text:
\begin{document}
\title[$p$-local homology of Coxeter groups]{
A vanishing theorem for \\ the $p$-local homology of Coxeter groups}
\author[T. Akita]{Toshiyuki Akita}
\address{Department of Mathematics, Hokkaido University,
Sapporo, 060-0810 Japan}
\email{akita@math.sci.hokudai.ac.jp}
\subjclass[2010]{Primary~20F55, 20J06; Secondary~55N91}
\keywords{Coxeter groups, Group homology}

\begin{abstract}
Given an odd prime number $p$ and a Coxeter group $W$ such that
the order of the product $st$ is prime to $p$ for every Coxeter generators
$s,t$ of $W$, we prove that the $p$-local homology groups $H_k(W,\Zp)$
vanish for $1\leq k\leq 2(p-2)$.
This generalize a known vanishing result for
symmetric groups due to Minoru Nakaoka.
\end{abstract}

\maketitle

\section{Introduction}
Coxeter groups are important objects in many branches of mathematics,
such as Lie theory and representation theory,
combinatorial and geometric group theory, topology and geometry.
Since the pioneering work of Serre \cite{serre}, Coxeter groups have
been studied in group cohomology as well. See the book by Davis \cite{davis-book}
and \S\ref{sec-known} of this paper for brief outlooks.
In this paper, we will study the $p$-local homology of Coxeter groups
for \emph{odd} prime numbers $p$.
For an arbitrary Coxeter group $W$,  its integral homology group $H_k(W,\Z)$ is
known to be a finite abelian group for all $k>0$,
and hence it decomposes into a \emph{finite} direct sum
of $p$-local homology groups each of which is a finite abelian $p$-group:
\[
H_k(W,\Z)\cong\bigoplus_p H_k(W,\Zp).
\]

According to a result of Howlett \cite{howlett},
the first and second $p$-local homology groups, $H_1(W,\Zp)$ and $H_2(W,\Zp)$,
are trivial for every odd prime number $p$.
On the other hand, the symmetric group of $n$ letters $\symm_n$ $(n\geq 2)$
is the Coxeter group of type $A_{n-1}$.
Much is known about the (co)homology of symmetric groups.
Most notably, in his famous two papers,
Nakaoka proved the homology stability for
symmetric groups \cite{nakaoka} and computed the stable mod $p$
homology \cite{nakaoka2}. As a consequence of his results,
$H_k(\symm_n,\Zp)$ vanishes
for $1\leq k\leq 2(p-2)$ (see Theorem \ref{nakaoka-result} below).
The purpose of this paper is to generalize vanishing of $H_k(\symm_n,\Zp)$ to all
Coxeter groups:
\begin{thm}
\label{main-thm}
Let $p$ be an odd prime number
and $W$ a $p$-free Coxeter group. Then $H_k(W,\Zp)=0$ holds for $1\leq k\leq 2(p-2)$.
\end{thm}

Here a Coxeter group $W$ is said to be $p$-free if  the order of the product $st$ is prime
to $p$ for every distinct Coxeter generators $s,t$ of $W$.
We should remark that, for $p\geq 5$, the $p$-freeness assumption is
necessary
and the vanishing range $1\leq k\leq 2(p-2)$ is best possible.
The situation is somewhat different for $p=3$. See \S\ref{sec-3-free}.

The proof of Theorem \ref{main-thm} consists of two steps,
a case by case argument for finite $p$-free Coxeter groups
with relatively small rank, and the induction on the number of generators.
The induction is made possible by means of the equivariant homology of
Coxeter complexes and the Leray spectral sequence converging to the
equivariant homology.
Now we will introduce the content of this paper very briefly.
In \S\ref{subsec-defn}, we will recall definitions and
relevant facts concerning of Coxeter groups.
Known results about homology of Coxeter groups and their consequences
will be reviewed more precisely in \S\ref{sec-known}.
After the consideration of the equivariant homology of Coxeter complexes in \S\ref{sec-equivariant}.
the proof of Theorem \ref{main-thm} will be given in \S\ref{sec-proof}.
The final section \S\ref{sec-vanishing} consists of miscellaneous results.
There we will give some classes of Coxeter groups such that all the $p$-local homology
groups vanish.

\begin{notation}
Throughout this paper, $p$ is an \emph{odd} prime number unless otherwise stated.
$\Zp$ is the localization of $\Z$ at the prime $p$ (the ring of $p$-local integers).
For a finite abelian group $A$, its $p$-primary component is denoted by $A_{(p)}$.
Note that $A_{(p)}\cong A\otimes_{\Z}\Zp$.
For a group $G$ and a (left) $G$-module $M$, the co-invariant of $M$ is denoted by $M_G$
(see \cite{brown}*{II.2} for the definition). For a prime number $p\geq 2$,
we denote the cyclic group of order $p$ and the field with $p$ elements
by the same symbol $\Z/p$.
\end{notation}

\section{Preliminaries}\label{sec-defn}
\subsection{Coxeter groups}\label{subsec-defn}
We recall definitions and relevant facts concerning of Coxeter groups.
Basic references are \cites{book-building,bourbaki,davis-book,humphreys}.
See also \cite{MR777684} for finite Coxeter groups.
Let $S$ be a finite set and $m:S\times S\rightarrow \N\cup\{\infty\}$
a map satisfying the following conditions:
\begin{enumerate}
\item $m(s,s)=1$ for all $s\in S$
\item $2\leq m(s,t)=m(t,s)\leq\infty$ for all distinct $s,t\in S$.
\end{enumerate}
The map $m$ is represented by the \emph{Coxeter graph} $\Gamma$ whose
vertex set is $S$ and whose edges are the unordered pairs $\{s,t\}\subset S$
such that $m(s,t)\geq 3$. The edges with $m(s,t)\geq 4$ are labeled
by those numbers.
The \emph{Coxeter system} associated to $\Gamma$
is the pair $(W,S)$ where $W=W(\Gamma)$ is the group generated by $s\in S$
and the fundamental relations $(st)^{m(s,t)}=1$ $(m(s,t)<\infty)$:
\[
W:=\langle s\in S\ |\ (st)^{m(s,t)}=1 (m(s,t)<\infty)\rangle.
\]
The group $W$ is called the \emph{Coxeter group} of type $\Gamma$,
and elements of $S$ are called \emph{Coxeter generators} of $W$.
The cardinality of $S$ is called the \emph{rank} of $W$
and is denoted by $|S|$ or $\rank\,W$.
Note that the order of the product $st$ is precisely $m(s,t)$.
For a subset $T\subseteq S$ (possibly $T=\varnothing$),
the subgroup $W_T:=\langle T\rangle$ of $W$ generated
by elements $t\in T$ is called a \emph{parabolic subgroup}.
In particular, $W_S=W$ and $W_\varnothing=\{1\}$.
It is known that $(W_T,T)$ is a Coxeter system.

A Coxeter group $W$ is called \emph{irreducible} if its defining graph
$\Gamma$ is connected,
otherwise called \emph{reducible}.
For a reducible Coxeter group $W(\Gamma)$ of type
$\Gamma$, if $\Gamma$ consists of the connected components
$\Gamma_1,\Gamma_2,\dots,\Gamma_r$, then $W(\Gamma)$
is the direct product of parabolic subgroups
$W(\Gamma_i)$'s $(1\leq i\leq r)$, each of which is irreducible:
\[
W(\Gamma)=W(\Gamma_1)\times W(\Gamma_2)\times\cdots\times W(\Gamma_r).
\]

Coxeter graphs for finite irreducible Coxeter groups are classified.
There are four infinite families $A_n\, (n\geq 1)$, $B_n\, (n\geq 2)$, $D_n\, (n\geq 4)$,
$I_2(q)\, (q\geq 3)$, and six exceptional graphs $E_6,E_7,E_8,F_4,H_3$ and $H_4$.
The subscript indicates the rank of the resulting Coxeter group.
See Appendix for the orders of finite irreducible Coxeter groups.
Here we follow the classification given in the book by Humphreys \cite{humphreys}
and there are overlaps $A_2=I_2(3),B_2=I_2(4)$.
Note that $W(A_n)$ is isomorphic to the symmetric group of $n+1$ letters,
while $W(I_2(q))$ is isomorphic to the dihedral group of order $2q$.

Finally,  given an odd prime number $p$,
we define a Coxeter group $W$ to be \emph{$p$-free} if $m(s,t)$ is prime to $p$
for all $s,t\in S$. Here $\infty$ is prime to all prime numbers by the convention.
For example, the Coxeter group $W(I_2(q))$ is $p$-free if and only if
$q$ is prime to $p$, while the Coxeter group $W(A_n)$ $(n\geq 2)$
is $p$-free for $p\geq 5$.
For every finite irreducible Coxeter group $W$,
the range of odd prime numbers $p$ such that $W$ is $p$-free can be found in Appendix.
Note that parabolic subgroups of $p$-free Coxeter groups are also $p$-free.
Henceforth, we omit the reference to the Coxeter graph $\Gamma$
and the set of Coxeter generators $S$
if there is no ambiguity.

\subsection{Known results for homology of Coxeter groups}\label{sec-known}
In this subsection, we will review some of known results concerning
the homology of Coxeter groups which are related to our paper.
A basic reference for (co)homology of groups is \cite{brown}.
In the beginning,
Serre \cite{serre} proved that every Coxeter group $W$ has
finite virtual cohomological dimension and is a group of type WFL
(see \cite{brown}*{Chapter VIII} for definitions).
This implies, in particular, that $H_k(W,\Z)$ is a
finitely generated abelian group for all $k$.
On the other hand, the rational homology of any Coxeter groups
are known to be trivial (see \cite[Proposition 5.2]{a-euler} or \cite[Theorem 15.1.1]{davis-book}).
Combining these results, we obtain the following result:

\begin{pro}\label{finite-abel}
For any Coxeter groups $W$, the integral homology group
$H_k(W,\Z)$ is a finite abelian group for all $k>0$.
\end{pro}

Consequently,
$H_k(W,\Z)$ $(k>0)$
decomposes into a finite direct sum
\begin{equation}\label{direct-sum}
H_k(W,\Z)\cong\bigoplus_p H_k(W,\Z)_{(p)}
\end{equation}
where $p$ runs the finite set of prime numbers dividing the order of
$H_k(W,\Z)$.
The universal coefficient theorem implies
\begin{equation}\label{univ-p-local}
H_k(W,\Z)_{(p)}\cong H_k(W,\Z)\otimes_\Z\Zp\cong H_k(W,\Zp)
\end{equation}
(see \cite[Corollary 2.3.3]{MR2378355}).
It turns out that the study of the integral homology groups of Coxeter groups reduces
to the study of the $p$-local homology groups.
Later, we will prove that $H_k(W,\Zp)=0$ $(k>0)$ if $W$ has
no $p$-torsion (Proposition \ref{p-tor-coho}).
The first and second integral homology of Coxeter groups are known.
\begin{pro}\label{low-homology}
For any Coxeter groups $W$, we have $H_1(W,\Z)\cong (\Z/2)^{n_1(W)}$
and $H_2(W,\Z)\cong (\Z/2)^{n_2(W)}$ for some non-negative integers
$n_1(W),n_2(W)$.
\end{pro}
The claim for $H_1(W,\Z)$ is obvious because $H_1(W,\Z)=W/[W,W]$ and
$W$ is generated by elements
of order $2$. The statement for $H_2(W,\Z)$ was proved by Howlett \cite{howlett}
(following earlier works by Ihara and Yokonuma \cite{ihara-yokonuma} and
Yokonuma \cite{yokonuma}).
The nonnegative integers $n_1(W),n_2(W)$ can be computed from the Coxeter graph
for $W$.
As for $n_1(W)$, let $\mathscr{G}_W$ be the graph whose vertices
set is $S$ and whose edges are unordered pair $\{s,t\}\subset S$ such that
$m(s,t)$ is a finite odd integer. Then it is easy to see that $n_1(W)$ agrees with
the number of connected components of $\mathscr{G}_W$.
In particular, $n_1(W)\geq 1$ and hence $H_1(W,\Z)=H_1(W,\Z_{(2)})\not=0$.
For the presentation of $n_2(W)$, see \cite{howlett}*{Theorem A} or \cite[\S8.11]{humphreys}.
As a consequence of Proposition \ref{low-homology}, we obtain the following result:

\begin{cor}\label{low-p-homology}
Let $p$ be an odd prime number.
For any Coxeter groups $W$, we have
$H_1(W,\Zp)=H_2(W,\Zp)=0$.
\end{cor}

The corollary does not hold for the third homology or higher.
Indeed, for the Coxeter group $W(I_2(q))$ of type $I_2(q)$, which is isomorphic to
the dihedral group of order $2q$ as mentioned before,
it can be proved that, if $p$ divides $q$, then
$H_k(W(I_2(q)),\Zp)\not=0$ whenever $k\equiv 3\pmod{4}$
(see \cite{MR1877725}*{Theorem 2.1} and \S\ref{sec:aspherical} below).
This observation also shows the necessity of the $p$-freeness assumption in our results
for $p\geq 5$.
Finally, we will recall a consequence of results of Nakaoka
\cites{nakaoka,nakaoka2} which was mentioned in the introduction.

\begin{thm}[Nakaoka \cites{nakaoka,nakaoka2}]\label{nakaoka-result}
Let $\symm_n$ be the symmetric group of $n$ letters. Then
$H_k(\symm_n,\Zp)=0$ $(1\leq k\leq 2(p-2))$ for all odd prime numbers $p$.
\end{thm}
\begin{proof}
In his paper \cite{nakaoka}, Nakaoka proved the homology stability for
symmetric groups.
Namely, for $2\leq m\leq n\leq\infty$, the homomorphism
$H_k(\symm_m,A)\rightarrow H_k(\symm_n,A)$ induced by
the natural inclusion $\symm_m\hookrightarrow\symm_n$ is injective for all $k$,
and is an isomorphism if $k<(m+1)/2$, where $A$ is an abelian group with the trivial
$\symm_n$-action, and $\symm_\infty$ is the infinite symmetric group
\cite[Theorem 5.8 and Corollary 6.7]{nakaoka}.
He also computed the mod $p$ homology of $\symm_\infty$ in
\cite[Theorem 7.1]{nakaoka2}, from which we deduce
that $H_k(\symm_\infty,\Z/p)=0$ for
$1\leq k\leq 2(p-2)$ and that $H_{2p-3}(\symm_\infty,\Z/p)\not=0$.
Combining these results, we see that $H_k(\symm_n,\Z/p)=0$
$(1\leq k\leq 2(p-2))$ for all $n$.
Applying the universal coefficient theorem, the theorem follows.
\end{proof}

Theorem \ref{main-thm}, together with Corollary \ref{low-p-homology} for $p=3$,
generalize Theorem \ref{nakaoka-result} to all Coxeter groups.
For further results concerning of (co)homology of Coxeter groups,
we refer the book by Davis \cite{davis-book} and
papers \cites{MR1334713,MR1603123,akita-quillen,MR1764318,MR1446573,pride-stohr,MR1877725,MR2709085}
as well as references therein.

\section{Coxeter complexes and their equivariant homology}\label{sec-equivariant}
\subsection{Coxeter complexes}\label{complex}
We recall the definition and properties of Coxeter complexes
which are relevant to prove Theorem \ref{main-thm}.
A basic reference for Coxeter complexes is \cite{book-building}*{Chapter 3}.
Given a Coxeter group $W$, the \emph{Coxeter complex}
$X_W$ of $W$ is the poset of
cosets $wW_T$ $(w\in W, T\subsetneq S)$,
ordered by reverse inclusion.
It is known that $X_W$ is actually an $(|S|-1)$-dimensional
simplicial complex (see \cite[Theorem 3.5]{book-building}).
The $k$-simplices of $X_W$ are the cosets $wW_T$ with $k=|S|-|T|-1$.
A coset $wW_T$ is a face of $w'W_{T'}$ if and only if $wW_T\supseteq w'W_{T'}$.
In particular, the vertices are cosets of the form $wW_{S\setminus\{s\}}$
$(s\in S,w\in W)$,
the maximal simplices are the singletons $wW_\varnothing=\{w\}$ $(w\in W)$,
and the codimension one simplices are cosets of the form
$wW_{\{s\}}=\{w,ws\}$ $(s\in S, w\in W)$.
In what follows, we will not distinguish between $X_W$ and its
geometric realization.

There is a simplicial action of $W$ on $X_W$ by left translation
$w'\cdot wW_T:=w'wW_T$. The isotropy subgroup of a simplex $wW_T$ is
precisely $wW_T w^{-1}$, which
fixes $wW_T$ pointwise.
Next, consider the subcomplex $\Delta_W=\{W_T\ |\ T\subsetneq S\}$ of $X_W$,
which consists of a single $(|S|-1)$-simplex $W_\varnothing$ and its faces.
Since the \emph{type function}  $X_W\rightarrow S$,
$wW_T\mapsto S\setminus T$ is well-defined
(see \cite[Definition 3.6]{book-building}), $\Delta_W$ forms the
set of representatives of $W$-orbits of simplices of $X_W$.
The following fact is well-known.
\begin{pro}
If $W$ is a finite Coxeter group, then $X_W$ is a triangulation of the
$(|S|-1)$-dimensional sphere $\sph^{|S|-1}$.
\end{pro}
See \cite[Proposition 1.108]{book-building} for the proof.
Alternatively, $W$ can be realized
as an orthogonal reflection group on the $|S|$-dimensional Euclidean space
$\R^{|S|}$ and hence it acts on the unit sphere $\sph^{|S|-1}$.
Each $s\in S$ acts on $\sph^{|S|-1}$
as an orthogonal reflection.
The Coxeter complex $X_W$ coincides
with the equivariant triangulation of $\sph^{|S|-1}$ cut out by the
reflection hyperplanes for $W$.
 In case $W$ is infinite, Serre proved the following result:
\begin{pro}[{\cite[Lemma 4]{serre}}]
\label{serre-cont} If $W$ is an infinite Coxeter group,
then $X_W$ is contractible.
\end{pro}

\subsection{Equivariant homology of Coxeter complexes}
Given a Coxeter group $W$,
let $H_k^W(X_W,\Zp)$ be the $k$-th equivariant homology group of $X_W$
(see \cite[Chapter VII]{brown} for the definition).
If $X_W$ is infinite, then $X_W$ is contractible so that the equivariant homology is
isomorphic to the homology of $W$:
\begin{pro}\label{compare0}
If $W$ is an infinite Coxeter group, then
\[H_k^W(X_W,\Zp)\cong H_k(W,\Zp)\]
for all $k$.
\end{pro}
If $W$ is finite, then $H_k^W(X_W,\Zp)$ may not be isomorphic to $H_k(W,\Zp)$,
however, they are isomorphic if $k$ is relatively small:
\begin{pro}\label{compare}
If $W$ is a finite Coxeter group, then
\[
H_k^W(X_W,\Zp)\cong H_k(W,\Zp)
\]
for  $k\leq \rank\,W-1$.
\end{pro}
\begin{proof}
Consider the spectral sequence
\[
E_{i,j}^2=H_i(W,H_j(X_W,\Zp))\Rightarrow H_{i+j}^W(X_W,\Zp)
\]
(see \cite[VII.7]{brown})
and note that $E_{i,0}^2\cong H_i(W,\Zp)$ for all $i$.
Since $X_W$ is homeomorphic to $\sph^{|S|-1}$, we have $E_{i,j}^2=0$ for
$j\not=0,|S|-1$. Hence $H_k^W(X_W,\Zp)\cong H_k(W,\Zp)$ for
$k\leq |S|-2$. Now
\[
E^2_{0,|S|-1}=H_0(W,H_{|S|-1}(X_W,\Zp))= H_{|S|-1}(X_W,\Zp)_W
\]
where the RHS is the co-invariant of $H_{|S|-1}(X_W,\Zp)$ as a $W$-module
(see \cite[III.1]{brown}).
Since each $s\in S$ acts on $X_W\approx\sph^{|S|-1}$ as an
orthogonal reflection as mentioned in \S\ref{complex},
it acts on $H_{|S|-1}(X_W,\Zp)\cong\Zp$ as the multiplication
by $-1$. It follows that the co-invariant $H_{|S|-1}(X_W,\Zp)_W$ is isomorphic to the quotient
group of $\Zp$ by the subgroup generated by $r-(-1)r=2r$ $(r\in\Zp)$.
But this subgroup is nothing but the whole group $\Zp$ because $2$ is invertible in $\Zp$.
This proves $E^2_{0,|S|-1}=0$ and hence
\[
H_{|S|-1}^W(X_W,\Zp)\cong H_{|S|-1}(W,\Zp)
\]
as desired.
\end{proof}

\section{Proof of Theorem \ref{main-thm}}\label{sec-proof}
We will prove Theorem \ref{main-thm} by showing the following two claims:
\begin{claim}\label{claim1}
If $W$ is a finite $p$-free Coxeter group with $\rank\,W\leq 2(p-2)$,
then $H_k(W,\Zp)=0$ for $1\leq k\leq 2(p-2)$.
\end{claim}
\begin{claim}\label{claim2}
Claim \ref{claim1} implies Theorem \ref{main-thm}.
\end{claim}
The first claim is equivalent to Theorem \ref{main-thm} for finite $p$-free
Coxeter groups with $\rank\,W\leq 2(p-2)$, and will be proved by a case by case argument.
The second claim will be proved by the induction on $\rank\,W$ by
using the equivariant homology of Coxeter complexes.
Let us prove Claim \ref{claim2} first.

\subsection{Proof of Claim \ref{claim2}}
For every Coxeter group $W$,
there is a spectral sequence
\begin{equation}\label{leray-ori}
E^1_{i,j}=\bigoplus_{\sigma\in\mathscr{S}_i}
H_j(W_\sigma,\Zp)\Rightarrow H_{i+j}^W(X_W,\Zp),
\end{equation}
where $\mathscr{S}_i$ is the set of representatives of $W$-orbits of $i$-simplices of $X_W$,
and $W_\sigma$ is the isotropy subgroup of an $i$-simplex $\sigma$
(see \cite[VII.7]{brown}).
It is the Leray spectral sequence for the natural projection
$EW\times_W X_W\rightarrow X_W/W$.
Note that $\Zp$ in $H_j(W_\sigma,\Zp)$ is the trivial $W_\sigma$-module
because $W_\sigma$ fixes $\sigma$ pointwise.
We may choose the subset
$\{W_T\ |\ T\subsetneq S, |T|=|S|-i-1\}$
(the set of $i$-simplices of $\Delta_W$) as $\mathscr{S}_i$,
and the spectral sequence can be rewritten as
\begin{equation}\label{leray}
E^1_{i,j}=\bigoplus_{\substack{T\subsetneq S \\ |T|=|S|-i-1}}
H_j(W_T,\Zp)\Rightarrow H_{i+j}^W(X_W,\Zp).
\end{equation}
\begin{lem}\label{base-term}
In the spectral sequence (\ref{leray}), $E^2_{i,0}=0$ for $i\not=0$ and
$E^2_{0,0}\cong\Zp$.
\end{lem}
\begin{proof}
We claim $E^2_{i,0}\cong H_i(\Delta_W,\Zp)$ for all $i$, which implies
the lemma because $\Delta_W$ is an $(|S|-1)$-simplex and hence contractible.
Although such a claim may be familiar to experts, we write down the proof
for completeness.
To show the claim, we recall the construction of the spectral sequence
(\ref{leray-ori}) given in \cite{brown}*{VII.7}.
At the first stage, the $E^1_{i,0}$-term of (\ref{leray-ori}) is given by
\[
E^1_{i,0}=H_0(W,C_i(X_W,\Zp))=C_i(X_W,\Zp)_W,
\]
which is isomorphic to the one in (\ref{leray-ori}) due to Eckmann-Shapiro lemma.
The differential $d^1:E^1_{i,0}\rightarrow E^1_{i-1,0}$
is the map induced by the boundary operator
$C_i(X_W,\Zp)\rightarrow C_{i-1}(X_W,\Zp)$.
On the other hand, the composition
\begin{equation}\label{fundom}
C_i(\Delta_W,\Zp)\hookrightarrow C_i(X_W,\Zp)
\twoheadrightarrow C_i(X_W,\Zp)_W
\end{equation}
is an isomorphism, where the first map is induced by the inclusion
$\Delta_W\hookrightarrow X_W$ and the second map is the natural
projection, because the subcomplex $\Delta_W$ forms the
set of representatives of $W$-orbits of simplices of $X_W$.
Moreover, the isomorphism (\ref{fundom}) is compatible
with the boundary operator of $C_i(\Delta_W,\Zp)$ and
the differential on $C_i(X_W,\Zp)_W$.
In other words, (\ref{fundom}) yields a chain isomorphism
of chain complexes
\[
(C_i(\Delta_W,\Zp),\partial)\rightarrow (C_i(X_W,\Zp)_W,d^1).
\]
The claim follows immediately.
\end{proof}

\begin{proof}[Proof of Claim \ref{claim2}]
We argue by the induction on $|S|$.
When $W$ is finite, we may assume $|S|> 2(p-2)$, for we suppose
that Claim \ref{claim1} holds.
Consider the spectral sequence (\ref{leray}).
Observe first that all $W_T$'s appearing in (\ref{leray}) are $p$-free and satisfy
$|T|<|S|$.
By the induction assumption, we have $H_j(W_T,\Zp)=0$
$(1\leq j\leq 2(p-2))$ for
all $T\subsetneq S$, which implies $E_{i,j}^1=E_{i,j}^2=0$ for $1\leq j\leq 2(p-2)$.
Moreover, $E^2_{i,0}=0$ for $i>0$ by Lemma \ref{base-term}. This proves
$H_k^W(X_W,\Zp)=0$ for $1\leq k\leq 2(p-2)$. Now Claim \ref{claim2} follows from
Proposition \ref{compare0} and \ref{compare}.
\end{proof}

\subsection{Proof of Claim \ref{claim1}}
Given an odd prime $p$,
if $W$ is a finite $p$-free Coxeter group with $\rank\,W\leq 2(p-2)$,
then $W$ decomposes into the direct product of
finite irreducible $p$-free Coxeter groups
$W\cong W_1\times\cdots\times W_r$
with $\Sigma_{i=1}^r\rank\,W_i=\rank\,W$.
Since $\Zp$ is PID, we may apply the K\"unneth theorem to conclude that
Claim \ref{claim1} is equivalent to the following claim:
\begin{claim}\label{claim3}
If $W$ is a finite irreducible $p$-free Coxeter group with $\rank\,W\leq 2(p-2)$,
then $H_k(W,\Zp)$ vanishes for $1\leq k\leq 2(p-2)$.
\end{claim}

We prove Claim \ref{claim3} for each finite irreducible Coxeter group.
Firstly, the Coxeter group $W(I_2(q))$ of type $I_2(q)$ is $p$-free if and only if
$q$ is prime to $p$. If so, $H_*(W(I_2(q)),\Zp)=0$ for $*>0$ because the order of
$W(I_2(q))$ is $2q$ and hence having no $p$-torsion.
Next, we prove the claim for the Coxeter group of type $A_n$.
To do so, we deal with cohomology instead of homology.
We invoke the following elementary lemma:
\begin{lem}\label{duality}
 Let $G$ be a finite group and $p\geq 2$ a prime. Then
$H_k(G,\Zp)\cong H^{k+1}(G,\Z)_{(p)}$ for all $k\geq 1$.
\end{lem}

Now Claim \ref{claim3} for $W(A_n)$ can be proved by applying standard arguments
in cohomology of finite groups:

\begin{lem}\label{case-symm}
$H_k(W(A_n),\Zp)=0$ $(1\leq k \leq 2(p-2))$ holds for all $n$ with
$1\leq n \leq 2(p-1)$.
\end{lem}
\begin{proof}
Recall that $W(A_n)$ is isomorphic to the symmetric group of
$n+1$ letters $\symm_{n+1}$.
If $n<p$ then $\symm_n$ has no $p$-torsion and hence
$H_k(\symm_n,\Zp)=0$ for all $k>0$.
Now suppose $p\leq n\leq 2p-1$, and let $C_p$ be a Sylow $p$-subgroup
of $\symm_n$, which is a cyclic group of order $p$. Then $H^*(C_p,\Z)\cong
\Z[u]/(pu)$ where $\deg u=2$.
Let $N_p$ be the normalizer
of $C_p$ in $\symm_n$.
It acts on $C_p$ by conjugation, and the induced map
$N_p\rightarrow\Aut (C_p)\cong (\Z/p)^\times$ is known to be surjective.
Consequently, the invariant $H^*(C_p,\Z)^{N_p}$ is the subring generated by $u^{p-1}$.
Since $C_p$ is abelian, the restriction $H^*(\symm_n,\Z)\rightarrow
H^*(C_p,\Z)$ induces the isomorphism
\[
H^k(\symm_n,\Z)_{(p)}\cong H^k(C_p,\Z)^{N_p}
\]
for $k>0$ by a result
of Swan \cite{swan}*{Lemma 1 and Appendix} (see also \cite{thomas-book}*{Lemma 3.4}).
This proves, for $p\leq n\leq 2p-1$, that $H^k(\symm_n,\Z)_{(p)}=0$ $(0<k<2p-2)$ and
$H^{2p-2}(\symm_n,\Z)_{(p)}\not=0$.
In view of Lemma \ref{duality}, the proposition follows.
\end{proof}

\begin{rem}\label{best-possible}
Since $H_{2p-3}(W(A_{p-1}),\Zp)\cong H^{2p-2}(\symm_p,\Z)_{(p)}\not=0$
for all prime numbers $p$ as was observed in the proof of Lemma \ref{case-symm},
the vanishing range $1\leq k\leq 2(p-2)$ in our theorem is best possible for $p\geq 5$.
\end{rem}

\begin{rem}
Of course, Lemma \ref{case-symm} is a direct consequence of
Theorem \ref{nakaoka-result}, however, we avoid the use of Theorem \ref{nakaoka-result}
for two reasons:
Firstly, by doing so, we provide an alternative proof of Theorem \ref{nakaoka-result}.
Secondly, the proof of Lemma \ref{case-symm} is much simpler than that of
Theorem \ref{nakaoka-result}, for the latter relies on the homology stability for
symmetric groups and the computation of $H_*(\symm_\infty,\Z/p)$.
\end{rem}

Claim \ref{claim3} for the Coxeter groups of type $B_n$ and $D_n$ follows from
Lemma \ref{case-symm} and the following proposition:
\begin{pro}\label{ABD} For any odd prime number $p$,
\[
H_*(W(B_n),\Zp)\cong H_*(W(A_{n-1}),\Zp)
\]
holds for all $n\geq 2$, and
\[
H_*(W(D_n),\Zp)\cong H_*(W(A_{n-1}),\Zp)
\]
holds for all $n\geq 4$.
\end{pro}
\begin{proof}
Recall that the Coxeter group $W(B_n)$ is isomorphic to the semi-direct product
$(\Z/2)^n \rtimes W(A_{n-1})$ (see \cite{davis-book}*{\S6.7} or \cite[\S1.1]{humphreys}).
In the Lyndon-Hochschild-Serre spectral sequence
\[
E^2_{i,j}=H_i(W(A_{n-1}),H_j((\Z/2)^n,\Zp))\Rightarrow H_{i+j}(W(B_n),\Zp),
\]
one has $E^2_{i,j}=0$ for $j\not=0$ since $H_j((\Z/2)^n,\Zp)=0$ for $j\not=0$.
This proves $H_*(W(B_n),\Zp)\cong H_*(W(A_{n-1}),\Zp)$.
On the other hand,
$W(D_n)$ is known to be isomorphic to the semi-direct product
$(\Z/2)^{n-1} \rtimes W(A_{n-1})$ (see loc.~cit.), and the proof for
$H_*(W(D_n),\Zp)\cong H_*(W(A_{n-1}),\Zp)$ is similar.
\end{proof}
These observations prove Claim \ref{claim3} for $p\geq 11$,
for all finite irreducible Coxeter groups of type other than $A_n,B_n,D_n$ and $I_2(q)$
have no $p$-torsion for $p\geq 11$.
The case $p=3$ follows from Corollary \ref{low-p-homology}.
Now we will prove the cases $p=5$ and $p=7$.
Observe that, apart from Coxeter groups of type $A_n$, $B_n$, $D_n$ and $I_2(q)$,
finite irreducible $p$-free Coxeter groups, with rank at most $2(p-2)$ and having $p$-torsion, are
$W(E_6)$ for $p=5$, $W(E_7)$ and $W(E_8)$ for $p=7$.
So the proof of Claim \ref{claim3} is completed by showing the following lemma:

\begin{lem}
$H_k(W(E_6),\Z_{(5)})$ vanishes for $1\leq k\leq 6$, while
$H_k(W(E_7),\Z_{(7)})$ and $H_k(W(E_8),\Z_{(7)})$ vanish for $1\leq k\leq 10$.
\end{lem}
\begin{proof}
The Coxeter group $W(A_4)$ is a parabolic subgroup of $W(E_6)$, and they have a common
Sylow $5$-subgroup $C_5$, which is a cyclic group of order $5$.
The transfer homomorphism to the Sylow $5$-subgroup
$H_k(W(E_6),\Z_{(5)})\rightarrow H_k(C_5,\Z_{(5)})$ is injective and factors
into a composition
of transfer homomorphisms
\[
H_k(W(E_6),\Z_{(5)})\rightarrow H_k(W(A_4),\Z_{(5)})\rightarrow H_k(C_5,\Z_{(5)}).
\]
In view of Lemma \ref{case-symm}, we conclude that $H_k(W(E_6),\Z_{(5)})=0$ for
$1\leq k\leq 6$, which proves the lemma for $W(E_6)$.
On the other hand, there is a sequence of parabolic subgroups $W(A_6)<W(E_7)<W(E_8)$,
and they have a common Sylow $7$-subgroup $C_7$, which is a cyclic group of order $7$.
The proof of the lemma for $W(E_7)$ and $W(E_8)$ is similar.
\end{proof}

\section{Coxeter groups with vanishing $p$-local homology}\label{sec-vanishing}
In this final section, we introduce some families of Coxeter groups such that
$H_k(W,\Zp)$ vanishes for all $k>0$.

\subsection{Aspherical Coxeter groups}\label{sec:aspherical}
A Coxeter group $W$ is called \emph{aspherical} in \cite{pride-stohr} if,
for all distinct Coxeter generators $s,t,u\in S$, the inequality
\[
\frac{1}{m(s,t)}+\frac{1}{m(t,u)}+\frac{1}{m(u,s)}\leq 1
\]
holds, where $1/\infty=0$ by the convention.
The inequality is equivalent to the condition that the parabolic subgroup
$W_{\{s,t,u\}}$ is of infinite order. The (co)homology groups of aspherical Coxeter
groups were studied by Pride and St{\"o}hr \cite{pride-stohr},
and the mod $2$ cohomology rings of aspherical Coxeter groups were
studied by the author \cite{akita-quillen}.
Among other things,
Pride and St{\"o}hr obtained the following exact sequence
\[
\cdots\rightarrow H_{k+1}(W,A)\rightarrow \bigoplus_{s\in S}H_k(W_{\{s\}},A)^{
\oplus n(s)}
\rightarrow\!\bigoplus_{\substack{\{s,t\}\subset S \\ m(s,t)<\infty}}\!
H_k(W_{\{s,t\}},A)\rightarrow H_k(W,A)\rightarrow\cdots
\]
terminating at $H_2(W,A)$, where $A$ is a $W$-module and $n(s)$ is a certain
nonnegative integer defined for each $s\in S$
\cite[Theorem 5]{pride-stohr}.
Since $W_{\{s\}}\cong\Z/2$, $H_k(W_{\{s\}},\Zp)=0$ for $k>0$.
Moreover, if $p$ does not divide $m(s,t)$, then $H_k(W_{\{s,t\}},\Zp)=0$ for $k>0$ either.
Here no prime numbers $p$ divide $\infty$ by the convention.
Hence we obtain the following result
(the statement for $k=1,2$ follows from Corollary  \ref{low-p-homology}):
\begin{pro}
For any aspherical Coxeter groups $W$, we have
\[
H_k(W,\Zp)\cong
\bigoplus_{\substack{\{s,t\}\subset S \\ p\,|\,m(s,t)}}
H_k(W_{\{s,t\}},\Zp)
\]
for all $k>0$. Furthermore, $H_k(W,\Zp)$ vanishes for all $k>0$
if and only if $W$ is $p$-free.
\end{pro}
Note that if $p$ divides $m(s,t)$, then
\[
H_k(W_{\{s,t\}},\Zp)\cong
\begin{cases}(\Z/m(s,t))_{(p)} & k\equiv 3\pmod{4} \\
0 & k\not\equiv 3\pmod{4}
\end{cases}
\]
for $k>0$, where $\Z/m(s,t)$ is the cyclic group of order $m(s,t)$
(see \cite{MR1877725}*{Theorem 2.1}).
\subsection{Coxeter groups without $p$-torsion}
Next we prove vanishing of the $p$-local homology of Coxeter groups without $p$-torsion.
Before doing so, we characterize such Coxeter groups in terms of their finite
parabolic subgroups.
\begin{pro}\label{p-tor}
Let $p$ be a prime number.
A Coxeter group $W$ has no $p$-torsion
if and only if every finite parabolic subgroup
has no $p$-torsion.
\end{pro}
\begin{proof}
According to a result of Tits,
every finite subgroup of $W$ is contained in
conjugate of some parabolic subgroup of finite order
(see \cite[Corollary D.2.9]{davis-book}).
The proposition follows at once.
\end{proof}
\begin{pro}\label{p-tor-coho}
If $W$ is a Coxeter group without $p$-torsion, then
$H_k(W,\Zp)=0$ for all $k>0$.
\end{pro}
\begin{proof}
The claim is obvious for finite Coxeter groups.
We prove the proposition for infinite Coxeter groups by the induction on $|S|$.
Let $W$ be an infinite Coxeter group without $p$-torsion and consider the
spectral sequence (\ref{leray}).
Every proper parabolic subgroup $W_T$ of $W$ has no $p$-torsion,
and hence $H_k(W_T,\Zp)=0$ $(k>0)$ by the induction assumption.
This implies $E^1_{i,j}=0$ for $j\not=0$. Moreover
$E^2_{i,j}=0$ for $(i,j)\not=(0,0)$ by Lemma \ref{base-term},
which proves the proposition.
\end{proof}
In view of the last proposition, the direct sum decomposition
(\ref{direct-sum}) can be replaced to the following:
\begin{cor}\label{cor-decomp}
For any Coxeter groups $W$ and $k>0$, we have
\[
H_k(W,\Z)\cong\bigoplus_p H_k(W,\Zp),
\]
where $p$ runs prime numbers such that $W$ has $p$-torsion.
\end{cor}
\begin{rem}
Proposition \ref{p-tor-coho} and Corollary \ref{cor-decomp}
should be compared with the following general results.
Namely, suppose that $\Gamma$ is a group having
finite virtual cohomological dimension $\vcd\,\Gamma$.
If $\Gamma$ does not have $p$-torsion, then $H^k(\Gamma,\Z)_{(p)}=0$
for $k>\vcd\,\Gamma$.
Consequently, we have the finite direct product decomposition
\[
H^k(\Gamma,\Z)\cong\prod_p H^k(\Gamma,\Z)_{(p)}
\]
which holds for $k>\vcd\,\Gamma$,
where $p$ ranges over the prime numbers such that
$\Gamma$ has $p$-torsion. See \cite[Chapter X]{brown}.
\end{rem}

\subsection{Right-angled Coxeter groups}
A Coxeter group is called \emph{right-angled} if $m(s,t)=2$ or $\infty$
for all distinct $s,t\in S$. The mod $2$ cohomology rings of
right-angled Coxeter groups were determined by Rusin \cite{rusin-thesis}
(see also \cite[Theorem 15.1.4]{davis-book}).
In this section, we prove vanishing of $p$-local homology for a class of Coxeter groups
which includes right-angled Coxeter groups.

\begin{pro}
If\, $W$ is a Coxeter group such that $m(s,t)$ equals to the power of 2 or $\infty$
for all distinct $s,t\in S$,
then $H_k(W,\Zp)=0$ $(k>0)$ for all odd prime numbers $p\geq 3$.
\end{pro}
\begin{proof}
The finite irreducible Coxeter groups satisfying the assumption
are $W(A_1)$ (of order $2$), $W(B_2)$ (of order $8$), and
$W(I_2(2^m))$'s (of order $2^{m+1}$).
Every finite parabolic subgroup of $W$ is isomorphic to a direct product
of copies of those groups and hence has the order the power of $2$.
Consequently, $W$ has no $p$-torsion by Proposition \ref{p-tor}.
Now the proposition follows from Proposition \ref{p-tor-coho}.
\end{proof}

\subsection{$3$-free Coxeter groups}\label{sec-3-free}
In this final section, we look into situations for $p=3$ more closely.
Firstly, according to Corollary \ref{low-p-homology},
$H_1(W,\Z_{(3)})=H_2(W,\Z_{(3)})=0$ for any Coxeter groups $W$.
This means that Theorem \ref{main-thm} remains true for $p=3$ without
$3$-freeness assumption.
On the other hand, the finite irreducible $3$-free Coxeter groups are
$W(A_1),W(B_2)$ and $W(I_2(q))$ such that $q$ is prime to $3$, all of which
have no $3$-torsion. Consequently, every $3$-free Coxeter group has
no $3$-torsion by Proposition \ref{p-tor}. Applying Proposition \ref{p-tor-coho}
we obtain the following result:
\begin{pro}\label{pro-3-free}
For every $3$-free Coxeter group,
$H_k(W,\Z_{(3)})=0$ holds for all $k>0$.
\end{pro}

\section*{Appendix}
The following is the table for the Coxeter graph $\Gamma$,
the order $|W(\Gamma)|$ of the corresponding
Coxeter group $W(\Gamma)$, the order $|W(\Gamma)|$ factored into primes, and
the range of odd prime numbers $p$ such that $W(\Gamma)$ is $p$-free.
\[
\begin{array}{cccc}
\Gamma & |W(\Gamma)| && \text{$p$-freeness} \\ \hline
A_1 & 2 && p\geq 3 \\
A_n\,(n\geq 2) & (n+1)! && p\geq 5 \\
B_2 & 8 && p\geq 3\\
B_n\,(n\geq 3) & 2^n n! && p\geq 5\\
D_n\,(n\geq 4) & 2^{n-1} n! && p\geq 5\\
E_6 & 72\cdot 6! & 2^7\cdot 3^4\cdot 5 & p\geq 5 \\
E_7 & 72\cdot 8! & 2^{10}\cdot 3^4\cdot 5\cdot 7 & p\geq 5 \\
E_8 & 192\cdot 10! & 2^{14}\cdot 3^5\cdot 5^2\cdot 7 & p\geq 5 \\
F_4 & 1152 & 2^7\cdot 3^2 & p\geq 5 \\
H_3 & 120 & 2^3\cdot 3\cdot 5 & p\geq 7 \\
H_4 & 14400 & 2^6\cdot 3^2\cdot 5^2 & p\geq 7 \\
I_2(q)\,(q\geq 3) & 2q && p\not|\ q
\end{array}
\]

\begin{ack}
This study started from questions posed by Takefumi Nosaka
concerning of the third $p$-local homology of Coxeter groups for
odd prime numbers $p$.
The author thanks to him for drawing our attention.
This study was partially supported by JSPS KAKENHI Grant Numbers
23654018 and 26400077.
\end{ack}

\end{document}


TITLE: Why must a function be closed for descent methods to work?
QUESTION [4 upvotes]: At the beginning of chapter 9 of Boyd & Vandenberghe's Convex Optimization, about unconstrained minimization, it is said:

The starting point  $x^{(0)}$ for a method must lie in $\mathbf{dom} f$, and in addition the sublevel set $$S = \{x \in \mbox{dom} \, f \mid f(x) \leq f(x^{(0)}) \}$$ must be closed. This is satisfied for all $x \in \mathbf{dom} f$ if the function is closed.

In other paper I've found an assumption, that a function $f$ must be convex, proper and closed in order that: minimize f is solvable. I understand why are the first two assumptions needed, but why is it important for the function to be closed so that problem is solvable?
I would appreciate any explanation for the intuition (and what does this mean for the epigrpah for the objective function).

REPLY [0 votes]: Another example is minimizing $f(x) = 1/x$. The minimum exists, and is 0, but the minimizer does not exist, and most standard methods behave oddly on this problem. The function's epigraph is
$$
\mathbf{epi}~f = \{(x,f(x)): x\in\mathbb R\}
$$
and does not include the point $(x,0)$ for any value of $x$ (but the sequence $(n,f(n))$ is in the epigraph and approaches the x axis). Therefore the epigraph is not closed, and the function is not closed.


TITLE: Disproof of an attempted disproof of the pythagorean theorem because of recursively defined path
QUESTION [0 upvotes]: Here is a video that attempts to state that the Pythagorean theorem is wrong:
https://www.youtube.com/watch?v=-rRaAgjU5kw
I know he must be making an incorrect assumption but I have gotten a lot of contesting reasons from other forums. So could someone provide a clear answer as to what is wrong with this?

REPLY [0 votes]: These curves used to be called pathological curves, now they are called fractal curves.
As you go to infinity, the stairwell curve never actually becomes a straight line, that is, it never becomes smooth. Because it's not really a straight line, the pythagorean theorem does not apply. The actual definition of smooth requires calculus and is complicated. There are even different degrees of smoothness.
Fractal curves are not smooth at all. They might look smooth, but that is a problem with the way our eyes and brains work and the actual practical matters involved in the way we graph functions.
Let't take a more technical look at this stair curve approximation to the diagonal. We need to show why, even after an infinite number of iterations, the resulting curve is not a straight line.

Just to make this discussion easier, let's assume that we have put a ruler against the diagonal so that the coordinate of the first point is $0_2 = \dfrac 01$ and the coordinate of the last point is $0.1_2 = \dfrac 12$.
The image above shows the first four iterations of the stair curve. The points that make of the stair curve are counted in base two from $0.0_2$ to $0.1_2$ (the radix is not shown) in increments of $2^{-n}$.
This implies that, even when we go to infinity, the number of points of the stair curve that are actually on the diagonal will be countable. Hence, even when we go to infinity, the stair curve will be different from the diagonal.
$$ \lim_{n \to \infty} S_n \ne \overline{A_0A_{\frac 12}} $$


TITLE: Prove $\nabla_X (w(Y)) = (\nabla_X w)(Y) + w(\nabla_X Y)$ directly
QUESTION [0 upvotes]: Suppose $w$ is one form and $X,Y$ are vectors. For affine connection $\nabla_X$, From its definition
\begin{align}
(\nabla_X T)_p = \lim_{s\rightarrow 0} \frac{1}{s} ( \tau_{s}^{-1} T_{\varphi(s)}-T_p)
\end{align}
where $T$ is a tensor type of $(p,q)$ and $\tau$ is the parallel translation along curve $\gamma$
From this definition I want to prove
\begin{align}
\nabla_X (w(Y)) = (\nabla_X w)(Y) + w(\nabla_X Y)
\end{align}
this via definition.
My trials are
\begin{align}
(\nabla_X (w(Y)))_p= \lim_{s\rightarrow 0} \frac{1}{s} ( \tau_{s}^{-1} w(Y)_{\varphi(s)}-w(Y)_p)
\end{align}
But having trouble in expressing this into right hand sides..

REPLY [1 votes]: You might be confused by the fact that you are denoting by the same symbol $\tau$ the parallel transport of any tensor along $\gamma$ while in reality you actually parallel transport tensors of different types differently.
Let's denote the parallel transport of tangent vectors along $\gamma$ from $\gamma(0)$ to $\gamma(s)$ by $\tau_s \colon T_{\gamma(0)} M \rightarrow T_{\gamma(s)} M$. Also denote the parallel transport of covectors along $\gamma$ by $\mu_s \colon T_{\gamma(0)}^{*} M \rightarrow T_{\gamma(s)}^{*} M$. The relation between $\mu_s$ and $\tau_s$ is given by
$$ \mu_s(\varphi) = \varphi \circ \tau_s^{-1} = \left( \tau_s^{-1} \right)^{*} \left( \varphi \right). $$
Hence, if $\gamma$ is a curve with $\gamma(0) = p$ and $\dot{\gamma}(0) = X_p$ then
$$ (\nabla_X w)(Y)|_{p} = \left( \lim_{s \to 0} \frac{\mu_s^{-1} \omega_{\gamma(s)} - \omega_{\gamma(0)}}{s} \right) \left( Y_p \right) = \left( \lim_{s \to 0} \frac{\tau_s^{*}(w_{\gamma(s)}) - \omega_{\gamma(0)}}{s} \right)(Y_p) \\
= \lim_{s \to 0} \frac{\omega_{\gamma(s)} \left( \tau_s(Y_p) \right) - \omega_{\gamma(0)}(Y_p)}{s}  \\
= \lim_{s \to 0} \frac{\omega_{\gamma(s)} (Y_{\gamma(s)}) - \omega_{\gamma(0)}(Y_{\gamma(0)}) + \omega_{\gamma(s)} \left( \tau_s(Y_p) \right) - \omega_{\gamma(s)}(Y_{\gamma(s)})}{s} \\
= \lim_{s \to 0} \frac{\omega_{\gamma(s)}(Y_{\gamma(s)}) - \omega_{\gamma(0)}(Y_{\gamma(0)})}{s} - \lim_{s \to 0} \left( \left( \omega_{\gamma(s)} \circ \tau_s \right) \left(  \frac{\tau_s^{-1} \left( Y_{\gamma(s)} \right) -  Y_{\gamma(0)}}{s} \right) \right) \\
\lim_{s \to 0} \frac{\omega_{\gamma(s)}(Y_{\gamma(s)}) - \omega_{\gamma(0)}(Y_{\gamma(0)})}{s} - \left( \lim_{s \to 0} \left( \omega_{\gamma(s)} \circ \tau_s \right) \right) \left( \lim_{s \to 0}  \frac{\tau_s^{-1} \left( Y_{\gamma(s)} \right) -  Y_{\gamma(0)}}{s} \right). $$
The first term tends by definition to $X(\omega(Y))|_{p}$ while the second tends to $\omega_{p}(\nabla_X Y|_{p})$ using the fact that $\omega_{\gamma(s)} \circ \tau_s \to \omega_{\gamma(0)} \circ \tau_0 = \omega_p$.


text:
\begin{document}

\maketitle

\begin{abstract}
Recent work has explored solver strategies for the linear system of equations arising from a spectral Galerkin
approximation of the solution of PDEs with parameterized (or stochastic) inputs. We consider the related problem of a
matrix equation whose matrix and right hand side depend on a set of parameters (e.g.\ a PDE with stochastic inputs
semidiscretized in space) and examine the linear system arising from a similar Galerkin approximation of the solution.
We derive a useful factorization of this system of equations, which yields bounds on the eigenvalues, clues to
preconditioning, and a flexible implementation method for a wide array of problems. We complement this analysis with
(i) a numerical study of preconditioners on a standard elliptic PDE test problem and (ii) a fluids application using
existing CFD codes; the \textsc{MATLAB} codes used in the numerical studies are available online.
\end{abstract}

\begin{keywords}
parameterized systems, spectral methods, stochastic Galerkin
\end{keywords}

\pagestyle{myheadings}
\thispagestyle{plain}
\markboth{P.~G. CONSTANTINE, D.~F. GLEICH, AND G. IACCARINO}{SPECTRAL GALERKIN FACTORIZATION}

\input{section-intro}
\input{section-galerkin}
\input{section-precondition}
\input{section-joe}
\input{section-summary}

\section{Acknowledgments}
This material is based upon work supported by the Department of Energy [National Nuclear Security Administration] under
Award Number NA28614.

\bibliographystyle{siam}
\bibliography{paulconstantine}

\end{document}

config: arxiv


TITLE: Understanding the role of Metatheories through an example
QUESTION [1 upvotes]: I've been studying for a course in set theory and I still have some problems in understading clearly the relation metatheory\theory. Being more specific, I'll present an example:
If we choose $\mathrm{ZFC}$ as our metatheory we can prove the completeness theorem for first order languages, right? But I have developed two different (?) interpretations about the nature of such "proof"

If we chooe $\mathrm{ZFC}$ as a metatheory that means that we are encoding our formal (first order) language (and theories) inside $\mathrm{ZFC}$, for example in $\mathrm{V}_\lambda$. Now formulae and formal proofs are elements of $\mathrm{V}_\lambda$ and $\mathrm{ZFC}$ can recognize and manipulate them. So the proof of completeness theorem will be a formal proof (inside $\mathrm{ZFC}$) so that: $$\mathrm{ZFC}\vdash \forall \ \ulcorner \mathrm{T}\urcorner( \mathrm{Con}(\ulcorner \mathrm{T}\urcorner)\longleftrightarrow \mathrm{Mod}(\ulcorner \mathrm{T}\urcorner) \neq \emptyset )$$
Choosing $\mathrm{ZFC}$ as a metatheory means that we simply take its axioms and work with them in the usual informal mathematical framework. So the proof of the completeness theorem will not be a formal (first order) proof as in the previous case, but a "standard" mathematical proof.

I'd lean towards the first one, although I sense that there is something missing. In this case Gödel's incompleteness theorems appears clearly to me to be mathematical results generating from taking certain theories simultaneously as both the object theory and the metatheory.
So, what is the issue here? Are both of them wrong? One of them is closer to being true?
Thanks

REPLY [1 votes]: Your interpretation of "ZFC as the metatheory" is right (except we don't need the $V_\lambda$-bit), at least ideally: to a formalist,

"Every consistent theory has a model"

is slang for

"ZFC proves [Every consistent theory has a model]$_{appropriately formulated}$" (and moreover we either have such a formal proof in hand or we're confident that the natural language proof of the completeness theorem is a satisfactory blueprint for a formal ZFC-proof of [Every consistent theory has a model]$_{appropriatelyformulated}$).

Why did I say "at least ideally?" Well, there are two subtelties here about how this is actually practiced. The first is the bit about what happens if we don't right now have a formal ZFC-proof in hand: there's an element of subjectivity here, since two different people can reasonably disagree over whether a given natural language proof really is a satisfactory blueprint. The second is about the "appropriately formulated" bit: there's an implicit claim that the formal $\{\in\}$-sentence $\chi$ which we look at is in fact an appropriate formulation of the original natural-language mathematical claim. This is a really slippery thing to pin down precisely. However, I don't know of any actual situation where there's any doubt on this point (and ultimately the formalist would say that if there's serious doubt then the original claim was too vague to be meaningful).
(Note that these subtleties mean that your second interpretation isn't really wrong; rather, what's going on is that your second interpretation describes what we actually do while your first interpretation describes what (according to a formalist) we ought to do.)

In this case Gödel's incompleteness theorems appears clearly to me to be mathematical results generating from taking certain theories simultaneously as both the object theory and the metatheory.

This isn't right - Godel's theorem, appropriately approached, is not foundationally subtle at all. Namely, blackboxing the subtleties mentioned above, ZFC proves
$(*)\quad$ "For any computably axiomatizable theory $T$ interpreting Robinson arithmetic, if $T$ is consistent then $T$ is incomplete."
(In fact, galactically less than ZFC is needed, and $(*)$ can be improved.)
Note that Godel's theorem as phrased this way doesn't involve any object/meta-theory relationship: it's just a perfectly concrete mathematical claim, proved like any other such claim. Moreover, ZFC proves straightforwardly that $(*)$ applies to ZFC in the sense that ZFC proves "If ZFC is consistent then ZFC is incomplete." Note that hypothesis: ZFC isn't proving its own incompleteness, it's proving its own conditional incompleteness.
It may help to read about approaches to formally proving the incompleteness theorem (see e.g. here).


text:
\begin{document}

\title{Dga models for moment-angle complexes}

\author{Matthias Franz}
\thanks{The author was supported by an NSERC Discovery Grant.}
\address{Department of Mathematics, University of Western Ontario,
  London, Ont.\ N6A\;5B7, Canada}
\email{mfranz@uwo.ca}

\subjclass[2020]{Primary 57S12; secondary 16E45, 55N10}

\begin{abstract}
  A dga model for the integral singular cochains on a moment-angle complex
  is given by the twisted tensor product of the corresponding Stanley--Reisner ring
  and an exterior algebra. We present a short proof of this fact and extend it
  to real moment-angle complexes. We also compare various descriptions
  of the cohomology rings of these spaces, including one stated without proof
  by Gitler and López de Medrano.
\end{abstract}

\maketitle

\section{Introduction}

Let \(\Sigma\) be a simplicial complex on the set~\([m]=\{1,\dots,m\}\),
containing the empty simplex~\(\emptyset\) and possibly having ghost vertices, and let
\begin{equation}
  \ZK(\Sigma) = \ZK_{\Sigma}(D^{2},S^{1}) = \bigcup_{\sigma\in\Sigma} (D^{2},S^{1})^{\sigma}
  \subset (D^{2})^{m}
\end{equation}
be the associated moment-angle complex, where
\begin{equation}
  (D^{2},S^{1})^{\sigma}
  = \bigl\{\,(z_{1},\dots,z_{m})\in (D^{2})^{m} \bigm| \text{\(z_{i}\in S^{1}\) if \(i\notin\sigma\)}\,\bigr\}.
\end{equation}
Moment-angle complexes play a central role in toric topology, see~\cite{BuchstaberPanov:2015}.
Replacing \((D^{2},S^{1})\) by~\((D^{n},S^{n-1})\) for any~\(n\ge1\) gives generalized moment-angle complexes.
Taking arbitrary CW~pairs leads to polyhedral products, which have gained a lot of attention in homotopy theory recently,
see \cite{BahriEtAl:2020} for a survey.

The moment-angle complex~\(\ZK(\Sigma)\) is homotopy-equivalent to the complement
of a complex coordinate subspace arrangement, which is a smooth toric variety.
The integral cohomology ring of~\(\ZK(\Sigma)\) was computed by the author~\cite[Sec.~4]{Franz:2003a}
(using the language of toric varieties)
and shortly afterwards by Baskakov--Buchstaber--Panov~\cite{BaskakovBuchstaberPanov:2004}.\footnote{
  The argument appearing in~\cite[Thm.~7.7]{BuchstaberPanov:2002} and earlier publications by the same authors is incorrect,
  compare~\cite[Sec.~1]{Franz:2018b}.}
The result is an isomorphism of graded rings
\begin{equation}
  \label{eq:iso-HZK-Tor}
  H^{*}(\ZK(\Sigma)) = \Tor_{\RC}(\Z,\Z[\Sigma]),
\end{equation}
where \(\RC=\Z[t_{1},\dots,t_{m}]\) and \(\Z[\Sigma]\) is the Stanley--Reisner ring of~\(\Sigma\)
with generators~\(t_{1}\),~\dots,~\(t_{m}\) of degree~\(2\).
Taking the Koszul resolution of~\(\Z\) over~\(\RC\),
one can describe the ring~\eqref{eq:iso-HZK-Tor} as the cohomology of the commutative
differential graded algebra (cdga)
\begin{equation}
  \label{eq:def-A}
  \AC(\Sigma) = \kk[\Sigma] \otimes \bigwedge(s_{1},\dots,s_{m}),
  \qquad
  d\,s_{i} = t_{i},
  \quad
  d\,t_{i} =0
\end{equation}
for~\(i\in[m]\), where each~\(s_{i}\) has degree~\(1\).
Dividing out out all squares~\(t_{i}^{2}\) as well as all terms~\(s_{i}\,t_{i}\),
one obtains a quasi-isomorphic dga~\(\BC(\Sigma)\).
As a cdga, \(\BC(\Sigma)\) is generated by the~\(s_{i}\) and~\(t_{i}=d\,s_{i}\) and has the relations \(s_{i}\,t_{i}=t_{i}\,t_{i}=0\) for~\(i\in[m]\)
as well as \(t_{i_{1}}\cdots t_{i_{k}}=0\) whenever \(\{i_{1},\dots,i_{k}\}\notin\Sigma\).

\begin{theorem}
  \label{thm:main}
  The singular cochain algebra~\(C^{*}(\ZK(\Sigma))\)
  is quasi-iso\-mor\-phic to the dgas~\(\AC(\Sigma)\) and~\(\BC(\Sigma)\),
  naturally with respect to inclusions of subcomplexes.
\end{theorem}

Recall that a dga~\(A\) is called an (integral) \newterm{dga model} for a space~\(X\) if \(A\) can be connected to~\(C^{*}(X)\)
via a zigzag of dga quasi-isomorphisms. In this language,
\Cref{thm:main} asserts that both~\(\AC(\Sigma)\) and~\(\BC(\Sigma)\) are dga models for~\(\ZK(\Sigma)\).

That \(C^{*}(\ZK(\Sigma))\) and~\(\AC(\Sigma)\) are quasi-isomorphic
is already implicit in the author's computation of~\(H^{*}(\ZK(\Sigma))\),
see~\cite[Sec.~4]{Franz:2003a}.
A different proof has recently been obtained by the author as a byproduct
of his work on the cohomology rings of partial quotients
of moment-angle complexes \cite[Prop.~6.1]{Franz:2018b}.
As remarked there, this result answers a question
posed by Berglund~\cite[Question~5]{Berglund:2010},
which was exactly whether \(\AC(\Sigma)\) is a dga model for~\(\ZK(\Sigma)\).
The aim of the present note is to give a much shorter proof for this model.
Like Baskakov--Buchstaber--Panov's calculation it is based on the dga~\(\BC(\Sigma)\).
The rational versions of~\(\AC(\Sigma)\) and~\(\BC(\Sigma)\) are
(analogously defined) cdga models for the polynomial differential forms
on~\(\ZK(\Sigma)\) by a result of Panov--Ray~\cite[Thm.~6.2]{PanovRay:2008}.

The proof of \Cref{thm:main} appears in the following section
and an adaptation to real moment-angle complexes in \Cref{sec:real}.
In the final section we relate the resulting cup product formulas for real and complex
moment-angle complexes with others appearing in the literature.
We in particular provide a proof that has been missing so far for a product formula
stated by Gitler and López de Medrano~\cite{GitlerLopezDeMedrano:2013}.

\begin{acknowledgements}
  I thank Don Stanley for his questions about dga models and Santiago López de Medrano for stimulating discussions.
  I also thank the organizers of the ``Thematic Program on Toric Topology and Polyhedral Products''
  at the Fields Institute for creating an environment favourable to research
  and an anonymous referee for comments that helped to improve the presentation of the paper.
\end{acknowledgements}

\section{Proof of Theorem~\ref{thm:main}}

We will obtain \Cref{thm:main} by dualizing the analogous homological result.
To state the latter, we need to introduce some terminology.
As already done in \Cref{thm:main}, we write \(C(-)\) and~\(C^{*}(-)\)
for \emph{normalized} singular (co)chains with integral coefficients.

Recall that the normalized singular chain complex of a space~\(X\)
is obtained from the usual non-normalized one by dividing out the subcomplex
of degenerate simplices. A singular \(n\)-simplex is called degenerated
if it factors through an \((n-1)\)-dimensional one via a map~\(\Delta^{n}\to\Delta^{n-1}\)
between standard simplices that in barycentric coordinates is given
by~\((t_{0},\dots,t_{n})\mapsto (t_{0},\dots,t_{i}+t_{i+1},\dots,t_{n})\)
for some~\(0\le i<n\). Projecting non-normalized to normalized singular chains
is a homotopy equivalence, compare~\cite[Sec.~VIII.6]{MacLane:1967}.
The normalized singular cochain complex~\(C^{*}(X)\) is the dual of~\(C(X)\) with differential
\begin{equation}
  \label{eq:def-d-dual}
  (d\,\gamma)(x) = - (-1)^{\deg{\gamma}}\,\gamma(d\,x)
\end{equation}
for~\(\gamma\in C^{n}(X)\) and a singular \((n+1)\)-simplex~\(x\).

The chain complex~\(C(X)\) is a differential graded coalgebra (dgc)
with diagonal and augmentation given by
\begin{equation}
  \Delta x = \sum_{k=0}^{n} x(0 \dots k) \otimes x(k \dots n)
  \qquad\text{and}\qquad
  \epsilon(x) = \begin{cases}
    1 & \text{if \(n=0\),} \\
    0 & \text{otherwise}
  \end{cases}
\end{equation}
for an \(n\)-simplex~\(x\). Here \(x(k_{1}\dots k_{2})\) denotes the face of~\(x\) with vertices~\(k_{1}\),~\dots,~\(k_{2}\).

We also recall part of the Eilenberg--Zilber theorem, compare~\cite[Sec.~17]{EilenbergMoore:1966}.
Given two spaces~\(X\) and~\(Y\), the shuffle map
\begin{equation}
  \shuffle\colon C(X)\otimes C(Y) \to C(X\times Y)
\end{equation}
is a homotopy equivalences of complexes, natural in~\(X\) and~\(Y\).
It moreover is associative and a morphism of dgcs.
Hence for any spaces~\(X_{1}\),~\dots,~\(X_{m}\) we have a natural quasi-isomorphism of dgcs
\begin{equation}
  C(X_{1})\otimes\dots\otimes C(X_{m}) \to C\bigl(X_{1}\times\dots\times X_{m}\bigr),
\end{equation}
again denoted by~\(\shuffle\).

Let \(\SRC{\Sigma}\) be the Stanley--Reisner coalgebra of~\(\Sigma\) dual to~\(\kk[\Sigma]\),
\cf~\cite[Sec.~8.2]{BuchstaberPanov:2015}.
The  canonical basis for~\(\SRC{\Sigma}\), considered as a \(\Z\)-module, are the monomials~\(u_{\alpha}\)
indexed by allowed multi-indices~\(\alpha\in\N^{m}\).
A multi-index~\(\alpha\) is \newterm{allowed} if it is supported on some simplex in~\(\Sigma\), that is, if
\begin{equation}
  \supp\alpha \coloneqq \{\, i\in[m] \mid \alpha_{i} > 0\,\} \in \Sigma.
\end{equation}

The degree of~\(u_{\alpha}\) is \(2(\alpha_{1}+\dots+\alpha_{m})\).
The structure maps are given by
\begin{equation}
  \Delta u_{\alpha} = \sum_{\beta+\gamma=\alpha} u_{\beta}\otimes u_{\gamma},
  \qquad
  \epsilon(u_{\alpha}) =
  \begin{cases}
    1 & \text{if \(\alpha=0\),} \\
    0 & \text{otherwise.}
  \end{cases}
\end{equation}

We consider the tensor product of graded coalgebras
\begin{equation}
  \KC(\Sigma) =  \SRC{\Sigma} \otimes \bigwedge(v_{1},\dots,v_{m}),
\end{equation}
where each~\(v_{i}\) is primitive of degree~\(1\). We turn~\(\KC(\Sigma)\) into a dgc by defining
\begin{equation}
  d(u_{\alpha}\otimes v_{\tau}) = \sum_{\alpha_{i}>0} u_{\alpha-i}\otimes v_{i}\wedge v_{\tau}
\end{equation}
for allowed multi-indices~\(\alpha\in\N^{m}\) and~\(\tau\subset[m]\).
Here we have written \(\alpha-i\) for the multi-index
that is obtained from~\(\alpha\) by decreasing the \(i\)-th component by~\(1\)
as well as \(v_{\tau}=v_{i_{1}}\wedge\dots\wedge v_{i_{k}}\) if~\(\tau=\{i_{1}<\dots<i_{k}\}\).
For~\(\sigma\in\Sigma\) we also write \(u_{\sigma}=u_{\alpha}\)
where \(\alpha\) is the indicator function of~\(\sigma\subset[m]\),
\begin{equation}
  \alpha_{i} =
  \begin{cases}
    1 & \text{if \(i\in\sigma\),} \\
    0 & \text{if \(i\notin\sigma\),}
  \end{cases}
\end{equation}
and we use the abbreviation~\(u_{\emptyset}=v_{\emptyset}=u_{\emptyset}\otimes v_{\emptyset}=1\).

Let \(\LC(\Sigma)\) be the sub-dgc of~\(\KC(\Sigma)\) spanned by all elements~\(u_{\sigma}\otimes v_{\tau}\)
with disjoint subsets~\(\sigma\in\Sigma\) and~\(\tau\subset[m]\).
The dual of~\(\KC(\Sigma)\) is the dga~\(\AC(\Sigma)\), and that of~\(\LC(\Sigma)\) is \(\BC(\Sigma)\).

\begin{theorem}
  \label{thm:main-hom}
  The dgcs \(C(\ZK(\Sigma))\),~\(\KC(\Sigma)\) and~\(\LC(\Sigma)\) are quasi-isomorphic,
  naturally with respect to inclusions of subcomplexes.
\end{theorem}

The proof is given in the remainder of this section.
Applying the universal coefficient theorem for cohomology then establishes \Cref{thm:main}.

\medbreak

The following two observations are immediate.
We write \(\restr{\Sigma}{i}\) for the restriction of~\(\Sigma\) to the single vertex~\(i\in[m]\).
It contains either the empty simplex only or additionally the \(0\)-simplex~\(\{i\}\).

\begin{lemma}
  \label{thm:KL-tensor}
  For any~\(\sigma\in\Sigma\) there are canonical isomorphisms of dgcs
  \begin{equation*}
    \KC(\sigma) = \bigotimes_{i=1}^{m} \KC(\restr{\sigma}{i}),
    \qquad
    \LC(\sigma) = \bigotimes_{i=1}^{m} \LC(\restr{\sigma}{i}).
  \end{equation*}
\end{lemma}

\begin{lemma}
  \label{thm:KL-MV}
  Let \(\Sigma_{1}\),~\(\Sigma_{2}\) be subcomplexes of~\(\Sigma\).
  There are short exact sequences
  \begin{gather*}
    0 \longrightarrow \KC(\Sigma_{1}\cap\Sigma_{2})
    \longrightarrow \KC(\Sigma_{1})\oplus \KC(\Sigma_{2})
    \longrightarrow \KC(\Sigma_{1}\cup\Sigma_{2})
    \longrightarrow 0,
    \\
    0 \longrightarrow \LC(\Sigma_{1}\cap\Sigma_{2})
    \longrightarrow \LC(\Sigma_{1})\oplus \LC(\Sigma_{2})
    \longrightarrow \LC(\Sigma_{1}\cup\Sigma_{2})
    \longrightarrow 0.
  \end{gather*}
\end{lemma}

Let \(y\) be the usual parametrization of~\(S^{1}\), considered as a singular \(1\)-simplex.
Choose a singular \(2\)-simplex~\(x\) in~\(D^{2}\) that
restricts to~\(y\) on the edge~\((12)\) and maps the other two edges~\((01)\) and~\((02)\) to the identity element~\(e\in S^{1}\). Then
\begin{align}
  \label{eq:y}
  d\,y &= 0, & \Delta y &= y\otimes e + e\otimes y, \\
  \label{eq:x}
  d\,x &= x(12)-x(02)+x(01) &\qquad\quad  \Delta x &= x\otimes e + x(01)\otimes x(12) + e\otimes x \\
  \notag
  &= y, &  &= x\otimes e + e\otimes x.
\end{align}
Here \(x(01)\) and~\(x(02)\) drop out because they are degenerate. For this
to hold it is crucial that we work with normalized chains.

We use the singular simplices~\(x\) and~\(y\) to define a dgc map
\begin{equation}
  \Psi(\Sigma)\colon \LC(\Sigma)\to C(\ZK(\Sigma)).
\end{equation}
For~\(m=1\) we map~\(u_{1}\mapsto x\),
\(v_{1}\mapsto y\) and~\(1\mapsto e\in C(S^{1})\);
this is well-defined by~\eqref{eq:y} and~\eqref{eq:x}. For~\(m>1\) and~\(\sigma\subset[m]\) we set
\begin{multline}
  \label{eq:def-Psi-sigma}
  \qquad
  \Psi(\sigma)\colon
  \LC(\sigma) = \bigotimes_{i=1}^{m} \LC(\restr{\sigma}{i})
  \xrightarrow{\bigotimes\Psi(\restr{\sigma}{i})} \bigotimes_{i=1}^{m} C(\ZK(\restr{\sigma}{i})) \\
  \xrightarrow{\;\;\shuffle\;\;} C\bigl(\ZK(\restr{\sigma}{1})\times\dots\times \ZK(\restr{\sigma}{m})\bigr)
  = C(\ZK(\sigma)),
  \qquad
\end{multline}
using \Cref{thm:KL-tensor} and the fact that the shuffle map is a morphism of dgcs.
In the general case \(\Psi(\Sigma)\) is determined by imposing
naturality with respect to inclusions of subcomplexes.
In other words, \(\Psi(\Sigma)\) agrees on~\(\LC(\sigma)\subset\LC(\Sigma)\) with~\(\Psi(\sigma)\),
followed by the inclusion~\(C(\ZK(\sigma))\hookrightarrow C(\ZK(\Sigma))\).

\begin{lemma}
  \label{thm:LC-CZK-quiso}
  The map~\(\Psi(\Sigma)\) is a quasi-isomorphism of dgcs.
\end{lemma}

\begin{proof}
  The case~\(m=1\) is settled by a direct verification; we therefore assume \(m>1\)
  and proceed by induction on the size of~\(\Sigma\).
  If \(\Sigma\) has a single maximal simplex~\(\sigma\), then \(\Psi(\Sigma)=\Psi(\sigma)\)
  is a quasi-isomorphism because so are the shuffle map and, by the Künneth theorem,
  the tensor product of the maps~\(\Psi(\restr{\sigma}{i})\).

  Otherwise we can split \(\Sigma\) up into two smaller complexes~\(\Sigma_{1}\) and~\(\Sigma_{2}\)
  with intersection~\(\Sigma_{12}=\Sigma_{1}\cap\Sigma_{2}\).
  The naturality of~\(\Psi\) gives us a map between the long exact sequence corresponding
  to the short exact sequence for~\(\LC\) from \Cref{thm:KL-MV} and the Mayer--Vietoris sequence
  for the CW~complex~\(\ZK(\Sigma)=\ZK(\Sigma_{1})\cup\ZK(\Sigma_{2})\),
  \begin{equation}
    \footnotesize
    \begin{tikzcd}[column sep=1em]
      \cdots \arrow{r} & H_{*+1}(\LC(\Sigma)) \arrow{d}{\Psi_{*+1}(\Sigma)} \arrow{r}
      & H_{*}(\LC(\Sigma_{12})) \arrow{d}{\Psi_{*}(\Sigma_{12})} \arrow{r}
      & H_{*}(\LC(\Sigma_{1})) \oplus H_{*}(\LC(\Sigma_{2})) \arrow{d}{(\Psi_{*}(\Sigma_{1}),\Psi_{*}(\Sigma_{2}))} \arrow{r}
      & H_{*}(\LC(\Sigma)) \arrow{d}{\Psi_{*}(\Sigma)} \arrow{r} & \cdots \\
      \cdots \arrow{r} & H_{*+1}(\ZK(\Sigma)) \arrow{r}
      & H_{*}(\ZK(\Sigma_{12})) \arrow{r}
      & H_{*}(\ZK(\Sigma_{1})) \oplus H_{*}(\ZK(\Sigma_{2})) \arrow{r}
      & H_{*}(\ZK(\Sigma)) \arrow{r} & \cdots \mathrlap{\;,}
    \end{tikzcd}
  \end{equation}
  where we have written \(\Psi_{*}(\Sigma)\) instead of~\(H_{*}(\Psi(\Sigma))\) etc.
  The five lemma together with induction implies
  that \(\Psi_{*}(\Sigma)\) is an isomorphism.
\end{proof}

An analogous argument shows that the inclusion map
\begin{equation}
  \label{eq:LC-KC-quiso}
  \LC(\Sigma)\hookrightarrow\KC(\Sigma)
\end{equation}
is a quasi-isomorphism of dgcs.\footnote{
  It is also a homotopy equivalence of complexes.
  See~\cite[Lemma~7.10]{BuchstaberPanov:2002} or~\cite[Lemma~3.2.6]{BuchstaberPanov:2015} for an explicit homotopy inverse.}
If \(\Sigma\) has a single maximal simplex, we combine \Cref{thm:KL-tensor} with the Künneth theorem.
Otherwise we again write \(\Sigma=\Sigma_{1}\cup\Sigma_{2}\) and compare the long exact sequences associated
to both short exact sequences in \Cref{thm:KL-MV}.

This completes the proof of \Cref{thm:main-hom}.

\begin{remark}
  \label{rem:generalized}
  Theorems~\ref{thm:main} and~\ref{thm:main-hom} remain valid for all generalized moment-angle complexes~\(\ZK_{\Sigma}(D^{n},S^{n-1})\) with even~\(n\ge 2\),
  up to the obvious degree shifts. In particular, the generators~\(s_{i}\) and~\(t_{i}\) in~\eqref{eq:def-A} are now of degrees~\(\deg{s_{i}}=n-1\) and~\(\deg{t_{i}}=n\).
  The singular \(n\)-simplex~\(x\) is obtained by collapsing all but the last facet of the standard \(n\)-simplex to a point, and \(y\) is this last facet.

  If \(n\ge3\) is odd, then \(\deg{y}\) is even and \(\deg{x}\) is odd. Proceeding as before, we get a quasi-isomorphism
  between~\(C^{*}(\ZK_{\Sigma}(D^{n},S^{n-1}))\) and the cdga~\(\BCt(\Sigma)\) with generators~\(s_{i}\) of degree~\(n-1\) and~\(t_{i}=d\,s_{i}\) of degree~\(n\)
  as well as relations
  \begin{equation}
    s_{i}\,s_{i} = s_{i}\,t_{i} = 0,
    \qquad\text{and}\qquad
    t_{i_{1}}\cdots t_{i_{k}}=0 \quad \text{if \(\{i_{1},\dots,i_{k}\}\notin\Sigma\).}
  \end{equation}
  Note that the Stanley--Reisner relations are monomial and therefore independent of the order
  of the anticommuting variables~\(t_{i}\).

  In general, such a quasi-isomorphism does not hold for the case~\(n=1\), which we treat in the following section.
\end{remark}

\section{Real moment-angle complexes}
\label{sec:real}

It is not difficult to adapt our approach to real moment-angle complexes
\begin{equation}
  \RZK(\Sigma) = \ZK_{\Sigma}(D^{1},S^{0}) \subset (D^{1})^{m}.
\end{equation}

We start with the homological setting.
As a chain complex, we define the analogue~\(\LR(\Sigma)\) of~\(\LC(\Sigma)\) as before, except that now the degrees are
\(\deg{u_{i}} = 1\) and~\(\deg{v_{i}}=0\) for all~\(i\in[m]\).

Let us consider the case~\(m=1\) first. Writing \(u=u_{1}\) and~\(v=v_{1}\),
we turn \(\LR(\Sigma)\) into a dgc via the augmentation~\(\epsilon(v)=\epsilon(u)=0\) and the diagonal
\begin{align}
  \Delta v &= v\otimes 1 + 1\otimes v + v\otimes v, \\
  \Delta u &= u\otimes 1 + 1\otimes u + u\otimes v.
\end{align}

Let \(x\) be the canonical path from~\(e=1\) to~\(g=-1\in S^{0}\), considered as a singular \(1\)-simplex in~\(D^{1}=[-1,1]\),
and let \(y=g-e\). Then
\begin{align}
  \label{eq:d-x-y}
  &\qquad d\, x = y, \qquad\qquad d\,y = 0, \\
  \label{eq:Delta-y}
  \Delta y &= g\otimes g - e\otimes e = y\otimes e + e\otimes y + y\otimes y, \\
  \label{eq:Delta-x}
  \Delta x &= x\otimes g + e\otimes x = x\otimes e + e\otimes x + x\otimes y.
\end{align}
Given that for~\(m=1\) we either have \(\Sigma=\{\emptyset\}\) or \(\Sigma=\{\emptyset,\{1\}\}\),
one verifies directly
that the map
\begin{equation}
  \label{eq:map-L-C-real}
  \PsiR(\Sigma)\colon \LR(\Sigma) \to C(\RZK(\Sigma)),
  \qquad
  1 \mapsto e, \quad v \mapsto y, \quad u \mapsto x
\end{equation}
is a quasi-isomorphism of dgcs. (Since this map is injective, one can also use it
to justify that \(\LR(\Sigma)\) with the diagonal given above is indeed a dgc.)

For~\(m>1\) we again proceed exactly as before.
We use the isomorphism of complexes
\begin{equation}
  \LR(\sigma) = \bigotimes_{i=1}^{m} \LR(\restr{\sigma}{i})
\end{equation}
analogous to \Cref{thm:KL-tensor} to define a dgc structure on~\(\LR(\sigma)\)
for each simplex~\(\sigma\in\Sigma\) as well as a dgc map~\(\LR(\sigma)\to C(\RZK(\sigma))\).
Then we extend both the dgc structure and the map to the colimit~\(\LR(\Sigma)\) of the~\(\LR(\sigma)\) over all~\(\sigma\in\Sigma\).
The same proof as for \Cref{thm:LC-CZK-quiso} shows that the resulting map
\begin{equation}
  \PsiR(\Sigma)\colon\LR(\Sigma)\to C(\RZK(\Sigma))
\end{equation}
is a quasi-isomorphism.
In this step one uses the obvious analogue of \Cref{thm:KL-MV} for~\(\LR\) instead of~\(\LC\).

We thus have established the following.

\begin{lemma}
  The map~\(\PsiR(\Sigma)\) is a quasi-isomorphism of dgcs.
\end{lemma}

\medskip

We now turn to cohomology.
The dual of the dgc~\(\LR(\Sigma)\) is the dga~\(\BR(\Sigma)\) with generators~\(s_{i}\) of degree~\(0\)
and \(t_{i}\) of degree~\(1\) satisfying the relations
\begin{gather}
  d\,s_{i}=-t_{i}, \qquad d\,t_{i} = 0, \\
  \label{eq:relations-s-t}
  s_{i}\,s_{i} = s_{i}, \qquad t_{i}\,s_{i} = t_{i}, \qquad s_{i}\,t_{i} = 0, \qquad t_{i}\,t_{i} = 0,
  \qquad \prod_{j\in\sigma} t_{j} = 0
\end{gather}
for any~\(i\in[m]\) and~\(\sigma\notin\Sigma\)
plus the rule that variables corresponding to \emph{distinct} subscripts commute in the graded sense.

The minus sign in~\(d\,s_{i} = t_{i}\) comes from the general definition
of the differential on the dual of a complex,
\cf~\eqref{eq:def-d-dual} and~\cite[eq.~(II.3.1)]{MacLane:1967}.
For~\(m=1\) we have
\begin{equation}
  (d\,s)(x) = -(-1)^{\deg{s}}\,s(d\,x) = - s(y) = -1 = -t(x).
\end{equation}
The minus sign could be removed by replacing \(t_{i}\) with~\(-t_{i}\), that is, by mapping \(u\) to~\(-x\).
The minus sign does not appear in~\cite[p.~512]{Cai:2017}
because of a different sign convention for the dual complex.

We can sum up our discussion as follows.

\begin{theorem}
  \label{thm:real-B}
  There is a quasi-isomorphisms of dgas
  \begin{equation*}
    C^{*}(\RZK(\Sigma)) \to \BR(\Sigma),
  \end{equation*}
  natural with respect to inclusions of subcomplexes.
\end{theorem}

We in particular recover Cai's isomorphism of graded rings~\cite[Secs.~3~\&~4]{Cai:2017}
\begin{equation}
  \label{eq:HRKZ-HBSigma}
  H^{*}(\RZK(\Sigma)) = H^{*}(\BR(\Sigma)).
\end{equation}
In fact, our proof shares some similarities with Cai's.
This would be even more so
if we worked with cubical singular chains, compare~\cite{Massey:1980}.
We also remark that in the case of real moment-angle complexes
it is not necessary to pass to normalized (singular) chains.
(The shuffle map is a morphism of dgcs for non-normalized chains already,
and the formulas~\eqref{eq:d-x-y}--\eqref{eq:Delta-x} do not need normalization, either.)

We discuss the dga~\(\AR(\Sigma)\) analogous to~\(\AC(\Sigma)\) only for coefficients in~\(\Z_{2}\).
General coefficients are considered in~\cite{Franz:realtorprod}
where \Cref{thm:real-B} is furthermore extended to real toric spaces, that is,
to quotients of~\(\RZK(\Sigma)\) by freely acting subgroups of~\((\Z_{2})^{m}\).

In characteristic~\(2\), the dga~\(\AR(\Sigma)\)
has the same generators~\(s_{i}\) and~\(t_{i}\) as~\(\BR(\Sigma)\) and the relations
\begin{gather}
  d\,s_{i}=t_{i}, \qquad d\,t_{i} = 0, \\
  \label{eq:relations-A-real}
  s_{i}\,s_{i} = s_{i}, \qquad t_{i}\,s_{i} = s_{i}\,t_{i} + t_{i},
  \qquad \prod_{j\in\sigma} t_{j} = 0
\end{gather}
for~\(i\in[m]\) and~\(\sigma\notin\Sigma\), again with the additional rule
that variables corresponding to different subscripts commute.
Observe that the ideal generated by the relations~\eqref{eq:relations-A-real} is closed under the differential,
so that \(\AR(\Sigma)\) is a well-defined dga.
The projection map~\(\AR(\Sigma)\to \BR(\Sigma)\otimes\Z_{2}\)
is again obtained by dividing out the ideal generated by the products~\(s_{i}\,t_{i}\) and~\(t_{i}^{2}\) for all~\(i\in[m]\),
and it can be seen to be a quasi-isomorphism by an argument analogous to the one given before or to~\cite[Lemma~7.10]{BuchstaberPanov:2002}.

The Stanley--Reisner ring~\(\Z_{2}[\Sigma]\), now with generators of degree~\(1\),
is contained in~\(\AR(\Sigma)\) as a sub-dga (with trivial differential). Moreover,
if \(\Sigma=[m]\) is the full simplex, then \(\AR(\Sigma)\) is the Koszul resolution of~\(\Z_{2}\) over~\(\RR=\Z_{2}[t_{1},\dots,t_{m}]\).
In general, \(\AR(\Sigma)\) is the tensor product of this resolution and~\(\Z_{2}[\Sigma]\) over~\(\RR\),
which gives the additive isomorphism
\begin{equation}
  H^{*}(\RZK(\Sigma);\Z_{2}) = \Tor_{\RR}(\Z_{2},\Z_{2}[\Sigma]).
\end{equation}
It is not multiplicative for the canonical product on the torsion product,
as can be seen for~\(\Sigma=\{\emptyset\}\) already, \cf~\cite[Sec.~10.3]{Franz:2018b}:
In this case we have \(\Z_{2}[\Sigma]=\Z_{2}\), so that the torsion product is a strictly exterior algebra.
On the other hand, \(\RZK(\Sigma)=(S^{0})^{m}\) is a finite set of points.
Hence any cohomology class on~\(\RZK(\Sigma)\) is a \(\Z_{2}\)-values function on these points and therefore squares to itself.

\section{Comparison of several product formulas}
\label{sec:compare}

The aim of this section is to relate the product formula in the cohomology of a (complex) moment-angle complex
with Baskakov's formula~\cite{Baskakov:2002} and also the formula for real moment-angle complexes
with one claimed by Gitler and López de Medrano~\cite{GitlerLopezDeMedrano:2013}
as well as the one given by Bahri--Bendersky--Cohen--Gitler~\cite{BahriEtAl:2012}
for arbitrary polyhedral products.\footnote{
  Another cup product formula
  has given by de~Longueville~\cite[Thm.~1.1]{DeLongueville:1999}
  for complements of real coordinate subspace arrangements.
  However, his formula is incorrect in general,
  see \cite[Sec.~3]{GitlerLopezDeMedrano:2013} for a counterexample.}
We note that another description for a class of polyhedral products including
all~\(\ZK_{\Sigma}(D^{n},S^{n-1})\) has been given by Zheng~\cite[Example~7.12]{Zheng:2016}.

We start with a variant of the generalized smash moment-angle complexes introduced in~\cite[Def.~2.2]{BahriEtAl:2012}.
For a closed subset~\(A\) of a compact Hausdorff space~\(X\) and a basepoint~\(*\in A\) we define the space
\begin{equation}
  \WK(X,A) = \bigl\{\, x \in \ZK(X,A) \bigm| \text{\(x_{i}=*\) for some~\(i\in[m]\)} \,\bigr\}
\end{equation}
and based on it the pair
\begin{equation}
  \hZK_{\Sigma}(X,A) = \bigl( \ZK_{\Sigma}(X,A), \WK_{\Sigma}(X,A) \bigr).
\end{equation}
We then have an isomorphism
\begin{equation}
  H^{*}(\hZK_{\Sigma}(X,A)) = \Hc^{*}\bigl(\ZK_{\Sigma}(X,A)\setminus\WK_{\Sigma}(X,A)\bigr)
  = \Hc^{*}\bigl(\ZK_{\Sigma}(X\setminus *,A\setminus *)\bigr),
\end{equation}
where \(\Hc^{*}(-)\) denotes cohomology with compact supports, \cf~\cite[Part~I]{Massey:1978}.

We now specialize to
\begin{equation}
  \label{eq:def-RhZK}
  \RhZK(\Sigma) = \hZK_{\Sigma}(D^{1},S^{0})
\end{equation}
(where the basepoint is~\(e=1\in S^{0}\)) and observe that

\begin{multline}
  \qquad
  \ZK_{\Sigma}(D^{1},S^{0})\setminus\WK_{\Sigma}(D^{1},S^{0}) =
  \ZK_{\Sigma}\bigl(D^{1}\setminus\{e\},S^{0}\setminus\{e\}\bigr) \\ = \ZK_{\Sigma}\bigl([-1,1),\{-1\} \bigr)
  \approx \ZK_{\Sigma}\bigl([0,\infty),\{0\} \bigr) = \CU \Sigma,
  \qquad
\end{multline}

where \(\CU \Sigma\) is the unbounded cone over the simplicial complex~\(\Sigma\).
We think of~\(\Sigma\) as embedded into the hyperplane of~\(\R^{m}\) with coordinate sum equal to~\(1\).

The analysis of~\(\RZK(\Sigma)\) in the preceding section carries over to the present case.
One simply ignores the element~\(e\in S^{0}\) and the counit~\(1\) in the cochain algebra.
(Recall that the cohomology with compact supports is a ring without unit in general.)
The result is as quasi-isomorphism between the relative cochain algebra~\(C^{*}(\RhZK(\Sigma))\)
and the multiplicatively closed subcomplex~\(\hB(\Sigma)\subset \BR(\Sigma)\) spanned by all \(m\)-fold products
\begin{equation}
  \label{eq:def-a1-am}
  a_{1}\cdots a_{m}
  \qquad
  \text{where each~\(a_{i}=s_{i}\) or~\(t_{i}\).}
\end{equation}
In particular, there is a multiplicative isomorphism
\begin{equation}
  \label{eq:iso-HcCU-HhB}
  \Hc^{*}(\CU \Sigma)  = H^{*}(\CB \Sigma,\Sigma) \cong H^{*}(\hB(\Sigma))
\end{equation}
where \(\CB \Sigma\) denotes the bounded cone over~\(\Sigma\) with base~\(\Sigma\).
Not surprisingly, \(\hB(\Sigma)\) does not have a unit unless~\(\Sigma=\{\emptyset\}\).

\medbreak

We now compare \(\hB(\Sigma)\) to the dgas~\(\BC(\Sigma)\) and~\(\BR(\Sigma)\) for complex and real moment-angle complexes, respectively.
In the complex case, we have a direct sum decomposition of complexes
\begin{equation}
  \BC^{*}(\Sigma) = \bigoplus_{\alpha\subset[m]} \hB^{*-\deg{\alpha}}(\Sigma_{\alpha})
\end{equation}
where \(\Sigma_{\alpha}\) is the full subcomplex of~\(\Sigma\) on the vertex set~\(\alpha\).
This gives Hochster's formula
\begin{equation}
  \label{eq:Hochster-ZK}
  H^{*}(\ZK(\Sigma)) = \bigoplus_{\alpha\subset[m]} \Hc^{*-\deg{\alpha}}(\CU \Sigma_{\alpha})
  = \bigoplus_{\alpha\subset[m]} \tilde H^{*-\deg{\alpha}-1}(\Sigma_{\alpha}),
\end{equation}
\cf~\cite[Thm.~3.2.7]{BuchstaberPanov:2002},
where we have used the additive isomorphism
\begin{equation}
  \Hc^{*}(\CU \Sigma) = \tilde H^{*-1}(\Sigma)
\end{equation}
between the reduced cohomology of the simplicial complex~\(\Sigma\) and
the cohomology with compact supports of the unbounded cone over it.
(Recall that \(\tilde H^{-1}(\emptyset)=\kk\).)

The additive isomorphism~\eqref{eq:Hochster-ZK} can be made multiplicative in the following way:
For~\(\alpha\cap\beta\ne\emptyset\), the product
\begin{equation}
  \Hc^{*}(\CU \Sigma_{\alpha}) \otimes \Hc^{*}(\CU \Sigma_{\beta}) \to \Hc^{*}(\CU \Sigma_{\alpha\cup\beta}),
\end{equation}
vanishes. For disjoint~\(\alpha\),~\(\beta\) we use the cross product via the composition
\begin{equation}
  \label{eq:cross-prod}
  \Hc^{*}(\CU \Sigma_{\alpha}) \otimes \Hc^{*}(\CU \Sigma_{\beta})
  \xrightarrow{\times} \Hc^{*}(\CU \Sigma_{\alpha}\times\CU \Sigma_{\beta})
  \xrightarrow{\iota^{*}} \Hc^{*}(\CU \Sigma_{\alpha\cup\beta}),
\end{equation}
where \(\iota\colon\CU \Sigma_{\alpha\cup\beta}\hookrightarrow\CU \Sigma_{\alpha}\times\CU \Sigma_{\beta}\) is the canonical inclusion.
This is Baskakov's formula~\cite{Baskakov:2002}, expressed in terms of Cartesian products of cones and cohomology with compact supports
instead of joins of simplices and reduced cohomology.

\medbreak

For a real moment-angle complex we again have a direct sum decomposition
\begin{equation}
  \BR(\Sigma) = \bigoplus_{\alpha\subset[m]} \hB^{*}(\Sigma_{\alpha}),
\end{equation}
hence also a Hochster formula
\begin{equation}
  \label{eq:Hochster-RZK}
  H^{*}(\RZK(\Sigma)) = \bigoplus_{\alpha\subset[m]} \Hc^{*}(\CU \Sigma_{\alpha})
  = \bigoplus_{\alpha\subset[m]} \tilde H^{*-1}(\Sigma_{\alpha}).
\end{equation}
Note that there are no degree shifts by~\(\deg{\alpha}\) this time.
The isomorphism becomes multiplicative
if one uses the following generalization of the product~\eqref{eq:cross-prod}.

Recall that for any open subsets~\(U=X\setminus A\) and~\(V=X\setminus B\) of a compact Hausdorff space~\(X\) there is a cup product
\begin{multline}
  \qquad
  \Hc^{*}(U) \otimes \Hc^{*}(V) = H^{*}(X,A) \otimes H^{*}(X,B) \\
  \xrightarrow{\;\;\cup\;\;}
  H^{*}\bigl(X, A\cup B\bigr) = \Hc^{*}(U\cap V),
  \qquad
\end{multline}
\cf~also~\cite[Sec.~7.4]{Massey:1978}.

Now let \(\pi_{\alpha}\colon\RZK(\Sigma_{\alpha\cup\beta})\to\RZK(\Sigma_{\alpha})\) be the (proper) projection
that sends the coordinates~\(z_{i}\) with~\(i\notin\alpha\) to the basepoint~\(e\),
and analogously for~\(\pi_{\beta}\colon\RZK(\Sigma_{\alpha\cup\beta})\to\RZK(\Sigma_{\beta})\).
We define the \(*\)-product as the composition
\begin{multline}
  \label{eq:cross-prod-general}
  \Hc^{*}(\CU \Sigma_{\alpha}) \otimes \Hc^{*}(\CU \Sigma_{\beta})
  \xrightarrow{\pi_{\alpha}^{*}\otimes\pi_{\beta}^{*}}
  \Hc^{*}(\pi_{\alpha}^{-1}(\CU \Sigma_{\alpha})) \otimes \Hc^{*}(\pi_{\beta}^{-1}(\CU \Sigma_{\beta})) \\
  \xrightarrow{\;\;\cup\;\;}
  \Hc^{*}\bigl(\pi_{\alpha}^{-1}(\CU \Sigma_{\alpha}) \cap \pi_{\beta}^{-1}(\CU \Sigma_{\beta})\bigr)
  = \Hc^{*}(\CU \Sigma_{\alpha\cup\beta}).
  \qquad\qquad
\end{multline}
In term of the isomorphism~\eqref{eq:iso-HcCU-HhB}, this exactly means to multiply representatives lying in~\(\hB(\Sigma_{\alpha})\)
and~\(\hB(\Sigma_{\beta})\) inside~\(\BR(\Sigma_{\alpha\cup\beta})\), which gives elements in~\(\hB(\Sigma_{\alpha\cup\beta})\).
Note that this construction reduces to the product~\eqref{eq:cross-prod} if \(\alpha\) and~\(\beta\) are disjoint.

The \(*\)-product is visibly graded commutative,
something that was not obvious from the multiplication rules~\eqref{eq:relations-s-t}.
Looking back, we can see that these asymmetric formulas arose from the non-commutativity of the Alexander--Whitney map
and the fact that only one of the two vertices of the singular \(1\)-simplex~\(x\) in~\(X=D^{1}\)
can be the basepoint~\(e\).

The product~\eqref{eq:cross-prod-general} coincides with the \(*\)-product
given by Bahri--Bendersky--Cohen--Gitler~\cite[Thm.~1.4]{BahriEtAl:2012}
because the former map can be thought of as induced by the partial diagonal
\begin{equation}
  \hat\Delta_{I}^{J,L}\colon \hat{Z}(K_{I})\to\hat{Z}(K_{J})\wedge\hat{Z}(K_{L})
\end{equation}
defined in~\cite[eq.~(1.5)]{BahriEtAl:2012} to construct the \(*\)-product.
In our notation we have \(K=\Sigma\), \(J=\alpha\),~\(L=\beta\) and~\(I=J\cup L=\alpha\cup\beta\).
For the comparison one uses that the compactly supported cohomology of
\begin{equation}
  \CU \Sigma = \ZK_{\Sigma}(X,A)\setminus\WK_{\Sigma}(X,A)
\end{equation}
with~\((X,A)=(D^{1},S^{0})\) is equal to the reduced cohomology of the quotient
\begin{equation}
  \hat{Z}(K)=\hat{Z}(K(X,A))=\ZK_{\Sigma}(X,A) \bigm/ \WK_{\Sigma}(X,A).
\end{equation}
considered in~\cite{BahriEtAl:2012}.

As remarked earlier, the product formula in~\cite{BahriEtAl:2012} is valid for general polyhedral products.
We can recover the version for complex moment-angle complexes
if we replace the pair~\(\RhZK(\Sigma) = \hZK_{\Sigma}(D^{1},S^{0})\) from~\eqref{eq:def-RhZK} by \(\hZK_{\Sigma}(D^{2},S^{1})\).
In this case, the distinction between disjoint and non-disjoint index sets~\(\alpha\) and~\(\beta\)
in Baskakov's formula is not necessary for the corresponding \(*\)-product
because two monomials of the form~\eqref{eq:def-a1-am} with overlapping index sets
always multiply to~\(0\) in~\(\BC(\Sigma)\).

\medbreak

We finally consider another description of~\(H^{*}(\ZK(\Sigma))\) in the polytopal case.
Let \(P\) be a simple polytope with \(m\)~facets, and let \(\Sigma\) be the boundary complex of the dual simplicial polytope.
For any subset~\(\alpha\subset[m]\), let \(P_{\alpha}\subset P\) be the union of the corresponding facets.

\begin{lemma}
  There is a ring isomorphism
  \begin{equation*}
    \Theta_{\alpha}\colon H^{*}(P,P_{\alpha}) \to \Hc^{*}(\CU \Sigma_{\alpha})
  \end{equation*}
  for any~\(\alpha\subset[m]\). Moreover, the diagram
  \begin{equation*}
    \begin{tikzcd}
      H^{*}(P,P_{\alpha}) \otimes H^{*}(P,P_{\beta}) \arrow{d}{\Theta_{\alpha}\otimes\Theta_{\beta}} \arrow{r}{\cup} & H^{*}(P,P_{\alpha\cup\beta}) \arrow{d}{\Theta_{\alpha\cup\beta}} \\
      \Hc^{*}(\CU \Sigma_{\alpha}) \otimes \Hc^{*}(\CU \Sigma_{\beta}) \arrow{r}{*} & \Hc^{*}(\CU \Sigma_{\alpha\cup\beta})
    \end{tikzcd}
  \end{equation*}
  commutes for all~\(\alpha\),~\(\beta\subset[m]\).
\end{lemma}

\begin{proof}
  Let \(\Sigma'\) be the barycentric subdivision of~\(\Sigma\), considered as a triangulation of~\(\partial P\).
  As a topological space, \(\Sigma_{\alpha}\) can be identified with a subcomplex of~\(\Sigma'\),
  hence \(\CB \Sigma_{\alpha}\) with a subcomplex of \(\CB \Sigma'\approx P\).
  We can also identify \(P_{\alpha}\) with the union of the closed blocks (or cells) in~\(\Sigma'\) dual to the vertices in~\(\alpha\),
  \cf~\cite[\S 64]{Munkres:84}.

  We claim that the canonical inclusion of pairs
  \begin{equation}
    \label{eq:map-barycentric-sudiv}
    (\CB \Sigma_{\alpha},\Sigma_{\alpha}) \to (\CB \Sigma',P_{\alpha})
  \end{equation}
  is a strong deformation retract.
  Similar to the proof of~\cite[Lemma~70.1]{Munkres:84}, we can define a strong deformation retraction
  that moves the vertex~\(v_{\sigma}\in C\,\Sigma'\) corresponding to a simplex~\(\sigma\in\Sigma\)
  to the vertex~\(v_{\sigma\cap\alpha}\in C\,\Sigma_{\alpha}\) along a straight line, which is inside~\(\sigma\) if \(\sigma\cap\alpha\ne\emptyset\).
  If \(\sigma\) has no vertex in~\(\alpha\), then \(v_{\sigma}\) is moved to the apex~\(v_{\emptyset}\) of the cone,
  and \(v_{\emptyset}\) is mapped to itself. We extend the map linearly to each simplex~\(\tau\in C\,\Sigma'\).
  If \(\tau\) is contained in~\(\sigma\in\Sigma\),
  then it is mapped to the cone over the simplex~\(\sigma\cap\alpha\in\Sigma_{\alpha}\)
  (with the empty simplex~\(\emptyset\) giving the apex).
  The deformation retraction restricts to one from~\(P_{\alpha}\) onto~\(\Sigma_{\alpha}\).
  We therefore get an isomorphism
  \begin{equation}
    \Theta_{\alpha}\colon H^{*}(P,P_{\alpha}) \to H^{*}(\CB \Sigma_{\alpha},\Sigma_{\alpha}) = \Hc^{*}(\CU \Sigma_{\alpha})
  \end{equation}
  in cohomology.

  To show that the above diagram commutes, we work on the chain level. We use simplicial chains for the left-hand side of~\eqref{eq:map-barycentric-sudiv},
  which canonically map to singular chains on the right. We choose a vertex ordering for~\(\CB \Sigma_{\alpha\cup\beta}\) such that
  all vertices smaller than the apex~\(v_{\emptyset}\) are in~\(\alpha\) and all greater ones in~\(\beta\). (Some may be in both.)
  To a simplex~\(\sigma\in \CB \Sigma_{\alpha\cup\beta}\)
  we have to apply the Alexander--Whitney diagonal and possibly the projections from~\(\CB \Sigma_{\alpha\cup\beta}\) to~\(\CB \Sigma_{\alpha}\)
  and~\(\CU \Sigma_{\beta}\), which send ``superfluous'' vertices to~\(v_{\emptyset}\). Afterwards we evaluate the resulting tensor product on~\(a\otimes b\)
  where \(a\),~\(b\in C(P)\) are cocycles vanishing on~\(P_{\alpha}\) and~\(P_{\beta}\), respectively.

  Because of the way we have ordered the simplices, the following happens:
  If \(\sigma\) does not contain \(v_{\emptyset}\), then the result is \(0\) for both ways of going through the diagram.
  Otherwise we obtain \((-1)^{\deg{b}\deg{\sigma'}}\,a(\sigma')\,b(\sigma'')\) for both ways where \(\sigma'\)
  is the front face of~\(\sigma\) ending in~\(v_{\emptyset}\) and \(\sigma''\) the back face starting there.
  Hence the diagram commutes in either case.
\end{proof}

As a consequence, we get a ring isomorphism
\begin{equation}
  H^{*}(\RZK(P)) = \bigoplus_{\alpha\subset[m]} H^{*}(P,P_{\alpha})
\end{equation}
where the multiplication on the right-hand side is given by the cup products
\begin{equation}
  H^{*}(P,P_{\alpha}) \otimes H^{*}(P,P_{\beta}) \to H^{*}(P,P_{\alpha\cup\beta})
\end{equation}
for all~\(\alpha\),~\(\beta\subset[m]\).
This description of the cohomology ring of a real moment-angle manifold
was stated without proof by Gitler and López de Medrano~\cite[p.~1526]{GitlerLopezDeMedrano:2013}.\footnote{
In~\cite[p.~489]{LopezDeMedrano:2021}, López de Medrano writes that
``at the end [of~\cite{GitlerLopezDeMedrano:2013}] we announced, prematurely,
a formula for the cohomology ring of any~[\(\RZK(P)\)], but the proof ran into some technical problems''.}

The Alexander-dual description for moment-angle manifolds,
\begin{equation}
  H^{*}(\ZK(\Sigma)) = \bigoplus_{\alpha\subset[m]} \tilde H_{d+m-\deg{\alpha}-*}(P_{\alpha})
\end{equation}
where~\(d=\dim P-1\), has been provided by Bosio--Meersseman~\cite[Thm.~10.1]{BosioMeersseman:2006},
with the product given up to sign by the intersection products
\begin{equation}
  \tilde H_{d-k}(P_{\alpha}) \otimes \tilde H_{d-l}(P_{\beta})
  \to \tilde H_{d-(k+l)}(P_{\alpha\cap\beta})
\end{equation}
for~\(\alpha\),~\(\beta\subset[m]\) with~\(\alpha\cup\beta=[m]\) and~\(k\),~\(l\ge0\).


TITLE: If $x\in\mathbb{F}_{q^2}$, then $x+x^q\in\mathbb{F}_{q}$.
QUESTION [0 upvotes]: I'm reading a paper right now that uses this property but they don't really explain it well. They just write that "it is clear", but it isn't exactly clear to me.
Their proof involves choosing a basis $\{1,\alpha\}$ of $\mathbb{F}_{q^2}$ over $\mathbb{F}_q$ (where $\mathbb{F}_q$ is the finite field of size $q$). What I have so far is:
Take $(m+\alpha n)\in\mathbb{F}_{q^2}$ where $m,n\in\mathbb{F}_q$. Then
\begin{align}
(m+\alpha n)+(m+\alpha n)^q &= m+\alpha n + m^q + \alpha^q n^q \\
&= 2m + n(\alpha+\alpha^q).
\end{align}
I know $2m\in\mathbb{F}_{q}$. What only remains is to show that $(\alpha+\alpha^q)\in\mathbb{F}_q$. Can anyone give me a hint on how to show this?

REPLY [4 votes]: The Frobenius map $F(x)=x^q$ generates the Galois group of $\mathbb{F}_{q^2}$ over $\mathbb{F}_q$, so an element $x\in\mathbb{F}_{q^2}$ is in $\mathbb{F}_q$ iff $F(x)=x$.  Applying this to an element of the form $x+x^q$ we have $$F(x+x^q)=x^q+x^{q^2}=x^q+x$$ (since $x^{q^2}=x$ for all $x\in\mathbb{F}_{q^2}$) so $x+x^q\in\mathbb{F}_q$.


TITLE: Why does this term in the Euler-Lagrange equation vanish?
QUESTION [1 upvotes]: I am reading Evans, in the book it is stated that we define the following function
$$i(\tau) = \int_U L(Du + \tau Dv, u + \tau v, x)dx$$
Then we compute:
$$i'(\tau) = \int_U L_{p_i}(Du + \tau Dv, u + \tau v, x)v_{x_i} + L_{z}(Du + \tau Dv, u + \tau v, x)vdx$$
Which is the derivative of the inside of the integral with respect to $\tau$. First confusion, how do we know that
$\frac{d}{d\tau}(\int_U L dx) = \int_U \frac{d}{d\tau}\ L dx$
From then we evaluate at 0 giving:
$$i'(0) = \int_U \sum L_{pi}(Du, u, x)v_{x_i} + L_z(Du, u, x)vdx$$
Then the book claims integration by parts yields:
$$\int_U\bigg[-\sum L_{p_i}(Du, u, x)_{x_i} + L_z(Du, u, x)\bigg]vdx$$
I just don't see how that's possible, lets separate:
$$i'(0) = \int_U \sum L_{pi}(Du, u, x)v_{x_i} dx + \int_u L_z(Du, u, x)vdx$$
Ignore the right for now and treat each term in the sum as its own intergral and set $H = L_{pi}, v' = v_{x_i}$:
$$\int_U L_{pi}(Du, u, x)v_{x_i} dx = \int_U Hv' dx$$
Integration by parts gives:
$$\int_U Hv' dx = [Hv]_U - \int_U H'v dx$$
Ok that right term is present on the equation, but the left term seems to vanish, how are we so certain that
$$[Hv]_U = L_{p_i}(Du, u, x)v = 0$$ for all $i$?

REPLY [1 votes]: You can interchange the derivative and integral if you have sufficient smoothness/finiteness assumptions. This is known as Leibniz's rule for differentiation under the integral sign. You can either search on Wikipedia/see one of the appendices for Evans more closely.

Regarding integration by parts,  once you do the product rule, you're left with the following term
\begin{align}
\int_U\sum_{i=1}^n\frac{\partial }{\partial x_i}\left(\frac{\partial L}{\partial p_i}\bigg|_{(Du(x), u(x), x)} \cdot v(x)\right)\, dx
\end{align}
This is the divergence of the vector field $F:U\to\Bbb{R}^n$, $F_i(x)=\frac{\partial L}{\partial p_i}\bigg|_{(Du(x), u(x), x)} \cdot v(x)$. So by the divergence theorem, $\int_U\text{div}F\,dx=\int_{\partial U}\langle F,\nu \rangle\, d\sigma$, where $\nu$ is the unit outward normal to $U$ and $d\sigma$ denotes the surface measure on the boundary. But notice that since $v$ vanishes on the boundary, so does $F$, so this term is zero.

Regarding point $2$, we actually don't even need to invoke the divergence theorem. We have the following elementary fact which follows from the fundamental theorem of calculus and Fubini (this is actually one of the steps used in proving the divergence/Stokes theorem).

Let $U\subset\Bbb{R}^n$ be open, $f:U\to\Bbb{R}$ a $C^1$  function with $\text{supp}(f)$ compact and contained in $U$. Then, $\int_U\frac{\partial f}{\partial x_i}\,dx=0$.

Here, there's no need to assume anything about the boundary of $U$, we also don't need to assume $U$ is a bounded set. The reason for assuming compactly supported functions is just to ensure integrability of everything.
To prove this, define for convenience $\phi:\Bbb{R}^n\to\Bbb{R}$ as $\phi(x)=f(x)$ if $x\in U$ and $\phi(x)=0$ otherwise. Since $f$ has support contained in $U$, it follows $\phi$ is $C^1$ on all of $\Bbb{R}^n$. Also,
\begin{align}
\int_U\frac{\partial f}{\partial x_i}\,dx&=\int_U\frac{\partial \phi}{\partial x_i}\,dx=\int_{\Bbb{R}^n}\frac{\partial \phi}{\partial x_i}\,dx.
\end{align}
Why? Now, we use Fubini's theorem to integrate first with respect to $x_i$:
\begin{align}
\int_{\Bbb{R}^n}\frac{\partial \phi}{\partial x_i}\,dx&=\int_{\Bbb{R}^{n-1}}\left[\int_{\Bbb{R}}\frac{\partial \phi}{\partial x_i}(x)\,dx_i\right]dx_1\cdots dx_{i-1}\,dx_{i+1}\cdots dx_n.
\end{align}
Now, use the fundamental theorem of calculus on the square brackets, and the fact that $\phi$ is compactly supported to deduce that it is $0$. Hence the entire integral vanishes.


TITLE: Using Quantifiers to express "At least two"
QUESTION [1 upvotes]: I have been stumped trying to solve problem e. I am confused as how to state "At least two students..." using quantifiers! I know if it were ONE student it would be: ∃x,y(Q(x,Jeopardy)).And all students would be:∀x,y(Q(x,Jeopardy)) So how can I expand on this?

REPLY [1 votes]: As another way to put this, this is the negation of the "there exists at most one" quantifier, which I usually denote as $\;\langle !x :: P(x) \rangle\;$.  Looking at another answer of mine and doing some basic manipulations, that negation would be
$$
    \langle \forall z :: \langle \exists x :: x \not= z \land P(x) \rangle \rangle
$$
where of course $\;P(x)\;$ stands for "$\;x\;$ has been on Jeopardy", and $\;x,z\;$ range over the school students.
Perhaps that looks counterintuitive, but it is correct: you could read it as, "Whichever $\;z\;$ you take away (even if $\;P\;$ holds for it), you will still have at least one $\;x\;$ left that satisfies $\;P\;$."
Note how this is the only formula which contains $\;P\;$ only once: the usual formulations have two occurrences of $\;P\;$.