# Explicit arithmetic Eisenstein cocycles: the toric case

Peter Xu

##  1\. Introduction

In [SV], the authors construct $1$-cocycles for $\text{GL}_{2}(\mathbb{Z})$
and its arithmetic subgroups, taking values in the Milnor $K$-groups
$K_{2}^{M}(k(\mathbb{G}_{m}^{2}))$ and $K_{2}^{M}(k(\mathcal{E}^{2}))$. In
this article, we aim to generalize their approach in the multiplicative
setting, by constructing analogous cocycles for (subgroups of)
$\text{GL}_{n}(\mathbb{Q})$ valued in $K_{n}^{M}(k(\mathbb{G}_{m}^{n}))$, or
various refinements/specializations thereof.

###  1.1. Relation to existing work

Eisenstein cocycles defined cohomologically (rather than automorphically) have
a long history in the literature; in lieu of naming dozens of works here, we
single out the pioneering work of Sczech [Scz] and Shintani [Shin]; indeed,
our use of spherical chain complexes is just a variation on the latter’s
approach using the eponymous “Shintani cones”.

In the motivic setting, we should also mention the work of Lim and Park [LP]
based off an idea of Stevens [Stevens], who construct a cocycle valued in
Milnor $K$-theory of a ring of “trigonometric functions” using an approach
similar to that of Shintani, and thus also to our parameterization by
spherical chain complexes. Their construction essentially is a version “at
infinite level” of ours, with their ring of trigonometric functions
corresponding to the functions on a pro-tower of algebraic tori over all
finite isogenies.

Our main inspiration is the work of Sharifi-Venkatesh [SV], who not only
construct the cocycle in the case $n=2$ via parameterization by spherical
chain complexes, but also show the link to the Gersten complex and
(speculatively) the equivariant polylogarithm, as well as compute the Hecke
action.

Finally, it is worth mentioning the recent work of Bergeron-Charollois-Garcia
[BCG3] which constructs the differential-form version of the cocycle in the
toric setting from the equivariant polylogarithm viewpoint, linking it via
explicit analytic formulas to their Eisenstein theta lift [BCG1]. The image of
our cocycle (3.18) under the regulator map

$d\log^{\wedge
n}:H^{n}(k(\mathbb{G}_{m}),\mathbb{Z}(n))\to\Omega^{n}_{k(\mathbb{G}_{m}^{n})}$

is

(1.1)
$(\gamma_{1},\ldots,\gamma_{n-1})\mapsto\begin{pmatrix}c_{1}&\ldots&c_{k}\end{pmatrix}_{*}\left(\left[\frac{1}{(\exp(2\pi
iz_{1})-1)\ldots(\exp(2\pi
iz_{n})-1)}-\frac{\phi(\underline{\gamma})}{z_{1}\ldots
z_{n}}\right]dz_{1}\wedge\ldots\wedge dz_{n}\right)$

(see the referenced section for notation). This coincides, after taking
suitable linear combinations (“stabilization”), with formula [BCG3, (8.8)].

###  1.2. Future work

The basic methods (parameterization by combinatorial/linear algebraic
complexes of symbols) of this article apply, though with some substantial
changes, to the setting of elliptic polylogarithms: for example, the action of
$\text{GL}_{n}/\mathbb{Q}$ on the $n$th power of the universal elliptic curve,
or $\text{GL}_{n}/K$ on the $n$th power of an elliptic curve with CM by $K$.
This elliptic case will be the subject of a sequel to this article. Working
analytically over $\mathbb{C}$, certain specializations of the former setting
yield (after pullback by torsion sections) the Siegel unit-valued modular
symbols constructed in the recent work of [BPPS]. The toric cocycle of the
present article is essentially the “degeneration to the cusp” of the elliptic
one. Also using symbol-complex methods, a topological version of the
construction in the present article is the subject of forthcoming joint work
with Marti Roset Julia, relating it to a generalization of the $p$-arithmetic
cocycles of [DPV1] and [DPV2].

There are many other directions to explore in this general setting: for
example, many natural generalizations or arithmetic applications of the
elliptic version of the construction suggest themselves; we omit their
discussion since they will be discussed in that article. To mention a
direction tied more to the toric setting, the relation between the present
article’s motivic cocycle and topologically-defined Eisenstein cocycles is
somewhat subtle and worth examining, reflecting in some sense the functional
equation of a Kronecker-Eisenstein series as well as a duality between
cotangent and Bernoulli-type sums in the sense of [Scz]. Further, the values
of these cocycles have rich connections to topological signature theory, which
we find very worthy of interest and hope to understand better.

##  2\. Preliminaries

###  2.1. Motivic cohomology

For a smooth equidimensional scheme $X$ over a discrete valuation ring (DVR)
$R$, we define the motivic cohomology

$H^{i}(X,\mathbb{Z}(n))$

as the Zariski hypercohomology

$\mathbb{H}^{i}(X,\underline{\mathbb{Z}(n)_{X}})$

of Bloch’s weight-$n$ _cubical_ complex of sheaves
$\underline{\mathbb{Z}(n)_{X}}$; this is the approach followed in [Tot] and
[GL], for example, over a field. It is more standard to use simplicial
language instead of cubical, but the two approaches are equivalent for formal
reasons, as proven in loc. cit. over a field, with the proof verbatim
translating to our setting. In simplicial language, this definition over a DVR
(even a Dedekind domain) was used by [Lev2] and [Geis].

######  Remark 2.1.

Due to its functoriality properties, this construction should perhaps more
properly be called Borel-Moore motivic homology; historically, this
construction was known as “higher Chow groups”. For smooth schemes over a
field, this construction agrees with the usual modern construction of motivic
cohomology defined via Voevodsky-style motivic complexes, thanks to the
results in [V] and [FS].

The Bloch complex of sheaves is defined as follows: let

$\tilde{z}^{n}(U,i):=Z^{n}(U\times\Box^{i})$

be the group of codimension-$n$ cycles on $U\times\Box^{i}$ meeting all faces
properly. Here, $\Box^{i}$ is the algebraic $i$-cube which we identify with

$(\mathbb{P}^{1}-\\{1\\})^{n}$

and the $j$th face map is given by the difference of the pullbacks to the
subvarieties cut out by $t_{j}=0$, respectively $t_{j}=\infty$; the
alternating sum of face maps gives, as usual, a differential from
$\tilde{z}^{n}(U,i)$ to $\tilde{z}^{n}(U,i-1)$.111In some conventions,
$\Box^{i}$ is identified with $mathbb{A}^{i}$, and the face maps are given by
$t_{j}=0,1$ instead of $0,\infty$. This differs from our convention by a
Möbius transformation; it is the same one used by [Tot], and we we find it to
be more convenient for later applications. It turns out that the resulting
complex splits into a direct sum

$\tilde{z}^{n}(U,i)=d^{n}(U,i)\oplus z^{n}(U,i)$

where the former summand consists of _degenerate_ cycles which can be pulled
back from one of the faces of $\Box^{i}$ given by $t_{j}=0$, and the latter
summand consists of the _reduced cycles_ which are in the kernel of of the
restriction to each face $t_{j}=0$. We define

$(\mathbb{Z}(n)_{X})^{i}(U):=z^{n}(U,2n-i).$

This complex is suitably functorial for flat pullbacks and proper
pushforwards.

By Zariski descent for the cohomology groups of this cycle complex over a DVR,
the natural map

$H^{i}(\mathbb{Z}(n)_{X})\to\mathbb{H}^{i}(X,\mathbb{Z}(n)_{X})$

is an isomorphism, where
$\mathbb{Z}(n)_{X}:=\Gamma_{X}(\underline{\mathbb{Z}(n)_{X}})$ is the global
sections on $X$ of the Bloch complex of sheaves; we will refer to this complex
as the _Bloch cycle complex_. This fact was first proven over a field [FS,
Corollary 12.2]; the case of DVRs is due to [Lev2]. (This is the reason we
work over DVRs rather than general Dedekind bases; over those, we cannot get
rid of the hypercohomology.)

It will also be useful for us to consider the _coniveau spectral sequence_ in
motivic cohomology given by the filtration on cycles by codimension [Geis,
§4], which is of the form

(2.1) $E^{p,q}_{1}=\bigoplus_{x\in
X^{(p)}}H^{q-p}(k(x),\mathbb{Z}(n-p))\Rightarrow H^{p+q}(X,\mathbb{Z}(n))$

where $X^{(p)}$ denotes the codimension-$p$ points.

####  2.1.1. The Gersten complex in Milnor $K$-theory

The _Gersten complex_ is a construction associated to any cohomology theories
with Zariski descent; one form of it is constructed in [Bl2]. The Gersten
complex for Milnor $K$-theory, the case of interest to us, is given by

(2.2) $Z_{Ger(K(n))}(X):=K_{n}(X)\to\bigoplus_{x\in
X^{(0)}}K_{n}(k(x))\to\bigoplus_{x\in
X^{(1)}}K_{n-1}(k(x))\to\ldots\to\bigoplus_{x\in X^{(d)}}K_{n-d}(k(x))\to 0$

for a scheme $X$, where the first map is the natural pullback and the
successive maps are the tame residue symbols. It has a map from the Bloch
complex:

######  Proposition 2.2.

There is a map of complexes

$\psi_{X}^{\bullet}:\mathbb{Z}(n)_{X}^{\bullet}\to
Z_{Ger(K^{n})}^{\bullet}(X)[-n]$

defined as follows: for the class of an irreducible closed subvariety $[Z]\in
z^{n}(U,2n-i)$, the projection of $Z$ to $U$ is at most codimension-$(i-n)$.
If it is strictly higher codimension, we set $\psi_{X}^{i}([Z])=0$. Otherwise,
$Z$ is dominant over a codimension-$(i-n)$ integral closed subscheme
$\tilde{Z}$ of $X$; write $p:Z\to\tilde{Z}$ for the projection map. We set

(2.3) $\psi_{X}^{i}(z):=\mathbf{N}_{p}([t_{1}\smile\ldots\smile t_{2n-i}])\in
K_{2n-i}(k(\tilde{Z}))$

where $\mathbf{N}$ is the norm222Also called transfer map or pushforward. in
Milnor $K$-theory (as defined in [BaTa, §5]) for the finite map $p$.333Recall
from the definition of the cubical Bloch complex that $t_{i}$ are the
coordinate functions on the algebraic cube. The map $\psi_{X}$ is functorial
for quasi-finite flat pullbacks and proper pushforwards.

###### Proof.

This map was first constructed simplicially and its properties proven in [La];
the cubical version was implicitly defined in [Tot]. ∎

######  Remark 2.3.

At the generic point, there is an inverse map $K_{k}(k(X))\to
H^{2k}(k(X),\mathbb{Z}(k))$ given by sending

$(k(Z),[f_{1}\otimes\ldots\otimes
f_{k}])\mapsto[\Gamma_{(f_{1},\ldots,f_{k})}]$

where $(f_{1},\ldots,f_{k}):Z\dashrightarrow\Box^{k}$ is a (rational) map from
the cycle $Z$ to the algebraic $k$-cube, and $\Gamma_{\bullet}$ is its graph.
Note, however, that these maps cannot be lifted to the level of cubical
cycles, and so do not furnish any kind of inverse to $\psi_{X}^{\bullet}$ at
the level of complexes.

####  2.1.2. Motivic cohomology of $\mathbb{G}_{m}^{n}$

We turn now to the motivic cohomology of powers of the multiplicative group

$\mathbb{G}_{m}=\text{Spec }R[z^{\pm}]$

over our base $\text{Spec }R$, which we give coordinates as

$\mathbb{G}_{m}^{n}=\text{Spec }R[z_{1}^{\pm},\ldots,z_{n}^{\pm}]$

This section is taken essentially entirely from [SV, §3.1,§4.1]; see there for
original attributions.

In particular, the following two (well-known) results are [SV, Proposition
3.1.1] and [SV, Corollary 3.1.2], respectively:

######  Proposition 2.4.

For any finite type quasi-projective scheme $Y/k$, there is a natural
isomorphism

$H^{i}(Y,\mathbb{Z}(j))\oplus H^{i-1}(Y,\mathbb{Z}(j-1))\cong
H^{i}(\mathbb{G}_{m}\times Y,\mathbb{Z}(j))$

for all $i,j\in\mathbb{Z}$ where inclusions of the two summands are
respectively by pullback by the projection map and left external product with

$-z\in H^{1}(\mathbb{G}_{m},\mathbb{Z}(1)).$

######  Corollary 2.5.

For any $n\geq 1$, and $i,j\in\mathbb{Z}$ we have a natural direct sum
decomposition

(2.4)
$H^{i}(\mathbb{G}_{m}^{n},\mathbb{Z}(j))\cong\bigoplus_{t=1}^{\min\\{n,j\\}}H^{i-t}(k,\mathbb{Z}(j-t))^{\binom{n}{t}}.$

by iteratively applying the isomorphism of the preceding proposition.

These were proven over fields, but once again the proofs hold verbatim over
DVRs.

Note in particular the corollaries:

######  Corollary 2.6.

The groups $H^{i}(\mathbb{G}_{m}^{n},\mathbb{Z}(j))$ vanish for $i>j$.

######  Corollary 2.7.

The natural map

$\psi_{n}:H^{n}(\mathbb{G}_{m}^{n},\mathbb{Z}(n))\to
K_{n}^{M}(k(\mathbb{G}_{m}^{n}))$

is injective.

###### Proof.

This follows from the previous corollary because there are never any incoming
arrows to the term $K_{n}^{M}(k(\mathbb{G}_{m}^{n}))$ in the coniveau spectral
sequence of [SV, Example 2.2.2]. ∎

We now wish to consider the action of the multiplicative monoid of positive
integers $\mathbb{N}^{\times}$ on $\mathbb{G}_{m}$, via the maps

$[a]:\mathbb{G}_{m}^{n}\xrightarrow{z_{i}\mapsto
z_{i}^{a}}\mathbb{G}_{m}^{n}.$

This induces, via finite pushforward, maps

$[a]_{*}:H^{i}(\mathbb{G}_{m}^{n},\mathbb{Z}(j))\to
H^{i}(\mathbb{G}_{m}^{n},\mathbb{Z}(j))$

on the associated motivic cohomology groups. We define the _trace-fixed part_

$H^{i}(\mathbb{G}_{m}^{n},\mathbb{Z}(j))^{(0)}$

to be the submodule fixed by $[a]_{*}$ for all $a\in\mathbb{N}^{\times}$. More
generally, on any module with an action of $\mathbb{N}^{\times}$, we write a
superscript $(0)$ for the trace-fixed part defined analogously.

###  2.2. Simplicial complexes from linear algebra

With the motivic background out of the way, we turn now to the other side of
the picture, constructing combinatorial “symbol complexes” we will use to
parameterize motivic data. These complexes, indexed by a dimension $n$
corresponding to the power $\mathbb{G}_{m}^{n}$, will have simple homology and
carry natural $\text{GL}_{n}(\mathbb{Q})$-actions.

Corresponding to the two kinds of motivic complexes we considered (i.e. the
Bloch cycle and Gersten complexes), we will consider two flavors of symbol
complexes, respectively:

  1. (1)

_Matroid complexes_ parameterizing linear algebraic data _with independence
conditions_.

  2. (2)

_Spherical chain complexes_ coming from the simplicial chain complexes of
profinite triangulations at rational vertices on spheres.

The latter type of complexes carry more interesting structure, and are
ultimately more suitable for explicitization. The reason for considering the
former type of complexes is that they are necessary if we wish to map to Bloch
complexes rather than Gersten complexes, which ends up being necessary for
various technical reasons which we will mention as they arise.

####  2.2.1. Matroid complexes

We begin by defining the matroid complexes.

######  Definition 2.8.

A (finite rank) _matroid_ is a set $M$ and a collection
$S\subset\mathcal{P}(M)$ of finite subsets of $M$ (i.e. elements of the power
set $\mathcal{P}(M)$), called independent sets, satisfying the properties:

  * •

There is at least one independent set.

  * •

If $s\in S$ is independent, every subset of $s$ is as well.

  * •

If $s$ and $t$ are in $S$ and $|t|>|s|$, then there exists an element $m\in t$
such that $t\cup\\{m\\}\in S$.

One can think of matroids as encoding the formal properties of independent
sets of vectors in some vector space, though in fact not every matroid is
realizable as such. Motivated by this description, the _rank_ of a matroid $M$
is then defined to be the maximal size of any independent set.

The complexes associated to matroids will be arise from a topological
construction derived from them:

######  Definition 2.9.

The _matroid complex_ $\mathbf{C}(M)$ of a matroid $M$ is a simplicial complex
whose vertices are parameterized by elements of $M$, where a set of vertices
$\\{v_{0},\ldots,v_{k}\\}\subset S$ form a $k$-simplex if and only if the
corresponding set is independent.

These matroid complexes are very simple from a homotopical point of view, as
proven in [Bj, §7]:

######  Proposition 2.10.

The matroid complex associated to a matroid of rank $d$ is homotopy equivalent
to a wedge of $d$-spheres.444Technically, the results of [Bj] apply only to
_finite_ matroids, but this is a trivial difference, since one can simply
write an infinite matroid of finite rank as a direct limit of finite matroids.

The simplicial homology of a simplicial complex $\mathbf{C}$ can be computed
as usual via the complex

(2.5)
$C_{\bullet}(\mathbf{C})=\ldots\to\mathbb{Z}[\mathbf{C}_{i-1}]\xrightarrow{\partial}\mathbb{Z}[\mathbf{C}_{i}]\xrightarrow{\partial}\mathbb{Z}[\mathbf{C}_{i-1}]\to\ldots$

but this is not the complex we are interested in: we want to keep track of the
_order_ of the vertices in a simplex. Building on this, there is a functor

$F:\textbf{SCmplx}\to\textbf{SSet}$

sending a simplicial complex to a simplicial set. The $k$-simplices
$(F\mathbf{C})_{k}$ are defined as the set of _ordered_ $(k+1)$-tuples
$(v_{0},\ldots,v_{k})$ of $0$-simplices of $\mathbf{C}$, such that any
repeated vertices are consecutive, and the set $\\{v_{0},\ldots,v_{k}\\}$
forms a $k$-simplex in $\mathbf{C}$. The $i$th face map forgets the $i$th
vertex, and the $j$th degeneracy map doubles the $j$th vertex; thus, the non-
degenerate simplices are precisely those for which all the $v_{i}$ are
distinct.

The homology of the geometric realization $|F\mathbf{C}|$, as with any
simplicial set, can be computed by the alternating face complex

(2.6)
$C_{\bullet}(F\mathbf{C})=\ldots\to\mathbb{Z}[(F\mathbf{C})_{i+1}^{nd}]\xrightarrow{\partial}\mathbb{Z}[(F\mathbf{C})_{i}^{nd}]\xrightarrow{\partial}\mathbb{Z}[(F\mathbf{C})_{i-1}^{nd}]\to\ldots$

where the $nd$ superscript indicates non-degenerate simplices.

We have the following acyclicity result:

######  Proposition 2.11.

The homology of $|F\mathbf{C}|$ vanishes in degree $0<i<d$ for any matroid
complex $\mathbf{C}$ of a matroid of rank $d$.

###### Proof.

By [WR, Proposition 2.14],555Thanks to Andy Putnam on MathOverflow for telling
me about this result. it suffices to prove that the link of of a $k$-simplex
$\sigma$ of $\mathbf{C}$ is $(d-k-1)$-connected for each $k$. Suppose $\sigma$
is associated to the independent set $\\{v_{0},\ldots,v_{k}\\}$ of vertices;
then the $j$-simplices in the link of $\sigma$ can be characterized as the
sets $\\{w_{0},\ldots,w_{j}\\}$ such that

$\\{v_{0},\ldots,v_{k},w_{0},\ldots,w_{j}\\}$

is independent. This condition on the $w_{i}$ gives the structure of a matroid
of rank $d-k$ on the $0$-simplices of $\mathbf{C}$ in the link of $\sigma$,
such that $\text{Link}(\mathbf{C},\sigma)$ is the associated matroid complex.
Thus, $\text{Link}(\mathbf{C},\sigma)$ is homotopy equivalent to a wedge of
$(d-k)$-spheres by Proposition 2.10, whence the result. ∎

Write $S^{n-1}(\mathbb{Q})$ for the set of unbounded rays from the origin in
an $n$-dimensional vector space $\mathbb{Q}^{n}$. Said differently, this is
the set of rational lines with orientation, thus yields a double cover of
$\mathbb{P}^{n-1}(\mathbb{Q})$ (hence the notation).

We now take as our matroid

$M(n)\subset\mathcal{P}(S^{n-1}(\mathbb{Q}))$

the sets of rays whose corresponding lines are linearly independent. By the
preceding results, the geometric realization of the simplicial set
$|F\mathbf{C}(M(n))|$ has trivial homology except possibly in degrees $n-1$
and $0$. In degree zero, this homology is just $\mathbb{Z}$, as
$\mathbf{C}(M(n))$ is certainly connected.

We thus conclude that if we define the modified homology complex

$\text{Mat}(n)_{\bullet}:=\left[C_{\bullet}(F\mathbf{C}(M(n)))/H_{n-1}(|F\mathbf{C}(M(n))|)\to\mathbb{Z}\right],$

it is fully exact; here, we take the grading convention that the augmentation
$\mathbb{Z}$ is in degree zero (so that the top homology being killed off is
in degree $n$). This is almost the complex which will map to the Bloch
complex; in fact, in the next section, we will use the spherical chain complex
to define a corank-$1$ submodule $\mathcal{R}\subset
H_{n-1}(|F\mathbf{C}(M(n))|)$ giving a slight modification of
$\text{Mat}(n)_{\bullet}$.

Via the left action of $\text{GL}_{n}(\mathbb{Q})$ on $\mathbb{Q}^{n}$ by the
defining representation, all above complexes carry natural left actions of
$\text{GL}_{n}(\mathbb{Q})$ as well as any arithmetic subgroups.

####  2.2.2. Spherical chain complexes

We shall construct a $\text{GL}_{n}(\mathbb{Q})$-chain complex
$\widetilde{\text{Chains}}(n)$, and show that it computes the reduced homology
of the sphere $S^{n-1}$ and is thus almost exact. We will also define an
equivariant map of complexes

$\widetilde{\Psi}:\widetilde{\text{Mat}}(n)\to\widetilde{\text{Chains}}(n)$

from a slight modification of the matroid complex of the previous section,
which induces an isomorphism on homology. In a later section,
$\widetilde{\text{Chains}}(n)$ will be endowed with a realization map to the
Gersten complex, and $\widetilde{\Psi}$ will correspond to the map $\psi$
between the Bloch and Gersten complexes.

The construction of this section directly generalizes [SV, §5], who consider
the case $n=2$, though we use slightly different conventions for the group
action: we are taking _left_ actions on spaces with a resulting pushforward
left action on cohomology groups, while they consider right actions on spaces.

To construct $\widetilde{\text{Chains}}(n)$, consider the poset $\mathcal{P}$
of finite geodesic triangulations $T$ of $S^{n-1}$ with vertices contained in

$S^{n-1}(\mathbb{Q})\subset S^{n-1},$

ordered by _refinement_ : that is to say, we say $T_{1}\geq T$ if every
simplex of $T$ is a union of simplices of $T_{1}$.

Here, by _geodesic triangulation_ , we mean a triangulation where every face
of every simplex is a regions of a $k$-dimensional great circle; thus, every
$k$-simplex is the intersection of a $S^{n-1}$ with dimension-$(k+1)$
simplicial polyhedral cones in $\mathbb{R}^{n}$ emanating from the origin
(given by connecting all points of the simplex with the origin). The
nomenclature is explained by the fact that these simplices are geodesic
simplices taken with respect to the Euclidean metric under the natural
inclusion

$S^{n-1}\subset\mathbb{R}^{n}.$

Notice that in particular, the great circle containing any given simplex can
be recovered as the intersection of $S^{n-1}$ with the subspace of
$\mathbb{R}^{n}$ spanned by the rays emanating to the vertices of the simplex.
As such, each of these corresponding subspaces are themselves rationally
defined, since the vertices are in $S^{n-1}(\mathbb{Q})$.

It is all well and good to define such objects, but we need to establish that
enough of them exist for our purposes. This is accomplished by the following
two propositions, which are both certainly well-known to the point of being
folklore, but which we state for the sake of completeness.

######  Proposition 2.12.

Given any geodesic $k$-simplex $\Delta$ with endpoints in
$S^{n-1}(\mathbb{Q})$ with $k\leq n-1$, there exists a triangulation
$T\in\mathcal{P}$ such that $\Delta$ is one of the triangles in $T$.

###### Proof.

For each face of $\Delta$, take the corresponding great circle on $S^{n-1}$;
these divide the sphere into polyhedral regions. In each region besides
$\Delta$ itself, choose a rational point, Pick a rational point in interior of
each polyhedral cell and subdivide it into simplices by joining the point to
each $(n-2)$-face of the cell with a $(n-1)$-simplex; we obtain a
triangulation with all vertices in $S^{n-1}(\mathbb{Q})$ such that $\Delta$ is
one of the triangles. ∎

######  Proposition 2.13.

Given any two triangulations $T_{1},T_{2}\in\mathcal{P}$, there exists a
common upper bound for them, i.e., a triangulation $T\in\mathcal{P}$ refining
both $T_{1}$ and $T_{2}$.

###### Proof.

Superimposing the triangulations $T_{1}$ and $T_{2}$, we obtain a
decomposition of $S^{n-1}$ into convex polyhedral cells. Each vertex of this
superimposition is still in $S^{n-1}(\mathbb{Q})$, because any new vertices
are the intersection points of rationally defined subspaces in
$\mathbb{R}^{n}$ (corresponding to the great circles forming the intersecting
faces) and hence themselves correspond to rational rays. Pick a rational point
in interior of each polyhedral cell and subdivide it into simplices by joining
the point to each $(n-2)$-face of the cell with a $(n-1)$-simplex; we obtain a
new triangulation which still has all vertices in $S^{n-1}(\mathbb{Q})$. ∎

Write $C_{T}$ for the augmented cellular chain complex of the simplicial
complex $T$ with

$C_{T,i}=\mathbb{Z}\\{T_{i}\\}$

for all $i\geq 0$, freely generated by the $i$-simplices endowed with their
orientation coming from the globally oriented $S^{n-1}$. As is customary, we
will also consider the oppositely-oriented version $\overline{\Delta}$ of a
simplex $\Delta$ to live in the chain complex, representing simply its
additive inverse

$[\overline{\Delta}]=-[\Delta].$

The boundaries $C_{T,i+1}\xrightarrow{\partial}C_{T,i}$ are given by the
alternating face maps for $i\geq 0$. The “augmented” in the definition means
that we further set $C_{T,-1}=\mathbb{Z}$, with its incoming boundary map
given by the degree map on $0$-simplices.

For each $T$,

$H_{n-1}(C_{T,\bullet})\cong\mathbb{Z}$

is the only nontrivial homology group, since each simplicial complex is
homotopy equivalent to $S^{n-1}$.

Associated to a refinement $T^{\prime}$ of $T$, there is a pullback map of
chain complexes

$C_{T}\to C_{T^{\prime}}$

inducing an isomorphism on homology; these maps are functorial for
compositions in the poset of triangulations. We hence define

$\widetilde{\text{Chains}}(n)_{i}:=\varinjlim C_{T,i}[1]$

as the colimit over the poset. By the exactness of directed limits,
$\widetilde{\text{Chains}}(n)$ inherits the property of being acyclic outside
degree $n$ (shifted from degree $n-1$), where it has homology $\mathbb{Z}$.

Via the left action of $\text{GL}_{n}(\mathbb{Q})$ on $S^{n-1}(\mathbb{Q})$
viewed as parameterizing rays in $\mathbb{R}^{n}$, the complex
$\widetilde{\text{Chains}}(n)$ takes a left action of
$\text{GL}_{n}(\mathbb{Q})$ via pushforward of simplices. In particular, if
$\gamma$ fixes a simplex $\Delta$ but $\det\gamma=-1$, then

$\gamma\cdot[\Delta]=[\overline{\Delta}]=-[\Delta].$

Let us give generators and relations for $\widetilde{\text{Chains}}(n)$. Given
any tuple of independent rays $(m_{1},m_{2},\ldots,m_{k})$ in
$S^{n-1}(\mathbb{Q})$ lying in the same (open) hemisphere - we will call such
a tuple _acyclic_ \- they span a unique oriented geodesic simplex (with
orientation corresponding to the order), whose $1$-frame is formed from the
shorter segment of the great $1$-circle through each pair of points. Note that
there always is a shorter segment: since all points lie in the same strict
hemisphere, we can never have a pair of antipodes.

For any triangulation $T$ containing the simplex so defined as a triangle,
write

$[\Delta(m_{1},m_{2},\ldots,m_{k})_{T}]\in C_{T,k-1}$

for the class in homology; under any triangulation $T^{\prime}$ which refines
$T$ with the same property, the class $[\Delta(m_{1},m_{2},\ldots,m_{k})_{T}]$
pushes forward to $[\Delta(m_{1},m_{2},\ldots,m_{k})_{T^{\prime}}]$. Thus, we
can speak unambiguously of a class in the direct limit

$[\Delta(m_{1},m_{2},\ldots,m_{k})]\in\widetilde{\text{Chains}}(n)_{k}.$

These classes generate $\widetilde{\text{Chains}}(n)_{i}$ for each $i$,
because any rational simplex can be subdivided into rational acyclic
simplices.

We claim that the relations between the generators corresponding to these
simplices are generated by subdivision of an acyclic simplex into
subsimplices. Indeed, suppose we have an relation of $k$-simplex generators

(2.7) $[\Delta_{1}]+\ldots+[\Delta_{t}]=0.$

Superimposing all the great circles corresponding to the faces of the
$\Delta_{i}$ and subdividing the resulting polyhedral cells as in the proof of
2.13, we can find a set of geodesic simplices
$\Delta_{1}^{\prime},\ldots,\Delta_{s}^{\prime}$, disjoint outside of their
faces, such that each $\Delta_{i}$ is a union

$\Delta_{i}=\bigsqcup_{j\in S_{i}}\Delta_{j}^{\prime}.$

Associated to this decomposition of simplices we have pushforward relation

(2.8) $[\Delta_{i}]=\sum_{j\in S_{i}}\Delta_{j}^{\prime}.$

Substituting these into (2.7), the resulting relation is between disjoint
simplices, on which the coefficient of each simplex therefore must be zero.
Hence relations of the form (2.8), coming from subdivision of an acyclic
simplex, generate all relations as we claimed.

In fact, we can go even further, and generate all acyclic simplicial
subdivision relations in terms of some easily classified ones:

######  Proposition 2.14.

Write $\mathbf{T}_{k}$ for the set of all acyclic positively oriented geodesic
$k$-simplices with rational faces.666As before, we will also freely use
negatively oriented simplices as generators, with the convention they are
simply $-1$ times the positively oriented simplex. The module
$\widetilde{\text{Chains}}(n)_{i}$ for $i>0$ is given by

$\mathbb{Z}\\{\mathbf{T}_{i-1}\\}/\mathcal{S}_{i-1}$

where $\mathcal{S}_{k}$ is the module of acyclic $k$-simplicial subdivision
relations, and is generated by _stellar subdivisions_ of rank $r$ for $2\leq
r\leq k$ given as follows: fix an acyclic simplex
$\Delta\in\Delta(m_{1},\ldots,m_{k})$ and a rational point $m$ lying in the
great circle corresponding to the face with vertices $(m_{1},\ldots,m_{r})$,
but sharing some hemisphere of the great circle with all of them.
Corresponding to this data we have the stellar subdivision relation

(2.9)
$[\Delta(m_{1},\ldots,m_{k})]=\sum_{i=1}^{r}[\Delta(m_{1},\ldots,\hat{m_{i}},m,m_{i+1},\ldots,m_{k})].$

###### Proof.

From the discussion preceding the proposition, it remains only to prove that
the stellar subdvision relations (3.11) generate all acyclic subdivision
relations.

First, we say that a subdivision of a simplex $\Delta$ is a _sequentially
stellar subdivision_ if it can be obtained by a sequence of stellar
subdivisions of $\Delta$ (and the resulting subsimplices at each sequential
step). The relation corresponding to a sequentially stellar subdivision of an
acyclic simplex is certainly generated by those of the implicated stellar
subdivisions by repeated substitution.

We observe, therefore, that it suffices to show that any acyclic subdivision

$\Delta=\Delta_{1}\sqcup\ldots\sqcup\Delta_{t}$

can be refined to a sequentially stellar subdivision of $\Delta$, which is
also a sequentially stellar subdivision of each $\Delta_{i}$ when restricted
to it.

The modern way of phrasing this problem in combinatorial topology is that a
subdivision of a simplex gives a piecewise linear (PL) homeomorphism between
the simplex and its subdivision, and any two PL-homeomorphic simplicial
complexes are related by a sequence of “stellar moves”. This result is
classical within that field, and was first proven in [New]; for a more modern
treatment, see [Lick]. ∎

In analogy with the matroid complex, we also write

$\text{Chains}(n):=\widetilde{\text{Chains}}(n)/H_{n}(\widetilde{\text{Chains}}(n))$

for the exact quotient of $\widetilde{\text{Chains}}(n)$ given by killing the
top homology.

####  2.2.3. The map $\widetilde{\Psi}$

We now turn to constructing the morphism of
$\text{GL}_{n}(\mathbb{Q})$-complexes

$\widetilde{\Psi}:\widetilde{\text{Mat}}(n)_{\bullet}\to\widetilde{\text{Chains}}(n).$

Here,

$\widetilde{\text{Mat}}(n)_{\bullet}:=\left[C_{\bullet}(F\mathbf{C}(M(n)))/\mathcal{R}\to\mathbb{Z}\right]$

for a corank-$1$ submodule $\mathcal{R}\subset H_{n-1}(|F\mathbf{C}(M(n))|)$
to be specified.

This map is defined in degree $0$ by the identity map on $\mathbb{Z}$. In
degree $k>0$, it is defined by sending

(2.10) $(m_{1},m_{2},\ldots,m_{k})\mapsto[\Delta(m_{1},\ldots,m_{k})].$

Notice that for _any_ $k$ independent rays, they lie in one strict hemisphere:
this is true for the standard basis and any subset thereof, so this follows by
moving the hemisphere around by the $\text{GL}_{n}(\mathbb{Q}))$-action until
(a subset of) the standard basis maps onto the $k$ rays in question.

We then _define_

$\mathcal{R}\subset H_{n-1}(|F\mathbf{C}(M(n))|)$

as the kernel of the map

$H_{n-1}(|F\mathbf{C}(M(n))|)\to
H_{n}(\widetilde{\text{Chains}}(n))\cong\mathbb{Z}$

Then tautologically, the definition (2.10) makes sense; it further is visibly
$\text{GL}_{n}(\mathbb{Q})$-equivariant and commutes with the boundary maps of
both complexes. We have the following:

######  Lemma 2.15.

The map

(2.11) $C_{n-1}(F\mathbf{C}(M(n)))\to\widetilde{\text{Chains}}(n)_{n}$

induces an isomorphism

(2.12) $H_{n}(\widetilde{\text{Mat}}(n))\cong
H_{n-1}(F\mathbf{C}(M(n)))/\mathcal{R}\xrightarrow{\sim}H_{n}(\widetilde{\text{Chains}}(n))\cong\mathbb{Z}.$

###### Proof.

The only thing to check is that the map (2.11) is surjective on top homology;
indeed, if we take tuples $\underline{m}_{1},\ldots,\underline{m}_{k}$ with

$\Delta(\underline{m}_{1})\sqcup\ldots\sqcup\Delta(\underline{m}_{k})=S^{n-1}$

with each simplex having positive orientation, then

$\sum[\Delta(\underline{m}_{i})]$

is the fundamental class of the sphere. ∎

. The map $\widetilde{\Psi}$ fits into a commutative diagram with the induced
map on exact quotient complexes

(2.13)
${\widetilde{\text{Mat}}(n)}$${\widetilde{\text{Chains}}(n)}$${\text{Mat}(n)}$${\text{Chains}(n)}$$\scriptstyle{\widetilde{\Psi}}$$\scriptstyle{\Psi}$

where the vertical maps are the quotients.

###  2.3. Combinatorial cohomology classes for $\text{GL}_{n}$

We now leverage the constructions of our complexes into Eisenstein cohomology
classes for $\text{GL}_{n}(\mathbb{Q})$. We use the following standard
construction in group cohomology, whose proof we sketch:

######  Lemma 2.16.

If a group $G$ acts on an exact complex $C_{\bullet}$ supported in degrees
$[0,n]$, then we have a natural map on cohomology

$C_{0}^{G}\to H^{n-1}(G,C_{n})$

inhomogenous cocycle representatives of which can be constructed as follows:
associated to $e\in C_{0}^{G}$, pick a lift $\ell_{1}$ of $e$ to $C_{1}$, and
consider the $1$-cochain

$\gamma\mapsto(\gamma-1)\ell_{1}\in C^{1}(G,C_{1}).$

By exactness, this is the boundary of an element $\ell_{2}\in C^{1}(G,C_{2})$;
we take the chain coboundary $\partial\ell_{2}\in C^{2}(G,C_{2})$ which again
lifts to $\ell_{3}\in C^{2}(G,C_{3})$, etc. The lift $\ell_{n}\in
C^{n-1}(G,C_{n})$ is a cocycle representing the image of $e$ in
$H^{n-1}(G,C_{n})$.

###### Proof.

The map $C_{0}^{G}\to H^{n-1}(G,C_{n})$ is the composition of connecting
homomorphisms

(2.14) $\displaystyle C_{0}$ $\displaystyle\to
H^{1}(G,\ker\partial_{1})=H^{1}(G,\text{im }\partial_{2})$ (2.15)
$\displaystyle\to H^{2}(G,\ker\partial_{2})=H^{2}(G,\text{im }\partial_{3})$
(2.16) $\displaystyle\to\ldots$ (2.17) $\displaystyle\to
H^{n-1}(G,\ker\partial_{n})=H^{n-1}(G,C_{n}).$

for the $G$-short exact sequences

$\ker\partial_{i}\xrightarrow{\partial_{i+1}}C_{i}\xrightarrow{\partial_{i}}\text{im
}\partial_{i}.$

Unwinding the definition of each of these maps yields precisely the process of
iterated lifting described in the lemma. ∎

We apply the lemma to the action of $\text{GL}_{n}(\mathbb{Q})$ on the exact
complexes $\text{Mat}(n)$ and $\text{Chains}(n)$, using the element
$e\in\mathbb{Z}$ in degree zero of each respective complex. This affords us
cohomology classes

(2.18) $\displaystyle\Theta_{mat}(n)\in
H^{n-1}(\text{GL}_{n}(\mathbb{Q}),\text{Mat}(n)_{n})$ (2.19)
$\displaystyle\Theta_{com}(n)\in
H^{n-1}(\text{GL}_{n}(\mathbb{Q}),\text{Chains}(n)_{n})$

with $\Psi_{*}\Theta_{mat}(n)=\Theta_{com}(n)$.

####  2.3.1. Explicit cocycle representatives

We now apply Lemma 2.16 to obtain explicit cocycle representatives for
$\Theta_{com}(n)$. In principle, one can do the same for $\Theta_{mat}(n)$,
but the extra structure on $\text{Chains}(n)$ compared to $\text{Mat}(n)$
makes the process much nicer.

To prepare to define the cocycle representative, we need the following
extension of the definition of $k$-simplices from earlier: recall that for a
tuple $(m_{1},\ldots,m_{k})$ of _independent_ rays, there is a unique acyclic
oriented $(k-1)$-simplex $\Delta(m_{1},\ldots,m_{k})$ whose faces are given by
forgetting each $m_{i}$ in turn, i.e. the collection of $k$ simplices
$\Delta(m_{1},\ldots,\hat{m_{i}},\ldots,m_{k})$.

To write down our explicit cocycle, it will be useful to specify an extension
of this notation to _any_ tuple $(m_{1},\ldots,m_{k})$. However, we do not
need to construct this extension on the level of geodesic simplices, but
simply chains in $\text{Chains}(n)_{\bullet}$. Thus, we define a
$\Delta$-_extension_ $E$ to be a collection of classes777This is somewhat
abusive notation, since we are not necessarily saying that
$[\Delta_{E}(m_{1},\ldots,m_{k})]$ is actually the class of the geodesic
simplex when the tuple is not linearly independent.

$[\Delta_{E}(m_{1},\ldots,m_{k})]\in\text{Chains}(n)_{k}$

for arbitrary tuples of rays $(m_{1},\ldots,m_{k})$ satisfying the following
properties:

  1. (1)

If the tuple $(m_{1},\ldots,m_{k})$ is independent, then

$[\Delta_{E}(m_{1},\ldots,m_{k})]=[\Delta(m_{1},\ldots,m_{k})]$

as we have previously defined it.

  2. (2)

$\gamma[\Delta_{E}(m_{1},\ldots,m_{k})]=[\Delta_{E}(\gamma m_{1},\ldots,\gamma
m_{k})]$ for all $\gamma\in\text{GL}_{n}(\mathbb{Q})$.

  3. (3)

The image of $[\Delta_{E}(m_{1},\ldots,m_{k})]$ under the face map is

$\sum_{i}(-1)^{i-1}[\Delta_{E}(m_{1},\ldots,\hat{m_{i}},\ldots m_{k})].$

  4. (4)

If $(m_{1},\ldots m_{k})$ is a dependent acyclic tuple, then
$[\Delta_{E}(m_{1},\ldots,m_{k})]$ is zero.

We will call the data of such an extension a _$\Delta$ -extension_; all
$\Delta$-extensions by definition agree on linearly independent tuples.

There are many such extensions and not just a unique one; the following
proposition shows this.

######  Proposition 2.17.

$\Delta$-extensions exist.

###### Proof.

We show how to construct such an $E$ inductively on the corank of
$(m_{1},\ldots,m_{k})$: that is, on the difference $k-r$ between the number of
rays and the rank of their span. When $k=r$, the definition of
$[\Delta_{E}(m_{1},\ldots,m_{k})]$ is forced on us by (1); these definitions
certainly satisfy (2) and (3), and (4) is not applicable. This furnishes the
base case.

We will now first complete the inductive step so as to fulfill (2), (3), then
return to analyze how one fulfills (4) in different inductive steps, as
fulfilling (4) cannot be analyzed uniformly across all steps.

Indeed, to satisfy (2), all we need to do is construct
$\Delta_{E}(m_{1},\ldots,m_{k})$ for an arbitrary representative of each
$\text{GL}_{n}(\mathbb{Q})$-orbit of tuples and extend by group translation,
since the conditions (3) and (4) are certainly translation-invariant. Thus,
assume that we have constructed $\Delta_{E}$ for corank up to $i-1$, and we
wish to construct it for tuples of corank $i$. Then the point is that

(2.20) $\sum_{i}(-1)^{i-1}[\Delta_{E}(m_{1},\ldots,\hat{m_{i}},\ldots m_{k})]$

has boundary zero, so by exactness of $\text{Chains}(n)$ it is possible to
pick some chain lifting it under the boundary map.

We claim that $(m_{1},\ldots,m_{k})$ is acyclic, we can pick the extension so
that (2.20) is identically zero. Indeed, in the first inductive step $i=1$,
(2.20) is precisely a stellar subdivision relation, so it vanishes and we can
pick the lift to be zero. When $i>1$, then inductively all the terms of (2.20)
are acyclic dependent tuples, which by the inductive hypothesis are zero, and
hence we can pick zero as a lift under the boundary map. ∎

To any $\Delta$-extension, there corresponds an explicit cocycle:

######  Theorem 2.18.

For any $(n-1)$-tuple of matrices
$\underline{\gamma}=(\gamma_{1},\ldots,\gamma_{n-1})$, write
$c_{i}(\underline{\gamma})$ for

$\gamma_{i}\gamma_{i-1}\ldots\gamma_{1}e_{1}$

for $i\geq 0$, where $e_{1}=(1,0,\ldots,0)$; this is equivalently the first
column of the product matrix written above. Now fix any $\Delta$-extension
$E$; for such an extension, we define a $(n-1)$-cochain $\theta_{com}(n)^{E}$
by

(2.21)
$\underline{\gamma}\to[\Delta_{E}[\underline{\gamma}]]:=[\Delta_{E}(c_{n-1}(\underline{\gamma}),c_{n-2}(\underline{\gamma}),\ldots,c_{0}(\underline{\gamma}))].$

(In particular, this is the zero class if the simplex in question is
degenerate.) Then $\theta_{com}(n)^{E}(n)$ is a cocycle representative for
$\Theta_{com}(n)$.

###### Proof.

This follows immediately from the properties of a $\Delta$-extension and the
lifting process of 2.16: we lift $1\in\mathbb{Z}$ to $e_{1}$, whose group
coboundary is the $1$-cochain $\gamma\mapsto(\gamma-1)e_{1}$ which lifts to

$\gamma\mapsto[\Delta_{E}(c_{1}(\gamma),c_{0}(\gamma))].$

The group coboundary of this lifts to the $2$-cochain

$\gamma\mapsto[\Delta_{E}(c_{2}(\gamma_{1},\gamma_{2}),c_{1}(\gamma_{1},\gamma_{2}),c_{0}(\gamma_{1},\gamma_{2}))]$

and so on. ∎

We note also the following corollary of Proposition 2.17 with respect to
vanishings of the cocycles $\theta^{E}_{com}(n)$:

######  Corollary 2.19.

If $\underline{\gamma}$ is a tuple such that the corresponding tuple
$(c_{0},\ldots,c_{n-1})$ (defined in the previous theorem) is acyclic
dependent, $\theta_{com}(n)^{E}(n)(\underline{\gamma})=0$.

####  2.3.2. Lifting the combinatorial cocycles

The cohomology classes $\Theta_{mat}(n)$ and its image $\Theta_{com}(n)$ are
valued in $\text{Mat}(n)_{n}$ and $\text{Chains}(n)_{n}$ respectively, which
are the top-degree parts of complexes with the entire top homology group
quotiented out: in particular, in the latter case, we have quotiented by the
fundamental class of the sphere. As we will see in section 3, the realizations
of these modules will live in a _quotient_ of motivic cohomology groups of
$\mathbb{G}_{m}^{n}$.

We will see that in order to have realizations valued in the motivic
cohomology groups themselves, one needs to lift the classes/cocycles to be
valued in $\widetilde{\text{Mat}}(n)_{n}$ or
$\widetilde{\text{Chains}}(n)_{n}$, i.e lift over the copy of the rank-$1$
free module $\mathbb{Z}$ corresponding to the fundamental class of $S^{n-1}$.
This lifting process is essentially identical for the matroid/spherical chains
settings. In this subsection we will describe it primarily in the latter
setting, as we chose to construct our explicit cocycles in that context.
However, it will be clear in the following discussion that analogous results
hold for the matroid complex.

To begin, note that applying the lifting process of Lemma 2.16 to the
length-$(n+2)$ exact complex of $\text{GL}_{n}(\mathbb{Q})$-modules

(2.22) $\mathbb{Z}(\text{sgn})\to\widetilde{\text{Chains}}(n)_{\bullet}$

in the same way as we did with $\text{Chains}(n)$ yields a cocycle
$\varepsilon_{n}^{E}\in
C^{n}(\text{GL}_{n}(\mathbb{Q}),\mathbb{Z}(\text{sgn}))$.

######  Proposition 2.20.

The cocycle $\varepsilon_{n}^{E}$ represents the Euler class for the standard
representation of $\text{GL}_{n}(\mathbb{Q})$.

###### Proof.

The complex $\widetilde{\text{Chains}}(n)$ is the reduced homology complex of
a pro-simplicial model of the $(n-1)$-sphere; since this is a closed manifold,
we can view it also by Poincaré duality, as computing cohomology in the
complementary degree. Then the double complex
$C^{\bullet}(\text{GL}_{n}(\mathbb{Q}),\widetilde{\text{Chains}}(n)_{\bullet})$
computes the equivariant cohomology of $S^{n-1}$. By Lemma 2.16, the sum of
the lifts associated to $E$

$\ell_{1}+\ldots+\ell_{n+1}$

(using the notation of the just-mentioned lemma) for the lifts of
$1\in\mathbb{Z}$ in the augmentation for this complex then is a representative
for the Thom class of the associated sphere bundle, since the Thom class is
dual to the zero section. Pulling back to the base (i.e. a point with the
trivial $\text{GL}_{n}(\mathbb{Q})$-action; equivalently,
$B\text{GL}_{n}(\mathbb{Q})$) by the zero section kills all terms except
$\varepsilon_{n}^{E}=\ell_{n+1}$, which is therefore a representative of the
Euler class. ∎

Suppose now that $\Gamma$ is an $S$-arithmetic subgroup of
$\text{GL}_{n}(\mathbb{Q})$, where $S$ is any subset of primes which has
nonempty complement.

######  Corollary 2.21.

The cocycle $\varepsilon_{n}$, restricted to $\Gamma$, is a coboundary after
inverting the greatest common denominator $d_{n,S}$ of $m^{n}(m^{n}-1)$ as $m$
ranges over all integers divisible only by primes not in $S$.

###### Proof.

This is a classical result of Sullivan on the Euler class of vector bundles
with integral structure [Sul]. ∎

The short exact sequence

$0\to\mathbb{Z}(\text{sgn})\to\widetilde{\text{Chains}}(n)_{n}\to\text{Chains}(n)_{n}\to
0$

yields a long exact sequence in cohomology

$\ldots\to H^{n-1}(\Gamma,\mathbb{Z}(\text{sgn}))\to
H^{n-1}(\Gamma,\widetilde{\text{Chains}}(n)_{n})\to
H^{n-1}(\Gamma,\text{Chains}(n)_{n})\to
H^{n}(\Gamma,\mathbb{Z}(\text{sgn}))\to\ldots$

After inverting $d_{n,S}$, the image of $\Theta_{com}(n)$ (i.e. the Euler
class) in the rightmost term vanishes, and hence $\Theta_{com}(n)$ lifts non-
uniquely to be valued in $\widetilde{\text{Chains}}(n)_{n}$. The set of lifts
of the $\Theta_{com}(n)$ to $\widetilde{\text{Chains}}(n)_{n}$ is therefore a
torsor under the image of
$H^{n-1}(\Gamma,\mathbb{Z}(\text{sgn})[d_{n,S}^{-1}])$. We thus find that:

######  Corollary 2.22.

After inverting $d_{n,S}$ and restricting to $\Gamma$, representatives for
these lifts are given by

$\underline{\gamma}\mapsto[\Delta_{E}(\underline{\gamma})]-\phi(\underline{\gamma})$

where $\phi(\underline{\gamma})$ ranges over primitives of
$\varepsilon_{n}^{E}$.

Write $\theta^{E,\phi}_{com}(n)$ for the cocycle corresponding to the
$(n-1)$-cochain $\phi$ transgressing the Euler cocycle; the corresponding
class

$\Theta^{E,\phi}_{com}(n)=[\theta^{E,\phi}_{com}(n)]\in
H^{n-1}(\Gamma,\widetilde{\text{Chains}}(n)_{n})[d_{n,S}^{-1}].$

######  Remark 2.23.

In [SV, §5], the authors construct a canonical lift of the analogue to the
cocycle $\theta_{com}(2)$ for $\Gamma=\text{SL}_{2}(\mathbb{Z})$. (When $n=2$,
the choice of $E$ is essentially immaterial, so we omit it from the notation.)
In this case, $H^{1}(\Gamma)=H^{2}(\Gamma)=0$ after inverting $6$, so there is
no ambiguity of lift, and the authors consequently find an explicit
distinguished primitive of the Euler class $\varepsilon_{2}$ to give them a
canonical lift. This raises the question whether one could find a more
canonical choice of lift in general even when $H^{n-1}(\Gamma)$ does not
vanish, which we do not in general know the answer to. Similarly to the case
of $\Delta$-extensions, we believe that there does exist additional structure
which could be arithmetically interesting in these lifts (as compared to the
coarser information of the Shintani-type quotients described in the following
subsection), but have not fully understood how to do this for $n>2$.

The entire preceding discussion on lifts, minus the specific cocycle
representatives, applies equally to the classes $\Theta_{mat}(n)$; after
inverting $d_{n,S}$ and restricting to $\Gamma$, they lift to
$\widetilde{\text{Mat}}(n)_{n}$-classes with an ambiguity which is a quotient
of $H^{n-1}(\Gamma,\mathbb{Z}(\text{sgn})[d_{n,S}^{-1}])$. We thus obtain

$\Theta^{\phi}_{mat}(n)\in
H^{n-1}(\Gamma,\widetilde{\text{Mat}}(n)_{n})[d_{n,S}^{-1}].$

######  Remark 2.24.

If $n$ is _odd_ , by inverting $2$ in the coefficients we can obtain fully
canonical lifts of $\Theta_{com}(n)$ and $\Theta_{mat}(n)$ for the whole group
$\text{GL}_{n}(\mathbb{Q})$ by demanding they be fixed also under $[-1]_{*}$,
as we obtain a projector

$\widetilde{\text{Chains}}(n)\to\widetilde{\text{Chains}}(n)^{[-1]_{*}=1},\Delta\mapsto\frac{1}{2}(\Delta+[-1]_{*}\Delta)$

from which it follows the latter complex is exact since this projector kills
the top homology (and similarly for the matroid complexes). The resulting
explicit cocycle lift of $\theta_{com}(n)^{E}$ sends

$(\gamma_{1},\ldots,\gamma_{n-1})\mapsto\frac{1}{2}([\Delta_{E}(\underline{\gamma})]+[-1]_{*}[\Delta_{E}(\underline{\gamma})]).$

####  2.3.3. The Steinberg quotient

We end this section by considering the following three questions:

  1. (1)

How can we modify our construction of $\theta_{com}(n)^{E}$ into a parabolic
cocycle?

  2. (2)

What is the precise relation between $\theta_{com}(n)^{E}$ and “Shintani
cocycles,” as defined, e.g. in [LP]?888We here mention the work of Lim and
Park since it is in a motivic setting close to ours, but of course the notion
of such cocycles is older, going back to the work of the namesake Shintani
[Shin].

  3. (3)

Can we obtain a more canonical version of $\theta_{com}(n)^{E}$ not relying on
the arbitrary choice of extension $E$?

All three questions are answered by the same modification of our construction,
which simultaneously elides the need to lift over the Euler class. We briefly
describe it now: let $\text{OS}(n)$ be the classical _Orlik-Solomon complex_
[OS] defined as follows: it is the graded-commutative algebra generated in
degree $1$ by symbols $[\ell]$ for $\ell\in\mathbb{P}^{n-1}(\mathbb{Q})$, with
relations generated multiplicatively by the dependence relations

$\partial([\ell_{1}]\wedge\ldots\wedge[\ell_{k}])=0$

for any lines $\ell_{1},\ldots,\ell_{k}$ spanning a space of rank strictly
less than $k$. Here, the differential-graded structure is defined via
$\partial[\ell]=1$ and extended by the graded Leibniz rule. The relations
generated as above are closed under the differential, and the resulting
differential-graded algebra is exact. Then we have a
$\text{GL}_{n}(\mathbb{Q})$-equivariant map of complexes

(2.23) $\text{Chains}(n)_{i}\twoheadrightarrow\text{OS}(n)_{i}$

sending

$\Delta(r_{1},\ldots,r_{k})\mapsto[\mathbb{Q}r_{1}]\wedge\ldots\wedge[\mathbb{Q}r_{k}].$

This map (2.23) is well-defined, since the Orlik-Solomon complex obeys
alternation in the vertices, and stellar subdivision relations simply become
dependence relations in the image.

In top degree, the map (2.23) corresponds to the
$\text{GL}_{n}(\mathbb{Q})$-equivariant quotient of the top spherical chains

(2.24)
$R_{St}:\text{Chains}(n)_{n}\twoheadrightarrow\text{St}(n):=\text{OS}(n)_{n}$

where $\text{St}_{n}$ is the _Steinberg module_. From the definition of the
Orlik-Solomon algebra, we see that it can be described as generated by symbols
$[\ell_{1}]\wedge\ldots\wedge[\ell_{n}]$ where
$\ell_{1},\ldots,\ell_{n}\in\mathbb{P}^{n-1}(\mathbb{Q})$, quotiented by the
relations

  1. (1)

$[\ell_{1}]\wedge\ldots\wedge[\ell_{n}]=0$ if the lines do not span
$\mathbb{Q}^{n}$, and

  2. (2)

For any $\ell_{0},\ldots,\ell_{n}$, the dependence relation

$\sum_{i=0}^{n}(-1)^{i}[\ell_{0}]\wedge\ldots\wedge[\hat{\ell_{i}}]\wedge\ldots\wedge[\ell_{n}]=0.$

From this description, one sees that the pushforward of $\theta_{com}(n)^{E}$
along (2.24) is therefore independent of the choice of $E$, since all
dependent tuples simply are sent to zero (as all independent tuples bound some
acyclic simplex). Hence we get a cocycle $(R_{St})_{*}\theta_{com}(n)^{E}$
representing a class

$\theta_{St}(n)\in H^{n-1}(\text{GL}_{n}(\mathbb{Q}),\text{St}(n))$

which one can see from relation (1) is parabolic.999After restricting to
$\text{SL}_{n}(\mathbb{Z})$, this cocycle is in fact the universal parabolic
cocycle coming from Bieri-Eckman duality, as the Steinberg module is the
dualizing module for $\text{SL}_{n}(\mathbb{Z})$. This answers questions (1)
and (3) above.

We need to consider realizations to fully answer question (2), i.e. construct
an equivariant map from $\text{St}(n)$ to a motivic group of interest. In the
next section, in particular subsection 3.2.1, we will discuss what we
sacrifice in taking this Steinberg quotient: one must quotient out by a
certain submodule of the target Milnor $K$-group, losing certain torsion
information for the pullbacks. This coarser realization corresponds to the
Shintani-style cocycle of [LP].

We conclude this section with the following description of the kernel of
(2.24), which will be useful when considering realizations:

######  Lemma 2.25.

The kernel of the map (2.24) is generated by “wedge” classes of the form

$\Delta(r_{1},\ldots,r_{n})-\Delta(-r_{1},r_{2},\ldots,r_{n})$

for independent tuples $(r_{1},\ldots,r_{n})$.

###### Proof.

The top-dimensional spherical chains are generated by acyclic simplices
together with stellar subdivision relations, while the Steinberg module is
generated by independent tuples modulo dependence relations. If one imposes
all the identification of spherical simplices

$\Delta(r_{1},\ldots,r_{n})\sim\Delta(\pm r_{1},\ldots,\pm r_{n}),$

for any combination of signs, then the relations resulting from the stellar
subdivision relations are precisely the dependence relations. These
identifications can be deduced, by (anti)symmetry, from the identifications

$\Delta(r_{1},\ldots,r_{n})\sim\Delta(-r_{1},r_{2},\ldots,r_{n})$

since we can then change one sign at a time and bootstrap to the general case.
∎

##  3\. Constructing cocycles in motivic cohomology

###  3.1. Symbols for the Bloch complex

In this section, we construct the realization maps

$\varphi_{\bullet}:\widetilde{\text{Mat}}(n)_{\bullet}\to\tau_{\geq
n}\mathbb{Z}(n)_{\mathbb{G}_{m}^{n}}^{2n-\bullet}$

where here $\tau_{\geq n}\mathbb{Z}(n)_{\mathbb{G}_{m}^{n}}^{2n-\bullet}$
means that we take the Bloch complex starting at degree $n$ up to degree $2n$
with the initial term $z^{n}(\mathbb{G}_{m}^{n}\times\Box^{n})$ replaced by

$z^{n}(\mathbb{G}_{m}^{n}\times\Box^{n})/\partial
z^{n}(\mathbb{G}_{m}^{n}\times\Box^{n+1})$

so that the complex computes the motivic cohomology of $\mathbb{G}_{m}^{n}$ in
degrees $[n,2n]$.

######  Theorem 3.1.

The map of complexes

(3.1)
$\varphi_{k}:(m_{1},\ldots,m_{k})\mapsto\begin{pmatrix}m_{1}&\ldots&m_{i}\end{pmatrix}_{*}\Gamma(1-z_{1},\ldots,1-z_{k})$

for $k>0$, and sending $1\in\mathbb{Z}$ to the class of the identity in
$z^{n}(\mathbb{G}_{m}^{n})$ for $i=0$, gives a well-defined map equivariant
for the $\text{GL}_{n}(\mathbb{Z})$ left pushforward action on the Bloch
complex (and the natural left action on rays). On cohomology, it sends a
generator of the rank-$1$ $\mathbb{Z}$-module
$H^{n}(\widetilde{\text{Mat}}(n))\cong\mathbb{Z}$ to the class of

$\\{-z_{1},\ldots,-z_{n}\\}\in H^{n}(\mathbb{G}_{m}^{n},\mathbb{Z}(n)).$

Here, the map

$\begin{pmatrix}m_{1}&\ldots&m_{k}\end{pmatrix}:\mathbb{G}_{m}^{i}\to\mathbb{G}_{m}^{n}$

is the map defined by the matrix with columns an integral generator of the
respective rays $m_{1},\ldots,m_{k}$, and we are taking its pushforward of the
graph of the function

$(1-z_{1},\ldots,1-z_{i}):\mathbb{G}_{m}^{i}\mapsto\Box^{i}.$

We will prove this theorem in stages throughout the course of this section.

First, note that the graphs $\Gamma(1-z_{1},\ldots,1-z_{n})$ all are visibly
closed cycles intersecting the cubical faces properly; their pushforwards
under finite maps therefore must be as well, so our definition makes sense. We
further note that the maps $\varphi_{i}$ are
$\text{GL}_{n}(\mathbb{Z})$-equivariant by construction, since the action of
$\text{GL}_{n}(\mathbb{Z})$ preserves the lattice
$\mathbb{Z}^{n}\subset\mathbb{Q}^{n}$ and hence sends integral generators of
rays to integral generators of rays. Furthermore, we claim they commute with
the differentials: indeed, this is certainly true for the differential of

$(m_{1},\ldots,m_{k})=(e_{1},\ldots,e_{k})$

where $e_{i}=(0,\ldots,1,\ldots,0)$ is the $i$th standard basis vector, as in
this case $\varphi_{i}$ simply sends

$(e_{1},\ldots,e_{k})\mapsto\Gamma(1-z_{1},\ldots,1-z_{k})$

viewed as a graph cycle in $z^{n}(\mathbb{G}^{n}_{m}\times\Box^{i})$, while
sending its boundary

$\sum_{i}(-1)^{i-1}(e_{1},\ldots,\hat{e_{i}},\ldots,e_{k})$

to

$\sum_{i}(-1)^{i-1}\Gamma(e_{1},\ldots,\hat{e_{i}},\ldots,e_{k})$

which is precisely the alternating sum of face maps of
$\Gamma(1-z_{1},\ldots,1-z_{k})$.

For general $(m_{1},\ldots,m_{k})$, then, we can use the fact that
intersection of algebraic cycles (and the face maps in particular) commutes
with finite pushforward to see that

(3.2) $\displaystyle\partial\varphi_{k}(m_{1},\ldots,m_{k})$
$\displaystyle=\partial\begin{pmatrix}m_{1}&\ldots&m_{k}\end{pmatrix}_{*}\Gamma(1-z_{1},\ldots,1-z_{k})$
(3.3)
$\displaystyle=\begin{pmatrix}m_{1}&\ldots&m_{k}\end{pmatrix}_{*}\partial\Gamma(1-z_{1},\ldots,1-z_{k})$
(3.4)
$\displaystyle=\begin{pmatrix}m_{1}&\ldots&m_{k}\end{pmatrix}_{*}\sum_{i}(-1)^{i-1}\Gamma(1-z_{1},\ldots,\widehat{1-z_{i}},\ldots,1-z_{k})|_{z_{i}=1}$
(3.5)
$\displaystyle=\sum_{i}(-1)^{i-1}\begin{pmatrix}m_{1}&\ldots&\widehat{m_{i}}&\ldots&m_{k}\end{pmatrix}_{*}\Gamma(1-z_{1},\ldots,1-z_{k})$
(3.6) $\displaystyle=\varphi_{k}[\partial(m_{1},\ldots,m_{k})]$

where the graph which appears up to the third line is viewed as having source
$\mathbb{G}_{m}^{k}$, and the graph on the fourth line as having source
$\mathbb{G}_{m}^{k-1}$.

This computation shows that (3.1) yields a well-defined map of complexes

$C_{i}(F\mathbf{C}(M(n)))\to\tau_{\geq
n}\mathbb{Z}(n)_{\mathbb{G}_{m}^{n}}^{2n-i-1}$

but it remains to check that this definition factors through the module of
relations $\mathcal{R}$: this is the hard part.

To do this, we will show that the map

(3.7) $H_{n-1}(F\mathbf{C}(M(n)))\to
H^{n}(\mathbb{G}_{m},\mathbb{Z}(n))=\ker\partial\subset
z^{n}(\mathbb{G}_{m}^{n}\times\Box^{n})/\partial
z^{n}(\mathbb{G}_{m}^{n}\times\Box^{n+1})$

factors through $\mathcal{R}$. First, we need an important lemma:

######  Lemma 3.2.

The map

(3.8) $(\varphi_{n})_{*}:H_{n-1}(F\mathbf{C}(M(n)))\to
H^{n}(\mathbb{G}_{m}^{n},\mathbb{Z}(n))$

actually takes values in the trace-fixed part

$H^{n}(\mathbb{G}_{m}^{n},\mathbb{Z}(n))^{(0)}$

and on this map with this target the $\text{GL}_{n}(\mathbb{Z})$-equivariance
of the map naturally extends to $\text{GL}_{n}(\mathbb{Q})$-equivariance,
factoring through through the positive part of the central torus
$(\mathbb{Q}_{+})^{\times}$.

###### Proof.

We first specify the natural $\text{GL}_{n}(\mathbb{Q})$-actions on source and
target: the source group, being generated by tuples of rays, already has an
obvious action by left pushforward factoring through
$(\mathbb{Q}_{+})^{\times}$. Meanwhile,
$H^{n}(\mathbb{G}_{m},\mathbb{Z}(n))^{(0)}$ has a left pushforward action of
$M_{2}(\mathbb{Z})$, which together with the (trivial, by definition) action
of $(\mathbb{Q}_{+})^{\times}$ assembles to an action of
$(\mathbb{Q}_{+})^{\times}$.

The key thing to note is that $(\psi_{n})_{*}$ is an isomorphism by Corollary
2.7. Thus, if we have a sum of tuples

$\sum_{\underline{m}\in S}(m_{1},\ldots,m_{n})\in H_{n-1}(F\mathbf{C}(M(n)))$

for some finite set $S$, then its image in
$H^{n}(\mathbb{G}_{m}^{n},\mathbb{Z}(n))$ can be described inside
$K_{n}^{M}(k(\mathbb{G}_{m}))$ as

$\sum_{\underline{m}\in
S}\begin{pmatrix}m_{1}&\ldots&m_{n}\end{pmatrix}_{*}\psi_{n}[\Gamma(1-z_{1},\ldots,1-z_{n})]=\sum_{\underline{m}\in
S}\begin{pmatrix}m_{1}&\ldots&m_{n}\end{pmatrix}_{*}\\{1-z_{1},\ldots,1-z_{n}\\}.$

Now we claim that it is no longer important to choose the column of the matrix
corresponding to each $m_{i}$ to be an integral generator: multiplying each
column by a scalar in $\mathbb{N}^{\times}$ does not affect the result.
Indeed, this is the same as precomposing by a diagonal integer matrix, so
follows from the fact that

$\begin{pmatrix}d_{1}&&&\\\ &d_{2}&&\\\ &&\ldots&\\\
&&&d_{n}\end{pmatrix}_{*}\\{1-z_{1},\ldots,1-z_{n}\\}=\\{1-z_{1},\ldots,1-z_{n}\\}$

as this can be broken into the $n$ separate computations

$(d_{i})_{*}(1-z_{i})=\prod_{j=1}^{d_{i}}(1-\zeta_{d_{i}}^{j}z_{i}^{1/d_{i}})=1-z_{i}.$

Since now each matrix we are pushing forward by has columns which can be any
vector in the ray, it is no longer important that the group action preserve
these generators of rays; thus, it follows formally that $(\varphi_{n})_{*}$
is equivariant for the $M_{n}(\mathbb{Z})$ action on both sides. Since both
source and target are also fixed by the trace action of $\mathbb{N}^{\times}$,
we obtain the $\text{GL}_{n}(\mathbb{Q})$-equivariance (factoring through
$\mathbb{Q}_{+}^{\times}$) we desire. ∎

Write $\mathbb{Z}^{\prime}=\mathbb{Z}[T^{-1}]$, where $T$ is the exponent of
the torsion in $H^{n}(\mathbb{G}_{m}^{n},\mathbb{Z}(n))^{(0)}$, which by
Proposition LABEL:prop:orientobstruct is either the characteristic of $k$ (if
$n$ is sufficiently large) or $1$. Then we see that the image of
$H_{n-1}(F\mathbf{C}(M(n)),\mathbb{Z}^{\prime})$ consists of multiples of the
class

$(-z_{1})\smile\ldots\smile(-z_{n}).$

The well-definedness, and thus Theorem 3.1, follows now from the following
proposition.

######  Proposition 3.3.

The map

$\varphi_{*}:H_{n-1}(F\mathbf{C}(M(n)),\mathbb{Z})\to
H^{n}(\mathbb{G}_{m},\mathbb{Z}(n))^{(0)}$

has kernel $\mathcal{R}$, and is surjective after extension of scalars to
$\mathbb{Z}^{\prime}$.

We will delay the full proof of this proposition to a corollary of Proposition
3.6. We introduce some concepts and notation in preparation: if we write

$\text{sgn}:\text{GL}_{n}(\mathbb{Q})\xrightarrow{\det}\mathbb{Q}^{\times}\xrightarrow{|x|/x}\\{\pm
1\\}$

for the sign character. Then as a $\text{GL}_{n}(\mathbb{Q})$-representation,

$H^{n}(\mathbb{G}_{m},\mathbb{Z}^{\prime}(n))^{(0)}=\mathbb{Z}^{\prime}(\text{sgn}).$

Meanwhile, we recall that $\mathcal{R}$ is the kernel of

$\widetilde{\Psi}_{*}:H_{n-1}(F\mathbf{C}(M(n)))\to
H_{n-1}(S^{n-1})=\mathbb{Z}(\text{sgn})$

so Proposition 3.3 says in particular that these two sgn-isotypic quotients
have the same kernel. In fact, we will exhibit in Proposition 3.6 that we have
a diagram of $\text{GL}_{n}(\mathbb{Q})$-representations

(3.9)
${H_{n-1}(F\mathbf{C}(M(n)))}$${H_{n}(\widetilde{\text{Chains}}(n))}$${H^{n}(\mathbb{G}_{m},\mathbb{Z}(n))^{(0)}}$${H^{n}(\mathbb{G}_{m},\mathbb{Z}(n))^{(0)}}$$\scriptstyle{\varphi_{*}}$$\scriptstyle{\widetilde{\Psi}_{*}}$$\scriptstyle{f_{*}}$

where

$f_{\bullet}:\widetilde{\text{Chains}}(n)_{\bullet}\to
Z_{Ger(K(n))}(\mathbb{G}_{m})^{n-\bullet}$

will be a map of complexes that will be defined in that section, inducing an
isomorphism on the right vertical map (between two copies of
$\mathbb{Z}(\text{sgn})$. This will show that $\mathcal{R}$ is the kernel of
the left vertical map. Further, $f_{*}$ will send the fundamental class of the
sphere to $\\{-z_{1},\ldots,-z_{n}\\}$, giving us the rest of the proposition,
as this element generates the group as a $\mathbb{Z}^{\prime}$-module.

######  Remark 3.4.

The complex of Bloch symbols is not strictly necessary to define the cocycles
of this article (though we certainly use the map from the Bloch to Gersten
complexes in the proof of Theorem 3.5), but we include it regardless, as Lemma
2.16 shows that the class $(\varphi_{n})_{*}\Theta_{mat}$ is essentially
represented by a highest-degree component (and hence the image under a
Hochschild-Serre edge map) of an equivariant cyclotomic polylogarithm, i.e.
the a lift of $1\in H^{0}(\text{Spec }R,\mathbb{Z}(0))^{\Gamma}$ under the
residue-at-the-identity map

$H^{2n-1}(\mathbb{G}_{m}^{n}-(0),\mathbb{Z}(n))\to H^{0}(\text{Spec
}R,\mathbb{Z}(0))^{\Gamma}.$

See the author’s thesis [X] for an explication of these polylogarithms in the
setting of abelian schemes. This fulfills the philosophy outlined in [SV,
§5.5]. We do not explain this in detail here for purposes of space, and the
fact that this perspective does not work as cleanly in the cyclotomic setting
as for abelian schemes: essentally due to the issue which arises in section
2.3.2), we do not know a “formal” algebraic construction of the equivariant
cyclotomic polylogarithm, so there is limited interest in simply constructing
it from the classes we have already defined in the present article. We will
explain the polylogarithmic perspective more in the elliptic setting, in the
sequel to this article.

###  3.2. Symbols for the Gersten complex

We now turn to constructing the symbol map for the Gersten complex. This
section will be devoted to proving the following theorem:

######  Theorem 3.5.

There exists a map of $\text{GL}_{2}(\mathbb{Q})$-complexes

$f_{\bullet}:\widetilde{\text{Chains}}(n)_{\bullet}\to\left[Z_{Ger(K(n))}(\mathbb{G}_{m}^{n})^{n-\bullet}\right]^{(0)}.$

defined in degree $k$ on generators by

(3.10)
$[\Delta(m_{1},\ldots,m_{k})]\mapsto\begin{pmatrix}m_{1}&\ldots&m_{k}\end{pmatrix}_{*}\\{1-z_{1},\ldots,1-z_{k}\\}\in
K_{k}^{M}(k(L))^{(0)}$

where $L$ is defined to be the image of the map

$\begin{pmatrix}m_{1}&\ldots&m_{k}\end{pmatrix}:\mathbb{G}_{m}^{k}\to\mathbb{G}_{m}^{n}.$

Here, it is no longer necessary to pick integral generators of rays; we can
allow the column corresponding to $m_{i}$ to be well-defined only up to the
multiplication-by-$a$ isogenies $[a]:\mathbb{G}_{m}\to\mathbb{G}_{m}$.

Further, the induced map on cohomology in top degree

$(f_{n})_{*}:H_{n}(\widetilde{\text{Chains}}(n))\to
H^{n}(\mathbb{G}_{m}^{n},\mathbb{Z}(n))^{(0)}$

is an isomorphism sending the fundamental class of $S^{n-1}$ to the
orientation obstruction

$(-z_{1})\smile\ldots\smile(-z_{n}).$

The hard part of the theorem is proving the relations, so we first view
$f_{k}$ as a map from $\mathbb{Z}\\{\mathbf{T}_{k+1}\\}$.

Similarly to what happened in top degree for the Bloch symbol map
$\varphi_{n}$, replacing $m_{i}$ by a scalar multiple is immaterial because
each $1-z_{i}$ is invariant under $[a]_{*}$, so pre-composing
$\begin{pmatrix}m_{1}&\ldots&m_{i}\end{pmatrix}_{*}$ with these isogenies does
not change the definition of $f$. The $\text{GL}_{n}(\mathbb{Q})$-equivariance
of the definition follows formally.

We note further that the element (3.10) can be identified with

$(\psi_{k}\circ\varphi_{k})[(m_{1},\ldots,m_{k})],$

so the commutation with boundary/tame symbol maps follows from the first part
of the proof of Theorem 3.1. Hence $f_{\bullet}$ does give a map of complexes.

It remains to check that the relations in $\mathcal{S}_{\bullet}$ between the
classes of simplices hold. By Proposition 2.14, it suffices to check the
acyclic stellar subdvision relations.

######  Proposition 3.6.

For each pair of integers $1\leq k\leq n$ and $2\leq r\leq k$, each acyclic
independent tuple $\underline{m}=(m_{1},\ldots,m_{k})$ of rays and ray $m$
lying on the great circle corresponding to the face spanned by
$(m_{1},\ldots,m_{r})$ and sharing some hemisphere with all of them, the
relation

(3.11)
$f_{k}[\Delta(m_{1},\ldots,m_{k})]=\sum_{i=1}^{r}f_{k}[\Delta(m_{1},\ldots,\hat{m_{i}},m,m_{i+1},\ldots,m_{k})]$

holds.

###### Proof.

Write $L\cong\mathbb{G}_{m}^{k}$ for the image of the map

$\begin{pmatrix}m_{1}&\ldots&m_{k}\end{pmatrix}:\mathbb{G}_{m}^{k}\to\mathbb{G}_{m}^{n}.$

We first note that the claimed relation is supposed to hold in
$K_{k}^{M}(k(L))$ and is in fact independent of the value of $n$; changing the
value of $n$ only changes which degree of the Gersten complex this Milnor
$K$-group is viewed as being a summand of. Thus, we may as well assume $k=n$.

We may further reduce to the case $r=k$ as follows: the claimed relation can
be written as

$\begin{pmatrix}m_{1}&\ldots&m_{k}\end{pmatrix}_{*}\\{1-z_{1},\ldots,1-z_{k}\\}=\sum_{i=1}^{r}\begin{pmatrix}m_{1}&\ldots&\hat{m_{i}}&m&\ldots&m_{k}\end{pmatrix}_{*}\\{1-z_{1},\ldots,1-z_{k}\\}.$

The left-hand side factors as the cup product

$\begin{pmatrix}m_{1}&\ldots&m_{r}\end{pmatrix}_{*}\\{1-z_{1},\ldots,1-z_{r}\\}\smile\begin{pmatrix}m_{r+1}&\ldots&m_{n}\end{pmatrix}_{*}\\{1-z_{r+1},\ldots,1-z_{k}\\}$

and the right-hand side as

$\left(\sum_{i=1}^{r}\begin{pmatrix}m_{1}&\ldots&\hat{m_{i}}&m&\ldots&m_{r}\end{pmatrix}_{*}\\{1-z_{1},\ldots,1-z_{r}\\}\right)\smile\begin{pmatrix}m_{r+1}&\ldots&m_{n}\end{pmatrix}_{*}\\{1-z_{r+1},\ldots,1-z_{k}\\}$

so it suffices to prove that

$\begin{pmatrix}m_{1}&\ldots&m_{r}\end{pmatrix}_{*}\\{1-z_{1},\ldots,1-z_{r}\\}=\sum_{i=1}^{r}\begin{pmatrix}m_{1}&\ldots&\hat{m_{i}}&m&\ldots&m_{r}\end{pmatrix}_{*}\\{1-z_{1},\ldots,1-z_{r}\\}$

in $K_{r}^{M}(k(L))$. But this is the pullback of a top-rank stellar relation
from the quotient $L\twoheadrightarrow L/K$, for $K\subset L$ the image of

$\begin{pmatrix}m_{r+1}&\ldots&m_{k}\end{pmatrix}:\mathbb{G}_{m}^{r}\to
L\subset\mathbb{G}_{m}^{n}.$

We therefore henceforth assume that $r=k=n$, and need to prove the relation

$\begin{pmatrix}m_{1}&\ldots&m_{n}\end{pmatrix}_{*}\\{1-z_{1},\ldots,1-z_{n}\\}-\sum_{i=1}^{n}\begin{pmatrix}m_{1}&\ldots&\hat{m_{i}}&m&\ldots&m_{n}\end{pmatrix}_{*}\\{1-z_{1},\ldots,1-z_{n}\\}=0.$

Identifying

$K_{n}^{M}(k(\mathbb{G}_{m}^{n}))\hookrightarrow
z^{n}(k(\mathbb{G}_{m}^{n})\times\Box^{n})/\partial
z^{n+1}(k(\mathbb{G}_{m}^{n})\times\Box^{n+1})$

we see that it suffices to show that

(3.12)
$\sum_{i=1}^{n+1}(-1)^{i}\begin{pmatrix}m_{1}&\ldots&\hat{m_{i}}&\ldots&m_{n+1}\end{pmatrix}_{*}\Gamma(1-z_{1},\ldots,1-z_{n})\in\partial
z^{n+1}(k(\mathbb{G}_{m}^{n})\times\Box^{n+1}).$

for all acyclic tuples of rays $(m_{1},\ldots,m_{n+1})$ such that each
sub-$n$-tuple is full rank. Formally, (3.12) is the image under the cubical
face maps of

(3.13)
$\begin{pmatrix}m_{1}&\ldots&m_{n+1}\end{pmatrix}_{*}\Gamma(1-z_{1},\ldots,1-z_{n+1})\in\partial
z^{n+1}(k(\mathbb{G}_{m}^{n})\times\Box^{n+1})$

where the matrix denotes the map

(3.14)
$\begin{pmatrix}m_{1}&\ldots&m_{n+1}\end{pmatrix}:\mathbb{G}_{m}^{n+1}\to\mathbb{G}_{m}^{n}.$

However, this seems not to use the acyclicity condition; what gives? The
problem is that (3.14) is not a finite map, so unlike in the definition of
$\varphi_{\bullet}$ it is no longer immediate that the pushed forward cycle
has proper intersection with the faces.

Indeed, we claim that (3.13) intersects all faces properly when
$(m_{1},\ldots,m_{n+1})$ is acyclic. We will prove this on the level of cycles
in

$\mathbb{G}_{m}^{n}\times\Box^{n+1}$

since this certainly implies it for the restriction to the generic point.

Write $M$ for the matrix $(m_{1},\ldots,m_{n+1})$, and define the
$(n+1)$-variable monomials

$p_{j}(z_{1},\ldots,z_{n+1})=\prod_{i=1}^{n+1}z_{i}^{M_{j,i}}$

whose exponents are the $j$th row of $M$. Then in
$\mathbb{G}_{m}^{n}\times\Box^{n+1}$, the cycle (3.13) can be described as the
closure of the locus

$(p_{1},p_{2},\ldots,p_{n},1-z_{1},\ldots,1-z_{n+1}).$

Its intersection with a codimension-$d$ cubical face is then indexed by a
labelled subset $I\in\\{1,\ldots,n+1\\}$ of cardinality $d$, with each element
$i$ of the subset labelled by $\ell_{I}(i)\in\\{0,\infty\\}$; this face is
given by the intersection of the cycle with the locus

$\bigcap_{i\in I}\\{1-z_{i}=\ell_{I}(i)\\}$

i.e. fixing each $z_{i}$ corresponding to the indexing set to be either $1$ or
$\infty$.

We wish to check, for each $I$, whether or not this intersection has the
correct codimension, i.e. codimension $n+|I|$; the cycle (3.13) meets all
faces properly if and only if all codimensions are correct.

First consider the case where $I$ has at least one label of $0$; without loss
of generality, we assume that it corresponds to $1\in I$, i.e. the relation
$1-z_{1}=1\Leftrightarrow z_{1}=0$. The intersection of (3.13) with
$\\{z_{1}=0\\}$ is then the closure of the locus

(3.15) $(p_{1}(z_{1}=0),\ldots,p_{n}(z_{1}=0),0,1-z_{2},\ldots,1-z_{n}).$

where the notation indicates that we plug in $0$ in the $i$th place.

Write $M^{\prime}$ for the submatrix of $M$ given by deleting the first
column; by assumption, it has full rank $n$. Viewing $M^{\prime}$ as
associated to a map

$\mathbb{G}_{m}^{n}\times\Box^{n}\to\mathbb{G}_{m}^{n}\times\Box^{n+1}$

by its natural action on the toric part, and by inclusion
$\Box^{n}\hookrightarrow\Box^{n+1}$ in the last $n$ coordinates, we see that
the locus (3.15) is then the finite pushforward

(3.16) $M^{\prime}_{*}\Gamma(1-z_{1},\ldots,1-z_{n}).$

The intersection of (3.13) with a face corresponding to $I$ is then the
intersection of (3.16) with the face corresponding to the labelled set
$I\setminus\\{1\\}$. But (3.16) is a finite pushforward of a cycle meeting all
faces properly, so we conclude that (3.13) meets the face corresponding to $I$
properly as well.

Thus, any face with at least one label of $0$ always intersects (3.13)
properly; it therefore suffices to check the intersection with faces labelled
only with $\infty$s.

We claim that if $(m_{1},\ldots,m_{n+1})$ is acyclic, this intersection is
always empty, and thus trivially proper.101010When $(m_{1},\ldots,m_{n+1})$
fails to be acyclic, the corresponding stellar simplicial relation should not
hold, and thus (3.13) must not meet all faces properly. The simplest example
of what happens in this case: if $n=1$, $m_{1}=1$, and $m_{2}=-1$, then the
locus (3.13) is the closure of $(z_{1}z_{2}^{-1},1-z_{1},1-z_{2})$. Its
intersection with the unique codimension-two face labelled with two $\infty$s
should therefore be codimension $3$, i.e. zero-dimensional. However, the
points of the curve parameterized by $(g,\infty,\infty)$ are in the closure
for all $g\in\mathbb{G}_{m}$, since the point corresponding to each fixed $g$
is in the closure (as $t\to\infty$) of the curve $z_{1}=gt$, $z_{2}=t$ with
free parameter $t$.

Note that the property of having empty intersection with the $\infty$-labelled
faces is invariant under left multiplication of $M$ by elements of
$\text{GL}_{n}(\mathbb{Q})\cap M_{n}(\mathbb{Z})$. Via left multiplication by
such a matrix, we can always turn the first $n$ columns of the matrix into
scalar multiples of the standard basis. Thus, it suffices to check matrices of
the form

$\begin{pmatrix}x_{1}&&&&y_{1}\\\ &x_{2}&&&y_{2}\\\ &&\ldots&&\ldots\\\
&&&x_{n}&y_{n}\end{pmatrix}$

where each of the $x_{i}$ and $y_{i}$ must be nonzero by the assumption that
every $n$-by-$n$ submatrix of $M$ is full rank, and
$\text{sgn}(x_{i})=\text{sgn}(y_{i})$ for at least one $i$ by the acyclicity
assumption.

In this case, suppose without loss of generality that $I=\\{1,2,\ldots,k\\}$
for some $k\leq n$, with $\ell(j)=\infty$ for each $j\in I$. Assume now for
the sake of contradiction that (3.13) intersects the face corresponding to $I$
nontrivially; in particular, suppose it contains a point with
$z_{1}=z_{2}=\ldots=z_{k}=\infty$ but

$p_{i}=z_{i}^{x_{i}}z_{n+1}^{y_{i}}\in\mathbb{G}_{m}(\overline{k})$

for $\text{sgn}(x_{i})=\text{sgn}(y_{i})$. If $i\in\\{1,2,\ldots,k\\}$, this
is impossible, since at this point we must have $z_{n+1}=0\Rightarrow
1-z_{n+1}=1$, which means our point fails to lie in the algebraic cube
$\Box^{n+1}$.

Otherwise, $x_{j}$ and $y_{j}$ have opposite signs for $j=1,2,\ldots,k$,
meaning that $z_{n+1}=\infty$ as well at this point. But then

$p_{i}=z_{i}^{x_{i}}z_{n+1}^{y_{i}}\in\mathbb{G}_{m}(\overline{k})$

implies that $z_{i}=0\Rightarrow 1-z_{i}=1$, again a contradiction. We
conclude the intersection with the face corresponding to $I$ is in fact empty,
as desired.

We therefore have exhibited that $f_{\bullet}$ gives a well-defined map

$\widetilde{\text{Chains}}(n)_{\bullet}\to(Z_{Ger(K(n))}(\mathbb{G}_{m}^{n})^{n-\bullet})^{(0)}.$

∎

It remains to show that the induced map

$(f_{n})_{\bullet}:H_{n-1}(S^{n-1})\to
H^{n}(\mathbb{G}_{m},\mathbb{Z}(n))^{(0)}$

sends the fundamental class to the generator

$(-z_{1})\smile\ldots\smile(-z_{n}).$

Indeed, we can decompose the fundamental class as a sum of simplicial orthants

$[S^{n-1}]=\sum_{I\in\\{\pm 1\\}^{n}}[\Delta(I)]$

where $\Delta(I)$ is the simplex corresponding to $\sigma(I)(I_{1}\cdot
e_{1},\ldots,I_{n}\cdot e_{n})$ where $\sigma(I)$ is an arbitrary even
permutation if $(I_{1}\cdot e_{1},\ldots,I_{n}\cdot e_{n})$ is positively
oriented, is an arbitrary odd permutation otherwise. Under $f_{n}$, this is
sent to the sum

$\sum_{I\in\\{\pm
1\\}^{n}}\sigma\\{1-z_{1}^{I_{1}},\ldots,1-z_{n}^{I_{n}}\\}=\left\\{\frac{1-z_{1}}{1-z_{1}^{-1}},\ldots\frac{1-z_{n}}{1-z_{n}^{-1}}\right\\}=\\{-z_{1},\ldots,-z_{n}\\}.$

This concludes the proof of Theorem 3.1. As a corollary, we also deduce
Proposition 3.3; in particular, the map in top cohomology is onto after
passing to $\mathbb{Z}^{\prime}$-coefficients.

Between its proof and the proof of Theorem 3.5, we obtain a commutative
diagram of $\text{GL}_{2}(\mathbb{Z})$-complexes

(3.17)
${\widetilde{\text{Mat}}(n)}$${\widetilde{\text{Chains}}(n)}$${\tau_{\geq
n}\mathbb{Z}(n)_{\mathbb{G}_{m}^{n}}}$${Z_{Ger(K(n))}(\mathbb{G}_{m}^{n})^{(0)}}$$\scriptstyle{\varphi}$$\scriptstyle{\widetilde{\Psi}}$$\scriptstyle{f}$$\scriptstyle{\psi}$

with the right vertical arrow also being
$\text{GL}_{n}(\mathbb{Q})$-equivariant.

With the symbols for the Gersten complex fully defined, we thus obtain the
following classes:

$\Theta_{\mathbb{G}_{m}}(n)=f_{*}\Theta_{com}(n)\in
H^{n-1}(\text{GL}_{n}(\mathbb{Q}),(K_{n}^{M}(k(\mathbb{G}_{m}^{n}))/\\{-z_{1},\ldots,-z_{n}\\})^{(0)}).$

and, if $\Gamma\subset\text{GL}_{n}(\mathbb{Q})$ is $S$-arithmetic for a co-
nonempty set of primes $S$, for any transgression $\phi$ of $\varepsilon_{n}$
a class

$\Theta_{\mathbb{G}_{m}}(n)^{\phi}=f_{*}\Theta_{com}(n)^{\phi}\in
H^{n-1}(\Gamma,K_{n}^{M}(k(\mathbb{G}_{m}^{n}))^{(0)})[d_{n,S}^{-1}].$

For any $\Delta$-extension $E$, the cocycle representative
$\theta_{\mathbb{G}_{m}}(n)^{E}:=f_{*}\theta_{com}(n)^{E}$ for the former
class is given by

$(\gamma_{1},\ldots,\gamma_{n-1})\mapsto\begin{pmatrix}c_{1}&\ldots&c_{k}\end{pmatrix}_{*}\\{1-z_{1},1-z_{2},\ldots,1-z_{n}\\}\in
K_{n}^{M}(k(\mathbb{G}_{m}))/\\{-z_{1},\ldots,-z_{n}\\}$

with $c_{i}=\gamma_{i}\ldots\gamma_{1}e_{1}$ whenever these columns are
independent; for any fixed $\Delta$-extension $E$, one can equally in
principle work out the image of any tuple with non-independent such $c_{i}$,
though we do not currently see a systematic way to do this. Similarly, the
latter class is represented by
$\theta_{\mathbb{G}_{m}}(n)^{E,\phi}:=f_{*}\theta_{com}(n)^{E,\phi}$ and under
the same assumptions sends

(3.18)
$(\gamma_{1},\ldots,\gamma_{n-1})\mapsto\begin{pmatrix}c_{1}&\ldots&c_{k}\end{pmatrix}_{*}\\{1-z_{1},1-z_{2},\ldots,1-z_{n}\\}-\phi(\underline{\gamma})\\{-z_{1},\ldots,-z_{n}\\}$

in $K_{n}^{M}(k(\mathbb{G}_{m}))[d_{n,S}^{-1}]$.

We note also that from Remark 2.24 that when $n$ is odd, by inverting $2$ we
can get a _canonical_ lift of $\Theta_{\mathbb{G}_{m}}(n)$ to
$K_{n}^{M}(k(\mathbb{G}_{m}^{n}))^{(0)}$ for the whole group
$\text{GL}_{n}(\mathbb{Q})$, represented by the
$\text{GL}_{n}(\mathbb{Q})$-cocycle sending

$(\gamma_{1},\ldots,\gamma_{n-1})\mapsto\frac{1}{2}\left[\begin{pmatrix}c_{1}&\ldots&c_{k}\end{pmatrix}_{*}\\{1-z_{1},1-z_{2},\ldots,1-z_{n}\\}+\begin{pmatrix}c_{1}&\ldots&c_{k}\end{pmatrix}_{*}\\{1-z_{1}^{-1},\ldots,1-z_{n}^{-1}\\}\right]$

######  Remark 3.7.

In [BCG3, §8], who produce a version of our cocycle “after regulators” as
mentioned in the introduction (1.1), they produce by analytic means a
seemingly canonical version of the regulator of one of the classes
$\Theta_{\mathbb{G}_{m}}^{\phi}$, without having to pick a specific
transgression of a representative of the Euler class. It would be interesting
to understand whether one could find a distinguished $\phi$ for the lift of
this analytic class under the $d\log$ regulator.

####  3.2.1. Shintani realization

We now describe the realization of the parabolic, Steinberg module-valued,
cocycle of section 2.3.3. This amounts to determining the image of the kernel
of (2.24) under the map $f$, which by Lemma 2.25, is generated by the
$\text{GL}_{2}(\mathbb{Q})$-orbit of

$\\{1-z_{1},1-z_{2},\ldots,1-z_{n}\\}-\\{1-z_{1}^{-1},1-z_{2},\ldots,1-z_{n}\\}=\\{-z_{1},1-z_{2},\ldots,1-z_{n}\\}.$

We thus see it suffices to quotient out by the degree-$n$ part of the ideal
$I$ of the Milnor $K$-theory ring generated by the symbols $-z_{i}\in
K_{1}^{M}(k(\mathbb{G}_{m}))$, whereupon we obtain a
$\text{GL}_{2}(\mathbb{Q})$-equivariant map

$f^{\prime}:\text{St}(n)\to(K_{n}^{M}(k(\mathbb{G}_{m}))/I)^{(0)}$

sending

$[\ell_{1}]\wedge\ldots\wedge[\ell_{n}]\mapsto\begin{pmatrix}\ell_{1}&\ldots&\ell_{k}\end{pmatrix}_{*}[\\{1-z_{1},1-z_{2},\ldots,1-z_{n}\\}]\in
K_{n}^{M}(k(\mathbb{G}_{m}))/I.$

From this modular symbol, one also deduces an explicit parabolic cocycle
representing the pushforward of $\Theta_{St}(n)$ in
$H^{n-1}(\text{GL}_{2}(\mathbb{Q}),(K_{n}^{M}(k(\mathbb{G}_{m}))/I)^{(0)}$:
namely, the pushforward $f^{\prime}_{*}\theta^{E}_{com}(n)$ is independent of
$E$, and is given by

$(\gamma_{1},\ldots,\gamma_{n-1})\mapsto\begin{pmatrix}c_{1}&\ldots&c_{k}\end{pmatrix}_{*}[\\{1-z_{1},1-z_{2},\ldots,1-z_{n}\\}]\in
K_{n}^{M}(k(\mathbb{G}_{m}))/I$

with $c_{i}=\gamma_{i}\ldots\gamma_{1}e_{1}$.

This is the analogue of the “classical” Shintani cocycle in our context, and
essentially coincides with that of [LP] after passing to infinite level (i.e.
taking an inverse limit of copies of $\mathbb{G}_{m}$).

###  3.3. Spreading out and pulled back cocycles

For parsimony of exposition, we have until now been using the Milnor
$K$-theory of the generic point as the target of the realization map $f$,
since it fits naturally with the existing notion of the Gersten complex.
However, we must do better in order to pull back our cocycles to motivic
cohomology of the base, since one cannot pull back from the generic point.

Let $\Gamma\subset\text{GL}_{n}(\mathbb{Q})$ be any subgroup. The proof of
Theorem 3.5 applies identically to show that the realization map factors, as a
$\Gamma$-map, through

$\widetilde{\text{Chains}}(n)\xrightarrow{f_{\Gamma}}Z_{\Gamma}(n)^{(0)}\to
Z_{Ger(K(n))}(\mathbb{G}^{n}_{m})^{(0)}$

where $Z_{\Gamma}(n)$ is defined by

(3.19) $Z_{\Gamma}(n)_{i}:=\varinjlim_{H}\bigoplus_{S\in
H_{i}}H^{i}(S_{i}^{\circ},\mathbb{Z}(i))$

and the direct limit ranges over hypersurface arrangements given by finite
subarrangements of the $\Gamma$-orbit of

$\mathbb{G}_{m}^{n-1}\subset\mathbb{G}_{m}^{n}$

embedded as the kernel of $1-z_{1}$, $H_{i}$ denotes the $i$-dimensional-
strata of such an arrangement, and $S_{i}^{\circ}$ is the open part of the
stratum $S$ (i.e. the complement of lower-dimensional strata inside the
stratum). In particular, when $i=n$, we just get a single summand

$\varinjlim_{H}H^{n}(\mathbb{G}_{m}^{n}-H,\mathbb{Z}(n)),$

since there is only one top-dimensional stratum. (This is the only degree of
the map which is particularly significant for our purposes.) Note that this
spreads out our cocycles in the “arithmetic direction” as well, i.e. over the
base $R$ and not just $k$.

######  Remark 3.8.

The complex (3.19) can be described as the $n$th row in the first page of the
weight-$n$ motivic cohomology descent/coniveau spectral sequence associated to
the poset of the hyperplane complements in question, as in [SV, §2]; if one
replaces the poset of hyperplane complements by the poset of all open subsets,
one obtains the usual Gersten complex.

####  3.3.1. Pullback by one section

Suppose now that $\Gamma$ fixes any a torsion section $x:\text{Spec
}R\to\mathbb{G}_{m}^{n}$ of order $N$, with $z_{1}(x)\neq 1$. This induces by
pullback a map

$Z_{\Gamma}(n)_{n}=\varinjlim_{H}H^{n}(\mathbb{G}_{m}^{n}-H,\mathbb{Z}(n))\to
H^{n}(\text{Spec }R,\mathbb{Z}(n))$

and therefore a corresponding specialization of our Eisenstein cocycle

$[x^{*}(f_{\Gamma})_{*}\theta_{com}]=x^{*}(f_{\Gamma})_{*}\Theta_{com}(n)\in
H^{n-1}(\Gamma,H^{n}(\text{Spec }R,\mathbb{Z}(n)))$

sending

$(\gamma_{1},\ldots,\gamma_{n-1})\mapsto
x^{*}\begin{pmatrix}c_{1}&\ldots&c_{k}\end{pmatrix}_{*}\\{1-z_{1},1-z_{2},\ldots,1-z_{n}\\}\in
H^{n}(\text{Spec }R,\mathbb{Z}(n))$

whenever the $c_{i}=\gamma_{i}\ldots\gamma_{1}e_{1}$ are independent.

If we write

$M=\begin{pmatrix}c_{1}&\ldots&c_{k}\end{pmatrix}_{*},$

this pullback can be written also as

$\sum_{x^{\prime}\in M^{-1}x}x^{\prime*}\\{1-z_{1},1-z_{2},\ldots,1-z_{n}\\}.$

One can equally make a “Shintani” version of this construction: the argument
of section 3.2.1 affords us a map

$f^{\prime}_{\Gamma}:\text{St}(n)\to\varinjlim_{H}H^{n}(\mathbb{G}_{m}^{n}-H,\mathbb{Z}(n))/I$

where $I$ is the part of the ideal generated by symbols $\\{-z_{i}\\}$ in
degree $n$. This affords us a modular symbol

$[\ell_{1}]\wedge\ldots\wedge[\ell_{n}]\mapsto\sum_{x^{\prime}\in
M^{-1}x}\\{1-z_{1},1-z_{2},\ldots,1-z_{n}\\}\in K_{n}^{M}(R)[N^{-1}]$

where

$M=\begin{pmatrix}\ell_{1}&\ldots&\ell_{k}\end{pmatrix},$

as well as the corresponding parabolic cocycle. Here, we have used the fact
that the pullback of the ideal $I$ by an $N$-torsion section consists of
$N$-torsion elements, since $I$ is generated by elements of the form $-z_{i}$
which upon pullback become $N$-torsion.

This same method applies identically to $\Gamma$-fixed combinations of torsion
sections which avoid bad hyperplanes. We do not go into more detail here,
because beyond the case $n=2$ (covered in [SV]), the norm residue isomorphism
theorem suggests that the pulled back cocycles are of limited arithmetic
interest.

####  3.3.2. Better integrality?

As we noted in the preliminary section 2.1, the technical details of descent
for motivic complexes force us to work over DVRs in order to prove the
relations needed for our cocycles: over a Dedekind domain, for example, we
cannot take compute the hypercohomology of the Bloch cycle complex on global
sections. In this section, we have seen that our methods allow us to obtain
classes on hyperplane complements in $\mathbb{G}_{m}$ defined over this DVR
base. However, the actual cocycles we get are built from functions on
$\mathbb{G}_{m}^{n}$ defined even over $\text{Spec }\mathbb{Z}$, raising the
question of whether everything can in fact be done over this base. We end this
section by devoting a few words to this question.

In general, the situation is this: if $X$ is a scheme over $\mathbb{Z}$, then
we have a residue long exact sequence

$\ldots\to\bigoplus_{p}H^{n-2}(X_{\mathbb{F}_{p}},\mathbb{Z}(n-1))\to
H^{n}(X,\mathbb{Z}(n))\to
H^{n}(X_{\mathbb{Q}},\mathbb{Z}(n))\to\bigoplus_{p}H^{n-1}(X_{\mathbb{F}_{p}},\mathbb{Z}(n-1))\to\ldots$

coming from a distinguished triangle of Bloch complexes, see e.g. the motivic
background section of the author’s thesis [X]. For $n=2$, the ambiguity term
does vanish since $n-2=0$, so in that case, covered in [SV], one can obtain
integral cocycles. However, for $n>2$ there is no reason to expect splitting
or degeneration; hence, one cannot easily lift the cocycles
$\Theta_{\mathbb{G}_{m}}$ and its cousins to be defined over $\mathbb{Z}$.

In fact, we do expect that one can show that the trace-fixed part of the
ambiguity term

$\left(\bigoplus_{p}H^{n-2}(X_{\mathbb{F}_{p}},\mathbb{Z}(n-1))\right)^{(0)}$

vanishes for a hyperplane complement $X=\mathbb{G}_{m}-H$ via an inductive
argument; this is an idea of the first author of [SV]. This would yield
integral cocycles for larger $n$. However, we do not pursue this idea in the
present article.

###  3.4. Kubota-Leopoldt distributions

We show how our geometric framework applies to give cocycles valued in
_distributions_ valued in Milnor $K$-theory of algebraic tori, since this is
done in the related work [LP] (as well as older treatments of Shintani-type
cocycles).

For convenience, we will write the results in this section in terms of Milnor
$K$-theory of the generic point, but all the arguments apply with only minor
modifications to the more spread-out cocycles of section 3.3.

Write $\mathbb{D}(\mathbb{Q}^{n},M)$ for the module of $M$-valued locally
constant (in the adelic topology) distributions on $\mathbb{Q}^{n}$, i.e. the
dual of the module generated by $M$-valued indicator functions on subsets of
the form $a+L$, where $L\subset\mathbb{Q}^{n}$ is a full-rank
$\mathbb{Z}$-lattice. If $M$ is a right $\text{GL}_{n}(\mathbb{Q})$-module, a
distribution $\mu\in\mathbb{D}(\mathbb{Q}^{n},M)$ takes a left
$\text{GL}_{n}(\mathbb{Q})$-action via

$(\gamma\mu)(f):=\gamma\cdot\mu((f\circ\gamma)).$

where $\gamma$ acts via its natural left action on $\mathbb{Q}^{n}$ (and thus
by a right pullback action on test functions). We write

$\mathbb{D}(\mathbb{Q}^{n},M)^{(0)}\subset\mathbb{D}(\mathbb{Q}^{n},M)$

for the scalar-invariant distributions, i.e. those for which the set $U$ has
the same measure as the set $qU$ for any $q\in\mathbb{Q}^{\times}$.

######  Proposition 3.9.

Suppose $R$ contains all roots of unity. There is a
$\text{GL}_{n}(\mathbb{Q})$-equivariant map

(3.20)
$\text{KL}:K_{n}^{M}(k(\mathbb{G}_{m}^{n}))^{(0)}\to\mathbb{D}(\mathbb{Q}^{n},K_{n}^{M}(k(\mathbb{G}_{m}^{n})))^{(0)},$

given by

$s\mapsto\left(1(a+\mathbb{Z}^{n})\mapsto t_{a}^{*}s\right)$

and extended by the scalar invariance property, where

$t_{a}:\mathbb{G}_{m}^{n}\to\mathbb{G}_{m}^{n}$

is the map given by translating by the vector of roots unity $\exp(2\pi ia)$.

###### Proof.

There are two things to check: that the distribution relation hold, and that
the map is equivariant.

Since we have specified the distribution via open sets of the form

$a+q\mathbb{Z}^{n}$

for scalars $q\in\mathbb{Q}$, it suffices to verify the distribution relations
of the form

$\mu(a+\mathbb{Z}^{n})=\sum_{a^{\prime}}\mu(a^{\prime}+k\mathbb{Z}^{n})=\sum\sum_{a^{\prime}}\mu\left(\frac{a^{\prime}}{k}+\mathbb{Z}^{n}\right)$

for any $\mu$ in the image of the map and any integer $k$, where $a^{\prime}$
runs over equivalence classes in $\mathbb{Q}^{n}/k\mathbb{Z}^{n}$ having image
$a\in\mathbb{Q}^{n}/\mathbb{Z}^{n}$. Here, the second equality follows from
scalar invariance.

If $\mu$ is the image of $s\in K_{n}^{M}(k(\mathbb{G}_{m}^{n}))^{(0)}$, this
relation follows from the equalities

$t_{a}^{*}s=t_{a}^{*}[k]_{*}s=\sum_{a^{\prime}}t_{a^{\prime}/k}^{*}s$

where the first equality is from trace invariance of $s$, and the second is an
equality of correspondences.

To show $\text{GL}_{n}(\mathbb{Q})$-equivariance, we see that for any $\gamma$
in the group,

(3.21) $\displaystyle\gamma s$
$\displaystyle\mapsto(1(a+\mathbb{Z}^{n})\mapsto t_{a}^{*}(\gamma_{*}s))$
(3.22)
$\displaystyle=\left(1(a+\mathbb{Z}^{n})\mapsto\gamma_{*}\sum_{a^{\prime}\in\gamma^{-1}a}t_{a^{\prime}}^{*}s\right)$
(3.23)
$\displaystyle=\left((1(a+\mathbb{Z}^{n})\circ\gamma)\mapsto\gamma_{*}t_{a}^{*}s\right)$

which is what we want. ∎

######  Corollary 3.10.

Via the pushforward by (3.20) of $\theta_{com}(n)^{E,\phi}$, we obtain a
cocycle which sends

$\underline{\gamma}\mapsto\left(1(a+\mathbb{Z}^{n})\mapsto
t_{a}^{*}\left[\begin{pmatrix}c_{1}&\ldots&c_{k}\end{pmatrix}_{*}\\{1-z_{1},1-z_{2},\ldots,1-z_{n}\\}-\phi(\underline{\gamma})\\{-z_{1},\ldots,-z_{n}\\}\right]\right)$

for tuples $\underline{\gamma}$ with linearly independent $c_{i}$ (notation as
previously), representing $\text{KL}_{*}\Theta_{com}(n)^{\phi}$.

As always, one can equally define Kubota-Leopoldt distributions for the other
versions of the cocycle we defined (including the Shintani modular symbol as
done complex-analytically in [LP]).

###  3.5. Hecke action

Let $S$ be a set of primes, and take $\Gamma\subset\text{GL}_{n}(\mathbb{Q})$
an arithmetic group. We wish to compute the action of the Hecke algebra of
$\Gamma$ on $\Theta_{\mathbb{G}_{m}}(n)$ (and its variants) at primes away
from $S$.

To define a Hecke action on group cohomology with coefficients in a
$\Gamma$-module $M$, one needs an action of not just $\Gamma$, but all of

$\Delta:=M_{n}(\mathbb{Z})\cap\text{GL}_{n}(\mathbb{Q}).$

We already have been acting on Milnor $K$-theory and motivic cohomology with
groups even larger than this, via pushforward. However, if we use the action
via pushforwards we have been using until now to define the Hecke action, we
do not obtain actions compatible with [SV] and more classical formulations; to
make the actions line up, we must take _pullback_ actions.

Since have been acting by $\gamma_{*}$, which on $\text{GL}_{n}(\mathbb{Z})$
coincides with $(\gamma^{-1})^{*}$, to extend the action to a pullback-type
action for $\gamma\in\Delta$ we need to define

(3.24) $\gamma\cdot x=(|\det\gamma|\gamma^{-1})^{*}x$

for any element $x$ in any of the various cohomology groups associated to
$\mathbb{G}_{m}^{n}$ or subvarieties thereof.

For any prime $p$, we now follow [RW] in defining the Hecke operator
$T_{p}^{(i)}$ to be the double coset operator

$\Gamma\text{diag}(p,\ldots,p,1,\ldots,1)\Gamma=\bigsqcup_{i=1}^{j}\Gamma\alpha$

with $i$ copies of $p$. For a $\Delta$-module $M$, it acts on $[c]\in
H^{d}(\Gamma,M)$ via

(3.25)
$(T_{p}^{(i)}c)(\gamma_{1},\ldots,\gamma_{d}):=\sum_{i=1}^{j}\alpha_{i}^{-1}c(\alpha_{i}).$

That this is well-defined and independent of chosen representatives is shown
in loc. cit. Further, associated to a short exact sequence of $\Delta$-modules

$0\to A\to B\to C\to 0$

we have the standard connecting homomorphisms

$H^{d}(\Gamma,C)\to H^{d+1}(\Gamma,A),$

whose description in Lemma 2.16 makes plain that they intertwine with the
Hecke action (3.25).

Consider the embedding of exact complexes

$\text{Chains}(n)\hookrightarrow\text{Chains}(n)\otimes\mathbb{Z}\\{Z_{0}(\mathbb{G}_{m}^{n}[p^{\infty}])\\}$

sending $x\mapsto x\otimes[1]$, where $[1]$ is the class of the identity
section, and $Z_{0}$ denotes the relative zero-dimensional cycles over $R$. By
the same argument as in the proof of Theorem 3.5, there is a realization map
extending $f_{\bullet}$ of the form

$\tilde{f}_{\bullet}:\text{Chains}(n)\otimes\mathbb{Z}\\{Z_{0}(\mathbb{G}_{m}^{n}[p^{\infty}])\\}\to\left[Z_{Ger(K(n))}(\mathbb{G}_{m}^{n})^{n-\bullet}\right]^{(0)}$

which sends

(3.26) $[\Delta(m_{1},\ldots,m_{k})]\otimes[x]\mapsto
t_{x}\circ\begin{pmatrix}m_{1}&\ldots&m_{k}\end{pmatrix}_{*}\\{1-z_{1},\ldots,1-z_{k}\\}\in
K_{k}^{M}(k(L))^{(0)}$

where the action of $[x]\in\mathbb{Z}\\{Z_{0}(\mathbb{G}_{m}^{n}[p^{\infty}])$
written on the left-hand side is the action of the correspondence

$t_{x}\subset\mathbb{G}_{m}^{n}\times_{R}\mathbb{G}_{m}^{n}$

given by preimage of $x$ under the division morphism

$\mathbb{G}_{m}^{n}\times\mathbb{G}_{m}^{n}\to\mathbb{G}_{m}^{n},(z_{1},z_{2})\mapsto
z_{1}/z_{2}.$

In particular, if $x$ consists of a single torsion section, $t_{x}$ acts by
translation by that torsion section.111111The reason for this artifice,
instead of simply considering translates by torsion sections, is that one may
not be able to separate the torsion over more general bases. The map
$\tilde{f}_{\bullet}$ clearly restricts to $f_{\bullet}$ on the identity
component, and is visibly $\Delta$-equivariant for the pullback action (3.24)
on both sides.

From this, we find the following theorem:

######  Theorem 3.11.

We have

$T_{p}^{(i)}\Theta_{\mathbb{G}_{m}}=\left(\binom{n-1}{i-1}_{p}[p]^{*}+\binom{n}{i}_{p}-\binom{n-1}{i-1}_{p}\right)\Theta_{\mathbb{G}_{m}}.$

where the $p$-binomial coefficients are given by

$\binom{a}{b}_{p}=\frac{a!_{p}}{b!_{p}(a-b)!_{p}}$

and the $p$-factorial by

$a!_{p}:=\prod_{k=1}^{a}(p^{k}-1).$

###### Proof.

By the preceding discussion, it suffices to show the equality of the operators
$T_{p}^{(i)}$ and

$\binom{n-1}{i-1}_{p}[p]^{*}+\binom{n}{i}_{p}-\binom{n-1}{i-1}_{p}$

acting on the identity $[1]\in\mathbb{Z}\\{\mathbb{G}_{m}^{n}[p^{\infty}]\\}$.
Since algebraic cycles satisfy fppf descent, this equality can be checked upon
base change to a base over which all the $p$-torsion of $\mathbb{G}_{m}^{n}$
is defined.

Under this assumption, this becomes a linear algebra problem: how many times
does a given $p$-torsion point appear in $T_{p}^{(i)}[1]$? For any non-
identity point, the answer is the number of $(n-i)$-planes in the
$n$-dimensional $\mathbb{F}_{p}$-vector space containing both that point, a
Schubert cell whose point count over $\mathbb{F}_{p}$ is
$\binom{n-1}{i-1}_{p}$. And for the identity, the answer is simply the total
number of such $(n-i)$-planes, which is simply the Grassmannian of $i$-planes
and whose point count is $\binom{n}{i}_{p}$. From this, the result follows. ∎

######  Remark 3.12.

This nice formula does not extend to the further-refined cocycles
$\Theta_{com}(n)^{\phi}$, as the additional term which is a multiple of the
orientation class certainly does not transform in the same way under the Hecke
operators.

######  Remark 3.13.

The approach of [SV] is to take the pullback action $\gamma^{*}$ from the
outset, making the treatment of Hecke operators slightly more natural. The
pullback they obtain is a left action (like our pushforward), as they take a
_right_ group action on $\mathbb{G}_{m}$. This approach, however, prevents
them from considering $S$-arithmetic actions as we did in this article. If one
examines the equation (3.24), this difference in conventions has the effect of
interchanging the Hecke operators $T_{p}^{(i)}$ and $T_{p}^{(n-i)}$. By the
symmetry relation

$\binom{a}{b}_{p}=\binom{a}{a-b}_{p},$

this actually has no effect on the resulting Hecke action.

## References

  * [BaTa] Hyman Bass and John Tate  “The Milnor ring of a global field.”  In _Algebraic $K$-theory II._ 342 Springer-Verlag, 1972
  * [BCG1] Nicolas Bergeron, Pierre Charollois and Luis Garcia  “Transgressions of the Euler class and Eisenstein cohomology of $\text{GL}_{N}(\mathbb{Z})$.”  In _Japanese Journal of Mathematics_ 15, 2020, pp. 311–379
  * [BCG3] Nicolas Bergeron, Pierre Charollois and Luis Garcia  “Cocycles de Sczech et arrangements d’hyperplans dans les cas additif, multiplicatif et elliptique.” In preparation.
  * [Bj] Anders Björner  “Homlogy and shellability of matroids and geometric lattices.”  In _Mathematische Annalen_ 372, 2018, pp. 781–844
  * [Bl2] Spencer Bloch  “Algebraic cycles and higher $K$-theory.”  In _Advances in Mathematics._ 61, 1986, pp. 267–304
  * [BPPS] Cecilia Busuioc, Jeehoon Park, Owen Patashnick and Glenn Stevens  “Modular symbols with values in Beilinson-Kato distributions.”, 2023  URL: https://arxiv.org/abs/2311.14620
  * [DPV1] Henri Darmon, Alice Pozzi and Jan Vonk  “Diagonal restrictions of $p$-adic Eisenstein families.”  In _Mathematische Annalen_ 379, 2021, pp. 503–548  URL: http://www.math.mcgill.ca/darmon/pub/Articles/Research/73.DPV1/DPV1.pdf
  * [DPV2] Henri Darmon, Alice Pozzi and Jan Vonk  “The values of the Dedekind-Rademacher cocycle at real multiplication points.” Preprint.
  * [FS] Eric Friedlander and Andre Suslin  “The spectral sequence relating algebraic $K$-theory to motivic cohomology.”  In _Annales Scientifiques de l’École Normale Supérieure._ 4, 2002, pp. 773–875
  * [Geis] Thomas Geisser  “Motivic cohomology over Dedekind rings.”  In _Mathematische Zeitschrift._ 248, 2004, pp. 773–794
  * [GL] Thomas Geisser and Marc Levine  “The $K$-theory of fields in characteristic $p$.”  In _Inventiones mathematicae._ 139, 2000, pp. 459–493
  * [La] Steven Landsburg  “Relative Chow groups.”  In _Illinois Journal of Mathematics._ 35.4, 1991
  * [Lev2] Marc Levine  “Techniques of localization in the theory of algebraic cycles.”  In _Journal of Algebraic Geometry._ 10, 2001, pp. 299–363
  * [Lick] William Lickorish  “Simplicial moves on complexes and manifolds”  In _Geometry and Topology Monographs._ 2, 1999, pp. 299–320
  * [LP] Sung Hyun Lim and Jehoon Park  “Milnor $K$-theory and the Shintani cocycle.”, 2019  URL: https://arxiv.org/abs/1909.03450
  * [New] Max Newman  “On the foundations of combinatorial Analysis Situs.”  In _Proceedings of the Royal Academy at Amsterdam._ 29, 1926, pp. 610–641
  * [OS] Peter Orlik and Louis Solomon  “Combinatorics and topology of complements of hyperplanes.”  In _Inventiones Mathematicae._ 56.2, 1980, pp. 167–189
  * [RW] Young Rhie and George Whaples  “Hecke operators in the cohomology of groups.”  In _Journal of the Mathematical Society of Japan._ 22, 1970, pp. 431–442
  * [Scz] Robert Sczech  “Eisenstein cocycles for $\text{GL}_{2}(\mathbb{Q})$ and values of $L$-functions in real quadratic fields.”  In _Commentarii Mathematici Helvetici._ 67, 1992, pp. 363–382
  * [Shin] Takuro Shintani  “On evaluation of zeta functions of totally real algebraic number fields at non-positive integers.”  In _Journal of the Faculty of Science, the University of Tokyo._ 23, 1976, pp. 393–417
  * [Stevens] Glenn Stevens  “$K$-theory and Eisenstein series.” Unfinished preprint., 2013  URL: http://math.bu.edu/people/ghs/preprints/Circular-KThry-11-07.pdf
  * [Sul] Vladimir Voevodsky, Andrei Suslin and Eric Friedlander  “La classe d’Euler réelle d’un fibré vectoriel à groupe structural $\text{SL}_{n}(\mathbb{Z})$ est nulle.”  In _Comptes Rendus de l’Académie des sciences de Paris._ 281, 1975
  * [SV] Romyar Sharifi and Akshay Venkatesh  “Eisenstein cocycles in motivic cohomology.” Preprint., 2020  URL: https://arxiv.org/pdf/2011.07241
  * [Tot] Burt Totaro  “Milnor $K$-theory is the simplest part of algebraic $K$-theory”  In _$K$ -Theory._ 6, 1992, pp. 177–189
  * [V] Vladimir Voevodsky  “Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic.”  In _International Mathematical Research Notices._ 7, 2002
  * [WR] Nathalie Wahl and Oscar Randal-Williams  “Homological stability for automorphism groups.” Preprint., 2014  URL: https://arxiv.org/abs/1409.3541
  * [X] Peter Xu  “Arithmetic Eisenstein theta lifts.”, 2023