F}^W \tilde\ast \tilde\bigast_{j=1}^k \cx{F}^{V_j}}_{hI(X)} .\] ::: :::{.remark} Thus \[ E_{p, q}^1 = \bigoplus _{A = (b, a_1,\cdots, a_{n-p-1})} \tilde H_q(D^0(A)_{hI(X)}l \ZZ\invert{2}) .\] ![](figures/2022-06-17_15-56-37.png) What this spectral sequence looks like: ![](figures/2022-06-17_15-59-40.png) The bottom row is \[ \tilde H_{d+1}( (\cx{F}^X_{hI(X)}) ) \to \bigoplus _{\abs A = 2} \tilde H_{d+1}(D^0(A)_{hI(X)}) \to \cdots \abuts P_*(X) .\] Upshot: get projection to the base. If $E_{p, q}^* \abuts G_{p+q}$ and $E_{*, n}'$ is the first nonzero row, get homomorphisms \[ \Theta_M: G_m \to E_{m-n, n}^1 .\] This produces a map \[ H_m( (\cof^{th} D)_{hI(X)}; \ZZ\invert{2} ) \mapsvia{\Theta_m} H_m(P_*(X)) ,\] so we reduce to computing the total homotopy cofiber here. ::: :::{.theorem title="?"} \[ (\cof^{th} D^0)_{hI(X)} \homotopic (S^0 \smasprod S^{n-1})_{hI(X)} .\] > Jonathan: this is a big deal! Giant cube with infinite wedges of spheres simplifies to a single sphere. > Maybe need to invert 2. ::: :::{.remark} \[ H_m(S^0\smashprod S^{n-1}; \ZZ\invert{2}) \cong \tilde H_{m+n}(I(X); \ZZ\invert{2}^t) \mapsvia{\Theta_n} \tilde H_m(P_*(X)) .\] ::: :::{.remark} Note that $\tilde H_m(P_*(X)) \to \RR$ by the Cheeger-Chern-Simons regulator, which agrees with the Borel regulator up to inverting 2, and the RHS maps via volume. ::: :::{.conjecture} $\Theta_m$ is an isomorphism. > Feeling: it is probably *not*. :::