# Diego: (Friday, June 17)

## Goncharov's Conjecture

:::{.remark}
Goncharov's conjectures:
let $V^n = \EE^n, \SS^n, \HH^n$ denote a geometry, and recall that the Dehn invariant is defined by
\[
P(V^n)&\to \oplus _{1\leq i\leq n-2} P(V^i) \tensor P(S^{n-i-1})\\
[P] &\mapsto \sum_{1\leq i\leq n-2} \sum_{i\dash\text{dimensional }faces} [A] \tensor [s(A)]
\]
where $S(A) \da s(A\perp) \intersect P$.

> Todo: missing some stuff

:::

:::{.theorem title="Goncharov"}
There exists a map
\[
\qty{ \ker \ro{ D_{\HH}^{2n-1} }{P(\HH^{2n-1}; \QQbar)} }_\QQ\to \qty{ \K_{2n-1}(\QQbar) \tensor \eps(n) }^-
.\]
and the LHS maps by volume to $\RR$ and the RHS to $\RR$ by the Borel regulator.
:::

:::{.remark}
Borel's theorem says the Borel regulator is injective up to torsion.
Note that this may not be true for spherical geometries, since the volume is not well-defined.
Note that we are working over $\QQbar$ and not just a number field; this is still open for $\CC$.
:::

:::{.remark}
Define the Dehn complex:
\[
D^*_V(n) = \bigoplus_{n\geq 1} \bigoplus_{\lambda \partition n} \bigotimes_{\lambda_i}P(V^{\lambda_i})
\]
where all $\lambda_i > 0$.
:::

:::{.conjecture}
There exists a map
\[
H^i(D^*(n))_\QQ \to \qty{ \gr_n^\gamma \K_{2n-i}(\CC)_\QQ \tensor \eps(n) }^-
\]
which is an isomorphism.
:::

:::{.remark}
Today we'll nook at $H^n$ and $H^{n-1}$.
:::

## Cohomology of Dehn Complexes

:::{.remark}
Let $E = \RR^n$ and $T(n)$ be the Tits building.
Then
\[
H_{n-2}(T(n); \QQ) = \St(n)
,\]
and $P(S^{n-1}) = H_0(\Orth_n; \St(n)^t)$.
Consider $[p] \mapsto \mcl_p(n)$, the $\qmod$ generated by $(E_1,\cdots, E_p)$ with $E_i\neq 0$ and $E = \bigoplus E_i$.
This yields a semisimplicial set $\vector E \mapsto (E_1,\cdots, E_i \oplus E_{i+1}, \cdots, E_p)$.
Then $H_n(\mcl_*(n)) = \St(n)$.
:::

:::{.remark}
Let $\AA = \bigoplus _{n, i} H_i(\Orth(n), \QQ^t)$.
Using Eilenberg-Zilber and Kunneth yields
\[
H_i(\Orth_n; \QQ^t) \tensor H_j(\Orth_m, \QQ^t)\to \cdots \to H_{i+j}(\Orth_n \times \Orth_m; \QQ^t\times \QQ^t)
.\]
:::

:::{.remark}
Coalgebra of homology of Steinberg modules:
\[
\HH\St = \bigoplus_{n, q} H_q(\Orth_n; \St(n)^t)
.\]
Define a map
\[
\mcl_p(n) &\to \bigoplus_{n_1 + n_2=n, p_1+p_2=n} \mcl_{p_1}(n_1) \tensor \mcl_{p_2}(n_2) \\
\vector E &\mapsto \sum \vector (E_1,\cdots, E_{p_1}) \tensor (E_{p_1+1},\cdots, E_{p})
,\]
which induces a map of complexes $\mcl_*(n) \to \bigoplus _{n_1+n_2} \mcl_*(n_1) \tensor \mcl_*(n_2)$.
Take homology on both sides; the LHS is $\St(n)$, and since the homology of the Tits building is only supported in one degree, the RHS is 
\[
\bigoplus _{n_1+n_2=n} \St(n_1) \tensor \St(n_2)
.\]
Use these as coefficients and apply Shapiro's lemma to obtain
\[
H_1(\Orth_n; \St(n)^t) \to \bigoplus _{n_1+n_2=n} H_q(\Orth_n; \bigoplus _{V_1 \oplus V_2 = V} \St(V_1)^t \tensor St(V_2^t)) \\
\cong
\bigoplus _{n_1+n_2=n} H_q(\Orth_{n_1} \times \Orth_{n_2}; \St(n_1)^t \tensor \St(n_2)^t )
.\]
Thus $\HH \St$ is a coalgebra.
We're interested in
\[
\HH_0 \St = \bigoplus _n H_0(\Orth_n, \St(n)^t) \cong \bigoplus _n P(S^{n-1})
.\]
A not-at-all obvious fact: this is an isomorphism of coalgebras.
:::

:::{.theorem title="Caithelineau"}
For every $n$, there exists a spectral sequence
\[
E^2_{p, q} = H_1^p(\HH \St; \QQ) \abuts H_{-p+q+n}(\Orth_n; \QQ^t)
,\]
where $n$ means the $n$th degree part in the cobar complex.
:::

:::{.proof title="Sketch"}
Consider the following tri-complex

\[
C_\alpha(\Orth_n, \qty{ \Omega_{\beta, \gamma} \mcl_*(n)}^t )
,\]
where we've used that $\mcl_*(n)$ has a comultiplication to take a cobar complex:
\[
\Omega_{\beta, -\gamma} \mcl_*(n) = \bigoplus _{n_i \partition n, \beta_{\gamma_i}\partition \beta} H_{p_1}(\Orth_{n_1}; \mcl_{p_1}(n_1)^t) \tensor \cdots
.\]

> Todo: laptop died. :-(

:::