#  Generalised Homotopy and Commutativity Principle

Ravi A. Rao and Sampat Sharma

###### Abstract.

In this paper, we study the action of special $n\times n$ linear (resp.
symplectic) matrices which are homotopic to identity on the right invertible
$n\times m$ matrices. We also prove that the commutator subgroup of
$\rm{O}_{2n}(R[X])$ is two stably elementary orthogonal for a local ring $R$
with $\frac{1}{2}\in R$ and $n\geq 3.$

###### Key words and phrases:

Keywords:

###### 1991 Mathematics Subject Classification:

2

020 Mathematics Subject Classification:13C10, 13H99, 19B14.

Homotopy, Classical groups, Local-Global Principle

Throughout this article we will assume $R$ to be a commutative ring with
$1\neq 0.$

##  1\. Introduction

In $($[15, Corollary 1.4]$)$, Suslin established the normality of the
elementary linear subgroup $\mbox{\rm E}_{n}(R)$ in $\mbox{\rm GL}_{n}(R)$,
for $n\geq 3$. This was a major surprise at that time as it was known due to
the work of Cohn in [6] that in general $\mbox{\rm E}_{2}(R)$ is not normal in
$\mbox{\rm GL}_{2}(R)$. This is the initial precursor to study the non-stable
$K_{1}$ groups ${\mbox{\rm SL}_{n}(R)}/{\mbox{\rm E}_{n}(R)},~{}n\geq 3$.

This theorem can also be got as a consequence of the local-global principle of
Quillen (for projective modules) in [9]; and its analogue for the linear group
of elementary matrices $\mbox{\rm E}_{n}(R[X])$, when $n\geq 3$ due to Suslin
in [15]. In fact, in [3] it is shown that, in some sense, the normality
property of the elementary group $\mbox{\rm E}_{n}(R)$ in $SL_{n}(R)$ is
equivalent to having a local-global principle for $\mbox{\rm E}_{n}(R[X])$.

In [2], Bak proved the following beautiful result:

######  Theorem 1.1.

$($Bak$)$ For an almost commutative ring $R$ with identity with centre $C(R)$.
The group ${\mbox{\rm SL}_{n}(R)}/{\mbox{\rm E}_{n}(R)}$ is nilpotent of class
atmost $\delta(C(R))+3-n$, where $\delta(C(R))<\infty~{}and~{}n\geq 3$, where
$\delta(C(R))$ is the Bass–Serre- dimension of $C(R)$.

This theorem, which is proved by a localization and completion technique,
which evolved from an adaptation of the proof of the Suslin’s $K_{1}$-analogue
of Quillen’s local-global principle was further investigated in [10]. In [10],
we proved that

######  Theorem 1.2.

Let $R$ be a local ring, and let $A=R[X]$. Then the group ${\mbox{\rm
SL}_{n}(A)}/{\mbox{\rm E}_{n}(A)}$ is an abelian group for $n\geq 3$.

This theorem is a simple consequence of the following principle:

######  Theorem 1.3.

$($[10, Theorem 2.19]$)$ $($Homotopy and commutativity principle$):$ Let $R$
be a commutative ring. Let $\alpha\in\mbox{\rm SL}_{n}(R)$, $n\geq 3$, be
homotopic to the identity. Then, for any $\beta\in\mbox{\rm SL}_{n}(R)$,
$\alpha\beta=\beta\alpha\varepsilon$, for some $\varepsilon\in\mbox{\rm
E}_{n}(R)$.

This principle is a consequence of the Quillen–Suslin’s local-global
principle; and using a non-symmetric application of it as done by Bak in [2].

Using Bak’s localization method, in [13], Stepanov proved the following result
for all simply connected Chevalley group of rank $>1$ :

######  Theorem 1.4.

Let $G$ be a simply connected group of rank $>1$ with $G(R)=E(R)$ when $R$ is
a local ring. Then for any commutative ring $R$ with 1,

$[\overset{\sim}{E}(R),G(R)]=E(R)$

where $\overset{\sim}{E}(R)=\bigcap_{(s_{1},\ldots s_{l})\in Um_{l}(R)}\prod
G(R,s_{1}R),\ldots,G(R,s_{l}R),$ denotes the extended elementary group.

In this paper, we generalise the homotopy and commutativity principle to any
$n\times m$ right invertible matrix over a commutative ring $R$. In
particular, we prove that :

######  Theorem 1.5.

$($Generalised homotopy and commutativity principle$)$ Let $R$ be a
commutative ring and $V\in\mbox{\rm Um}_{n,m}(R)$ with $m>n\geq
2~{}\mbox{or}~{}m=n\geq 3.$ Let $\delta\in\mbox{\rm SL}_{n}(R)$ be homotopic
to identity. Let $\delta(T)$ be a homotopy of $\delta.$ Then
$\exists~{}\sigma(T)\in\mbox{\rm SL}_{m}(R[T],(T))$ such that

$\delta(T)V=V\sigma(T)~{}\mbox{and}~{}\sigma(T)^{-1}(\delta(T)\perp
I_{m-n})\in\mbox{\rm E}_{m}(R[T],(T)).$

Moreover, if $\sigma(1)=\sigma$, then we have $\delta V=V\sigma$ and
$\sigma^{-1}(\delta\perp I_{m-n})\in\mbox{\rm E}_{m}(R).$

We also prove similar results in the case of symplectic groups (see theorem
3.13). We prove the similar statement in the case of orthogonal groups as well
with $m\geq n+2,n\geq 2$ (see theorem 4.10). As a consequence we prove that
linear and symplectic quotients are abelian, but in the case of orthogonal
quotients we could only establish the following:

######  Theorem 1.6.

Let $m\geq 3$, $R$ be a local ring, $\frac{1}{2}\in R.$ Then
$([\rm{O}_{2m}R[X],\rm{O}_{2m}R[X]]\perp I_{2})\subseteq\mbox{\rm
EO}_{2m+2}(R[X]).$

We do believe that orthogonal quotient groups are also abelian; as it is the
case when the base ring is regular local ring containing a field (see $($[10,
Corollary 4.21]$)$.

##  2\. Generalised Homotopy and Commutativity Principle for Linear Groups

Let $v=(a_{0},a_{1},\ldots,a_{r}),w=(b_{0},b_{1},\ldots,b_{r})$ be two rows of
length $r+1$ over a commutative ring $R$. A row $v\in R^{r+1}$ is said to be
unimodular if there is a $w\in R^{r+1}$ with $\langle
v,w\rangle=\Sigma_{i=0}^{r}a_{i}b_{i}=1$ and ${\mbox{\rm Um}}_{r+1}(R)$ will
denote the set of unimodular rows (over $R$) of length $r+1$.

The group of elementary matrices is a subgroup of $\mbox{\rm GL}_{r+1}(R)$,
denoted by $\mbox{\rm E}_{r+1}(R)$, and is generated by the matrices of the
form $e_{ij}(\lambda)=I_{r+1}+\lambda\mbox{\rm E}_{ij}$, where $\lambda\in
R,~{}i\neq j,~{}1\leq i,j\leq r+1,~{}\mbox{\rm E}_{ij}\in M_{r+1}(R)$ whose
$ij^{th}$ entry is $1$ and all other entries are zero. The elementary linear
group $\mbox{\rm E}_{r+1}(R)$ acts on the rows of length $r+1$ by right
multiplication. Moreover, this action takes unimodular rows to unimodular rows
: ${{\mbox{\rm Um}}_{r+1}(R)}/{\mbox{\rm E}_{r+1}(R)}$ will denote the set of
orbits of this action; and we shall denote by $[v]$ the equivalence class of a
row $v$ under this equivalence relation.

######  Definition 2.1.

An $\alpha\in M_{n\times m}(R)$ is said to be right invertible if
$\exists~{}\beta\in M_{m\times n}(R)$ such that $\alpha\beta=I_{n}.$ We will
denote set of all $n\times m$ right invertible matrices by ${\mbox{\rm
Um}}_{n,m}(R).$

######  Definition 2.2.

An $R$-module $P$ is said to be stably free of type $n$, if $P\oplus R^{n}$ is
a free module.

To every $\alpha\in{\mbox{\rm Um}}_{n,m}(R)$, we can associate a stably free
module $P$ of type $n$, in the following way:

Since $\alpha\in{\mbox{\rm Um}}_{n,m}(R)$, it gives rise to a surjective map
$R^{m}\overset{\alpha}{\longrightarrow}R^{n}$. Let $P=\mbox{Ker}({\alpha})$,
then we have a short exact sequence

$0\longrightarrow P\longrightarrow R^{m}\longrightarrow R^{n}\longrightarrow
0.$

Since $R^{n}$ is a free module, the above short exact sequence splits and we
have $P\oplus R^{n}\simeq R^{m}.$

To every stably free module $P$ of type $n$, we can associate an element
$\alpha$ of ${\mbox{\rm Um}}_{n,m}(R)$, for some $m$, in the following way:

Since $P$ is stably free, we have a short exact sequence

$0\longrightarrow P\longrightarrow R^{m}\longrightarrow R^{n}\longrightarrow
0.$

Let $\alpha$ to be the matrix of the map $R^{m}\longrightarrow R^{n}$. Since
$R^{n}$ is a free module, the above short exact sequence splits and we have
$\alpha\in{\mbox{\rm Um}}_{n,m}(R).$

######  Lemma 2.3.

$($[8, Chapter 1, Proposition 4.3]$)$ An $\alpha\in{\mbox{\rm Um}}_{n,m}(R)$
is completable to an invertible matrix of determinant $1$ if and only if the
corresponding stably free module is free.

######  Lemma 2.4.

Let $R$ be a local ring and $V\in{\mbox{\rm Um}}_{n,m}(R)$ for
$m~{}(\mbox{or}~{}n)\geq 2$. Then $V$ is completable to an elementary matrix.

${{\bf Proof:}}$ Every $V\in\mbox{Um}_{n,m}(R)$ corresponds to a stably free
module $P.$ Since a projective module over a local ring is free, $P$ is free.
In view of lemma 2.3, V is completable to a matrix $W\in\mbox{\rm
SL}_{m}(R)=\mbox{\rm E}_{m}(R).$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

######  Definition 2.5.

Let R be a ring. A matrix $\alpha\in\mbox{\rm SL}_{n}(R)$ is said to be
homotopic to identity if there exists a matrix $\gamma(X)\in\mbox{\rm
SL}_{n}(R[X])$ such that $\gamma(0)=\textit{Id}~{}and~{}\gamma(1)=\alpha$.

######  Proposition 2.6.

Let $R$ be a local ring and $V\in\mbox{\rm Um}_{n,m}(R)$ for $m>n\geq 2$ or
$m=n\geq 3.$ Let $\delta\in\mbox{\rm SL}_{n}(R)$ be homotopic to identity. Let
$\delta(T)$ be a homotopy of $\delta.$ Then there exists,
$\sigma(T)\in\mbox{\rm SL}_{m}(R[T])$ with $\sigma(0)=\textit{Id}$ and
$\sigma(T)^{-1}(\delta(T)\perp I_{m-n})\in\mbox{\rm E}_{m}(R[T])$ such that

$\delta(T)V=V\sigma(T).$

${{\bf Proof:}}$ In view of lemma 2.4, $V$ is completable to a matrix
$W\in\mbox{\rm SL}_{m}(R)$. Since $R$ is a local ring, $W\in\mbox{\rm
E}_{m}(R).$ By $($[15, Corollary 1.4]$)$, $\mbox{\rm
E}_{m}(R[T])\trianglelefteq\mbox{\rm SL}_{m}(R[T])$, for $m\geq 3.$ Thus there
exists $\varepsilon_{1}(T)\in\mbox{\rm E}_{m}(R[T])$ such that

$(\delta(T)\perp I_{m-n})W(\delta(T)\perp I_{m-n})^{-1}=\varepsilon_{1}(T).$

Thus we have $(\delta(T)\perp
I_{m-n})W=\varepsilon_{1}(T)W^{-1}W(\delta(T)\perp I_{m-n}).$ Again by
normality of $\mbox{\rm E}_{m}(R[T])$ in $\mbox{\rm SL}_{m}(R[T]$ for $m\geq
3,$ there exists $\varepsilon(T)\in\mbox{\rm E}_{m}(R[T])$ such that

$(\delta(T)\perp I_{m-n})W=W(\delta(T)\perp I_{m-n})\varepsilon(T).$

Note that $\varepsilon(0)=\textit{Id}.$ Upon taking $\sigma(T)=(\delta(T)\perp
I_{m-n})\varepsilon(T)$ and multiplying above equation
by$\begin{bmatrix}I_{n}&0\\\ 0&0\\\ \end{bmatrix}$, we gets desired result.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

######  Theorem 2.7.

Let $R$ be a commutative ring and $V\in\mbox{\rm Um}_{n,m}(R)$ with $m>n\geq
2~{}\mbox{or}~{}m=n\geq 3.$ Let $\delta\in\mbox{\rm SL}_{n}(R)$ be homotopic
to identity. Let $\delta(T)$ be a homotopy of $\delta.$ Then
$\exists~{}\sigma(T)\in\mbox{\rm SL}_{m}(R[T],(T))$ such that

$\delta(T)V=V\sigma(T)~{}\mbox{and}~{}\sigma(T)^{-1}(\delta(T)\perp
I_{m-n})\in\mbox{\rm E}_{m}(R[T],(T)).$

Moreover , if $\sigma(1)=\sigma$, then we have $\delta V=V\sigma$ and
$\sigma^{-1}(\delta\perp I_{m-n})\in\mbox{\rm E}_{m}(R).$

${{\bf Proof:}}$ Define,

$\displaystyle J=$ $\displaystyle\\{s\in
R~{}|~{}\delta(T)_{s}V_{s}=V_{s}\sigma(T)~{}\mbox{for
~{}some}~{}\sigma(T)\in\mbox{\rm SL}_{m}(R_{s}[T],(T))~{}$
$\displaystyle\mbox{with}~{}\sigma(T)^{-1}(\delta(T)_{s}\perp
I_{m-n})\in\mbox{\rm E}_{m}(R_{s}[T])\\}.$

$\bf{Claim:}$ $J$ is an ideal.

For $s\in J,\lambda\in R$, clearly $\lambda s\in J$. So we need to prove that
if $s_{1},s_{2}\in J$ then $s_{1}+s_{2}\in J.$ Since $s_{1},s_{2}\in J$, we
have $(s_{1}+s_{2})s_{1}$, $(s_{1}+s_{2})s_{2}\in J.$ We rename
$R_{s_{1}+s_{2}}$ by $R$, now it suffices to show that

$\displaystyle\delta(T)V$
$\displaystyle=V\sigma(T)~{}\mbox{for}~{}\mbox{some}~{}\sigma(T)\in\mbox{\rm
SL}_{m}(R[T],(T))$ $\displaystyle\mbox{with}~{}\sigma(T)^{-1}(\delta(T)\perp
I_{m-n})\in\mbox{\rm E}_{m}(R[T],(T))~{}\mbox{provided
that}~{}s_{1}+s_{2}=1~{}\mbox{ and}$ (1)
$\delta(T)_{s_{1}}V_{s_{1}}=V_{s_{1}}\sigma_{1}(T)~{}\mbox{with}~{}\sigma_{1}(T)^{-1}(\delta(T)_{s_{1}}\perp
I_{m-n})\in\mbox{\rm E}_{m}(R_{s_{1}}[T],(T)),$ (2)
$\delta(T)_{s_{2}}V_{s_{2}}=V_{s_{2}}\sigma_{2}(T)~{}\mbox{with}~{}\sigma_{2}(T)^{-1}(\delta(T)_{s_{2}}\perp
I_{m-n})\in\mbox{\rm E}_{m}(R_{s_{2}}[T],(T)).$

Let

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sigma_{1}(T)(\delta(T)_{s_{1}}\perp
I_{m-n})^{-1}=\varepsilon_{1}(T)\in\mbox{\rm E}_{m}(R_{s_{1}}[T],(T)),$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sigma_{2}(T)(\delta(T)_{s_{2}}\perp
I_{m-n})^{-1}=\varepsilon_{2}(T)\in\mbox{\rm E}_{m}(R_{s_{2}}[T],(T)).$

Now, $\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\
\end{bmatrix}\varepsilon_{1}(T)_{s_{2}}\varepsilon_{2}(T)_{s_{1}}^{-1}=\begin{bmatrix}V_{s_{1}s_{2}}\\\
0\\\ \end{bmatrix}.$ Let
$\theta(T)=\varepsilon_{1}(T)_{s_{2}}\varepsilon_{2}(T)_{s_{1}}^{-1}.$ By
Quillen’s splitting property, $\mbox{for}~{}b\in(s_{2}^{N}),~{}N>>0$, we have

(3) $\theta(T)=\theta(bT)\\{\theta(bT)^{-1}\theta(T)\\}$

$~{}\mbox{with}~{}\theta(bT)\in\mbox{\rm
E}_{m}(R_{s_{1}}[T]),~{}\mbox{and}~{}\theta(bT)^{-1}\theta(T)\in\mbox{\rm
E}_{m}(R_{s_{2}}[T]).$

Since $\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\
\end{bmatrix}\theta(T)=\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\ \end{bmatrix}$.
We have, $\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\
\end{bmatrix}\theta(bT)=\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\ \end{bmatrix}$
and
$\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\
\end{bmatrix}\theta(bT)^{-1}\theta(T)=\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\
\end{bmatrix}.$

Define, $\theta(bT)=\eta_{1}(T)^{-1},~{}\theta(bT)^{-1}\theta(T)=\eta_{2}(T)$.
Thus we have $V_{i}\eta_{i}(T)=V_{i}$ with $\eta_{i}(0)=\textit{Id}$ and
$\eta_{i}(T)\in\mbox{\rm E}_{m}(R_{s_{i}}[T])$ for $i=1,2.$

In view of equation 3, we gets

(4)
$(\eta_{1}(T)\varepsilon_{1}(T))_{s_{2}}=(\eta_{2}(T)\varepsilon_{2}(T))_{s_{1}}.$

Now, by equation 3 and equation 4,

$\delta(T)_{s_{1}}V_{s_{1}}=V_{s_{1}}\eta_{1}(T)\sigma_{1}(T)$
$\delta(T)_{s_{2}}V_{s_{2}}=V_{s_{2}}\eta_{2}(T)\sigma_{2}(T).$

In view of equation 4, we have
$(\eta_{1}(T)\sigma_{1}(T))_{s_{2}}=(\eta_{2}(T)\sigma_{2}(T))_{s_{1}}$. Since
$s_{1}+s_{2}=1$, $\exists~{}\sigma(T)\in\mbox{\rm SL}_{m}(R[T])$ such that
$\sigma(T)_{s_{1}}=\eta_{1}(T)\sigma_{1}(T)$ and
$\sigma(T)_{s_{2}}=\eta_{2}(T)\sigma_{2}(T)$ with
$\sigma(T)_{s_{i}}^{-1}(\delta(T)_{s_{i}}\perp I_{m-n})\in\mbox{\rm
E}_{m}(R_{s_{i}}[T])$ for $i=1,2.$ Since $s_{1}$ and $s_{2}$ are comaximal, by
Suslin’s local-global principle $($[15, Theorem 3.1]$)$, we have

$\displaystyle\delta(T)V$
$\displaystyle=V\sigma(T)~{}\mbox{for}~{}\mbox{some}~{}\sigma(T)\in\mbox{\rm
SL}_{m}(R[T],(T))~{}$
$\displaystyle\mbox{with}~{}\sigma(T)^{-1}(\delta(T)\perp I_{m-n})\in\mbox{\rm
E}_{m}(R[T],(T)).$

This proves that $J$ is an ideal.

In view of Proposition 2.6, for every maximal ideal $\mathfrak{m}$ of $R$, we
have

$\delta(T)_{\mathfrak{m}}V_{\mathfrak{m}}=V_{\mathfrak{m}}\sigma^{{}^{\prime}}(T)~{}\mbox{with}~{}\sigma^{{}^{\prime}}(T)^{-1}(\delta(T)_{\mathfrak{m}}\perp
I_{m-n})\in\mbox{\rm E}_{m}(R_{\mathfrak{m}}[T],(T)).$

Thus there exists $s\in R\setminus\mathfrak{m}$, such that

$\delta(T)_{s}V_{s}=V_{s}\sigma^{{}^{\prime}}(T)~{}\mbox{with}~{}\sigma^{{}^{\prime}}(T)^{-1}(\delta(T)_{s}\perp
I_{m-n})\in\mbox{\rm E}_{m}(R_{s}[T],(T)).$

Therefore $J\nsubseteq\mathfrak{m}$, for any maximal ideal $\mathfrak{m}$ of
$R$ i.e. $1\in J.$ Thus $\exists~{}\sigma(T)\in SL_{m}(R[T],(T))$ such that

$\delta(T)V=V\sigma(T)~{}\mbox{with}~{}\sigma(T)^{-1}(\delta(T)\perp
I_{m-n})\in\mbox{\rm E}_{m}(R[T],(T)).$

Now put $T=1$, and take $\sigma(1)=\sigma$ to get the desired result.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

######  Corollary 2.8.

$($[10, Theorem 2.19]$)$ Let $n\geq 3$ and $\alpha,\beta\in\mbox{\rm
SL}_{n}(R).$ Let either $\alpha$ or $\beta$ be homotopic to identity. Then
$\alpha\beta=\beta\alpha\varepsilon,~{}~{}\mbox{for~{}
some}~{}\varepsilon\in\mbox{\rm E}_{n}(R).$

${{\bf Proof:}}$ Let us assume that $\alpha$ is homotopic to identity, so
there exists $\delta(T)\in\mbox{\rm SL}_{n}(R[T])$ such that
$\delta(0)=\textit{Id}~{}\mbox{and}~{}\delta(1)=\alpha.$ By theorem 2.7, there
exists $\varepsilon(T)\in\mbox{\rm E}_{n}(R[T])$ with
$\varepsilon(0)=\textit{Id}$ such that

$\delta(T)\beta=\beta\delta(T)\varepsilon(T).$

Put $T=1$ to get the desired result.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

######  Corollary 2.9.

$($Vaserstein$)$ Let $\delta\in\mbox{\rm SL}_{n}(R)$ and $V\in\mbox{\rm
Um}_{n,m}(R),m>n\geq 2~{}\mbox{or}~{}n=m\geq 3.$ Then $\delta V=V\sigma$ for
some $\sigma\in\mbox{\rm SL}_{m}(R)$ with
$(\sigma\perp\delta^{-1})\in\mbox{\rm E}_{n+m}(R).$

${{\bf Proof:}}$ By Whitehead’s Lemma, $(\delta\perp\delta^{-1})\in\mbox{\rm
E}_{2m}(R).$ Since every elementary matrix is homotopic to identity, thus by
theorem 2.7,

$(\delta\perp\delta^{-1})(V\perp I_{n})=(V\perp
I_{n})\sigma^{{}^{\prime}},~{}\mbox{with}~{}\sigma^{{}^{\prime}}\in\mbox{\rm
E}_{n+m}(R).$

Write $\sigma^{{}^{\prime}}=\begin{bmatrix}\alpha&\beta\\\ \gamma&\zeta\\\
\end{bmatrix}$ where $\alpha\in M_{m\times m}(R),~{}\beta\in M_{m\times
n}(R),~{}\gamma\in M_{n\times m}(R),~{}\zeta\in M_{n\times n}(R)$. Thus we
have,

$(\delta V\perp\delta^{-1})=\begin{bmatrix}V\alpha&V\beta\\\ \gamma&\zeta\\\
\end{bmatrix}.$

Upon compairing both sides we gets $\gamma=0$ and $\zeta=\delta^{-1}.$
Therefore

$\begin{bmatrix}\alpha&\beta\\\ 0&\delta^{-1}\\\ \end{bmatrix}\in\mbox{\rm
E}_{n+m}(R).$

Now, take $\alpha=\sigma$, so we have $(\sigma\perp\delta^{-1})\in E_{n+m}(R)$
and $\delta V=V\sigma.$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

######  Lemma 2.10.

$($Suslin$)$ $($[16, Lemma 2.8]$)$ Let $r\geq 3$ and
$v_{1},~{}v_{2},~{}w~{}\in M_{1,r}(R)$ be such that $\langle
v_{1},w\rangle=\langle v_{2},w\rangle=1$, then $v_{1}\overset{\mbox{\rm
E}}{\sim}v_{2}.$

######  Corollary 2.11.

Let $n\geq 3$ and $\alpha=\begin{bmatrix}a_{1},&\ldots,&a_{n}\\\
b_{1},&\ldots,&b_{n}\\\ \end{bmatrix}\in\mbox{\rm Um}_{2,n}(R).$ Then,

$(a_{1},\ldots,a_{n})\overset{\mbox{\rm E}}{\sim}(b_{1},\ldots,b_{n}).$

${{\bf Proof:}}$ Since $\alpha\in\mbox{Um}_{2,n}(R)$,
$\exists~{}\beta=\begin{bmatrix}c_{1}&d_{1}\\\ \vdots&\vdots\\\
c_{n}&d{{}_{n}}\\\ \end{bmatrix}\in M_{n,2}(R),$ such that
$\alpha\beta=I_{2}.$ Let
$w=(c_{1}+d_{1},\ldots,c_{n}+d_{n})$. Since
$\langle(a_{1},\ldots,a_{n}),w\rangle=\langle(b_{1},\ldots,b_{n}),w\rangle=1.$
Thus by lemma 2.10,

$(a_{1},\ldots,a_{n})\overset{E}{\sim}(b_{1},\ldots,b_{n}).$

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

######  Corollary 2.12.

$($Roitman$)$ $($[11, Theorem 8]$)$ Let $(x_{0},\ldots,x_{n})\in\mbox{\rm
Um}_{n+1}(R),~{}n\geq 2$ and $0\leq k\leq n-1,~{}y_{i}\in
R~{}\mbox{for}~{}k\leq i\leq n.$ Let I be an ideal of $R$ generated by
$2\times 2$ minors of the matrix

$\alpha=\begin{bmatrix}x_{k},&\ldots,&x_{n}\\\ y_{k},&\ldots,&y_{n}\\\
\end{bmatrix}.$

Assume that $Rx_{0}+\ldots+Rx_{k-1}+I=R.$ Then

$(x_{0},\ldots,x_{k-1},x_{k},\ldots,x_{n})\overset{E}{\sim}(x_{0},\ldots,x_{k-1},y_{k},\ldots,y_{n}).$

${{\bf Proof:}}$ Consider the ring $\overset{-}{R}=R/Rx_{0}+\ldots+Rx_{k-1}$,
by hypothesis we have $\overset{-}{R}=\overset{-}{I}$, therfore
$\overset{-}{\alpha}\in\mbox{Um}_{2,n-k+1}(\overset{-}{R}).$ Thus by corollary
2.11, $\exists~{}\overset{-}{\varepsilon}\in\mbox{\rm
E}_{n-k+1}(\overset{-}{R})$ such that
$(\overset{-}{x}_{k},\ldots,\overset{-}{x}_{n})\overset{-}{\varepsilon}=(\overset{-}{y}_{k},\ldots,\overset{-}{y}_{n}).$
Let $\varepsilon\in\mbox{\rm E}_{n-k+1}(R)$ be a lift of
$\overset{-}{\varepsilon}.$ Therefore,

$(x_{k},\ldots,x_{n})\varepsilon=(y_{k}+a_{k},\ldots,y_{n}+a_{n}),~{}\mbox{for~{}some}~{}a_{i}\in
Rx_{0}+\ldots+Rx_{k-1}.$

Thus we have,
$(x_{0},\ldots,x_{k-1},x_{k},\ldots,x_{n})(I_{k}\perp\varepsilon)=(x_{0},\ldots,x_{k-1},y_{k}+a_{k},\ldots,y_{n}+a_{n}).$
$\mbox{Since}~{}a_{i}\in Rx_{0}+\ldots+Rx_{k-1}$, we have

$(x_{0},\ldots,x_{k-1},x_{k},\ldots,x_{n})\overset{E}{\sim}(x_{0},\ldots,x_{k-1},y_{k},\ldots,y_{n}).$

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

##  3\. Generalised Homotopy and Commutativity Principle for Symplectic Groups

######  Notation 3.1.

Let $\psi_{1}=\begin{bmatrix}0&1\\\ -1&0\\\
\end{bmatrix},~{}~{}\psi_{n}=\psi_{n-1}\perp\psi_{1};$ for $n>1$.

######  Notation 3.2.

Let $\sigma$ be the permutation of the natural numbers given by
$\sigma(2i)=2i-1$ and $\sigma(2i-1)=2i$.

######  Notation 3.3.

$\mbox{\rm E}_{ij}(\lambda)$ will denote a matrix whose $ij^{th}$ entry is
$\lambda$ and all other entries are $0.$

######  Definition 3.4.

Symplectic group $\mbox{\rm Sp}_{2m}(R):$ The subgroup of $\mbox{\rm
GL}_{2m}(R)$ consisting of all $2m\times 2m$ matrices $\\{\alpha\in\mbox{\rm
GL}_{2m}(R)~{}\mid\alpha^{t}\psi_{m}\alpha=\psi_{m}\\}$.

######  Definition 3.5.

Elementary symplectic group $\mbox{\rm ESp}_{2m}(R)$: We define for $1\leq
i\neq j\leq 2m,~{}z\in R,$

$se_{ij}(z)=\begin{cases}I_{2m}+zE_{ij},~{}~{}~{}\textit{if}~{}i=\sigma(j);\\\
I_{2m}+zE_{ij}-(-1)^{i+j}zE_{\sigma(j)\sigma(i)},~{}~{}~{}\textit{if}~{}i\neq\sigma(j).\par\end{cases}$

It is easy to verify that all these matrices belong to $\mbox{\rm
Sp}_{2m}(R)$. We call them the elementary symplectic matrices over $R$. The
subgroup generated by them is called the elementary symplectic group and is
denoted by $\mbox{\rm ESp}_{2m}(R)$.

######  Notation 3.6.

$\mbox{\rm Sp}\mbox{\rm Um}_{2n,2m}(R)=\\{V\in\mbox{\rm
Um}_{2n,2m}(R)|V\psi_{m}V^{t}=\psi_{n}\\}.$

######  Lemma 3.7.

$($Rao-Swan$)$ Let $n\geq 2$ and $\varepsilon\in\mbox{\rm E}_{2n}(R)$. Then
there exists $\rho\in\mbox{\rm E}_{2n-1}(R)$ such that
$\varepsilon(1\perp\rho)\in\mbox{\rm ESp}_{2n}(R).$

${{\bf Proof:}}$ For a proof see $($[5, Lemma 4.4]$).$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

######  Lemma 3.8.

$($ Vaserstein$)$ $($[19, Lemma 5.5]$)$ For an associative ring $R$ with
identity, and for any natural number $m$

$e_{1}\mbox{\rm E}_{2m}(R)=e_{1}(\mbox{\rm Sp}_{2m}(R)\cap\mbox{\rm
E}_{2m}(R)).$

######  Remark 3.9.

It was observed in $($[4, Lemma 2.13]$)$ that Vaserstein’s proof actually
shows that $e_{1}\mbox{\rm E}_{2m}(R)=e_{1}\mbox{\rm ESp}_{2m}(R).$

######  Theorem 3.10.

(Local-Global principle for the symplectic groups) $($[7, Theorem 3.6]$)$ Let
$m\geq 2$ and $\alpha(X)\in\mbox{\rm Sp}_{2m}(R[X])$, with
$\alpha(0)=\textit{Id}$. Then $\alpha(X)\in\mbox{\rm ESp}_{2m}(R[X])$ if and
only if for any maximal ideal $\mathfrak{m}\subset R$, the canonical image of
$\alpha(X)\in\mbox{\rm Sp}_{2m}(R_{\mathfrak{m}}[X])$ lies in $\mbox{\rm
ESp}_{2m}(R_{\mathfrak{m}}[X])$.

######  Lemma 3.11.

Let $R$ be a local ring, $m\geq n\geq 1$ and $V\in\mbox{\rm Sp}\mbox{\rm
Um}_{2n,2m}(R).$ Then $V$ is completable to an elementary symplectic matrix.

${{\bf Proof:}}$ We will proceed by induction on $n.$ Since $\mbox{\rm
Sp}_{2}(R)=\mbox{\rm ESp}_{2}(R),$ we are done for the case $m=n=1.$ Let us
assume that $n=1,m>1$, since $V\in\mbox{\rm Sp}\mbox{\rm
Um}_{2,2m}(R)\subseteq\mbox{\rm Um}_{2,2m}(R)$ and $R$ is a local ring, there
exists $\varepsilon\in\mbox{\rm E}_{2m}(R)$ such that

$V\varepsilon=\begin{bmatrix}1&0&\cdots&0\\\ 0&1&\cdots&0\\\ \end{bmatrix}.$

In view of Rao-Swan Lemma, there exists $\rho\in\mbox{\rm E}_{2n-1}(R)$ such
that $\varepsilon(1\perp\rho)\in\mbox{\rm ESp}_{2n}(R),$ therefore

$V\varepsilon(1\perp\rho)=\begin{bmatrix}1&0&\cdots&0\\\
0&b_{2}&\cdots&b_{2m}\\\ \end{bmatrix}\in\mbox{\rm Sp}\mbox{\rm
Um}_{2,2m}(R),~{}\mbox{for~{}some}~{}b_{i}\in R,~{}2\leq i\leq 2m.$

Now,

$\begin{bmatrix}1&0&\cdots&0\\\ 0&b_{2}&\cdots&b_{2m}\\\
\end{bmatrix}\psi_{m}\begin{bmatrix}1&0\\\ 0&b_{2}\\\ \vdots&\vdots\\\
0&b_{2m}\\\ \end{bmatrix}=\psi_{1}.$

Upon comparing coefficients we gets $b_{2}=1.$ Therefore $V\overset{\mbox{\rm
ESp}}{\sim}\begin{bmatrix}1&0&0&\cdots&0\\\ 0&1&b_{3}&\cdots&b_{2m}\\\
\end{bmatrix}.$ Now take $\alpha=\prod_{k=3}^{2m}se_{2,k}(-b_{k})\in\mbox{\rm
ESp}_{2m}(R).$ Then $V\overset{\mbox{\rm
ESp}}{\sim}\begin{bmatrix}1&0&0&\cdots&0\\\ c&1&0&\cdots&0\\\
\end{bmatrix}~{}\mbox{for~{}some}~{}c\in A.$ Now take $\beta=se_{21}(-c)$,
then we gets, $V\overset{\mbox{\rm ESp}}{\sim}\begin{bmatrix}1&0&\cdots&0\\\
0&1&\cdots&0\\\ \end{bmatrix}.$ Therefore, $V$ is completable to an elementary
symplectic matrix.

Now assume that $n>1$, since $R$ is a local ring, $V\in\mbox{\rm Sp}\mbox{\rm
Um}_{2n,2m}(R)\subseteq\mbox{\rm Um}_{2n,2m}(R)$, there exists
$\varepsilon\in\mbox{\rm E}_{2m}(R)$ such that

$V\varepsilon=\begin{bmatrix}1&0&\cdots&0&\cdots&0\\\ 0&1&\cdots&0&\cdots&0\\\
&&\vdots\\\ 0&0&\cdots&1&\cdots&0\end{bmatrix}.$

In view of Rao-Swan Lemma, there exists $\rho\in\mbox{\rm E}_{2m-1}(R)$ such
that $\varepsilon(1\perp\rho)\in\mbox{\rm ESp}_{2m}(R),$ therefore

$V\varepsilon(1\perp\rho)=\begin{bmatrix}1&0&\cdots&0\\\
0&b_{2}&\cdots&b_{2m}\\\ \hskip 36.135pt&W&\hskip 36.135pt\\\
\end{bmatrix}\in\mbox{\rm Sp}\mbox{\rm Um}_{2n,2m}(R),$

$~{}\mbox{for~{}some}~{}b_{i}\in R,~{}2\leq i\leq 2m,W\in\mbox{\rm
Sp}\mbox{\rm Um}_{2n-2,2m}(R).$

Repeating the process done in $n=1$ case, there exists
$\varepsilon_{1}\in\mbox{\rm ESp}_{2m}(R)$ such that

$V\varepsilon_{1}=\begin{bmatrix}1&0&\cdots&0\\\ 0&1&\cdots&0\\\ \hskip
36.135pt&V^{\prime}&\hskip 36.135pt\\\
\end{bmatrix}~{}\mbox{for~{}some}~{}V^{\prime}\in\mbox{\rm Sp}\mbox{\rm
Um}_{2(n-1),2m}(R).$

Since $V\varepsilon_{1}\in\mbox{\rm Sp}\mbox{\rm Um}_{2n,2m}(R),$
$(V\varepsilon_{1})\psi_{m}(V\varepsilon_{1})^{t}=\psi_{n}.$ Therefore upon
comparing the coefficients on the both side of the equation, one gets
$V^{\prime}=(0,V^{\prime\prime})$ for some $V^{\prime\prime}\in\mbox{\rm
Sp}\mbox{\rm Um}_{2(n-1),2(m-1)}(R).$

By induction hypothesis $V^{\prime}$ is completable to an elementary
symplectic matrix, therefore $V$ is completable to an elementary symplectic
matrix.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

######  Proposition 3.12.

Let $R$ be a local ring and $V\in\mbox{\rm Sp}\mbox{\rm Um}_{2n,2m}(R)$ for
$m>n\geq 2$ or $m=n\geq 3.$ Let $\delta\in\mbox{\rm Sp}_{2n}(R)$ be symplectic
homotopic to identity. Let $\delta(T)$ be a homotopy of $\delta.$ Then there
exists, $\sigma(T)\in\mbox{\rm Sp}_{2m}(R[T])$ with $\sigma(0)=\textit{Id}$
and $\sigma(T)^{-1}(\delta(T)\perp I_{2m-2n})\in\mbox{\rm ESp}_{2m}(R[T])$
such that

$\delta(T)V=V\sigma(T).$

${{\bf Proof:}}$ In view of Lemma 3.11, $V$ is completable to a matrix
$W\in\mbox{\rm Sp}_{2m}(R)$. Since $R$ is a local ring, $W\in\mbox{\rm
ESp}_{2m}(R).$ By $($[7, Corollary 1.11]$)$, $\mbox{\rm
ESp}_{2m}(R[T])\trianglelefteq\mbox{\rm Sp}_{2m}(R[T])$, for $m\geq 3,$ there
exists $\varepsilon_{1}(T)\in\mbox{\rm ESp}_{2m}(R[T])$ such that

$(\delta(T)\perp I_{2m-2n})W(\delta(T)\perp
I_{2m-2n})^{-1}=\varepsilon_{1}(T).$

Thus we have $(\delta(T)\perp
I_{2m-2n})W=\varepsilon_{1}(T)W^{-1}W(\delta(T)\perp I_{2m-2n}).$ Again by
normality of $\mbox{\rm ESp}_{2m}(R[T])$ in $\mbox{\rm Sp}_{2m}(R[T]$ for
$m\geq 3,$ there exists $\varepsilon(T)\in\mbox{\rm ESp}_{2m}(R[T])$ such that

$(\delta(T)\perp I_{2m-2n})W=W(\delta(T)\perp I_{2m-2n})\varepsilon(T).$

Note that $\varepsilon(0)=\textit{Id}.$ Upon taking $\sigma(T)=(\delta(T)\perp
I_{2m-2n})\varepsilon(T)$ and multiplying above equation
by$\begin{bmatrix}I_{2n}&0\\\ 0&0\\\ \end{bmatrix}$, we gets desired result.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

######  Theorem 3.13.

Let $R$ be a commutative ring and $V\in\mbox{\rm Sp}\mbox{\rm Um}_{2n,2m}(R)$
with $m>n\geq 2~{}\mbox{or}~{}m=n\geq 3.$ Let $\delta\in\mbox{\rm Sp}_{2n}(R)$
be symplectic homotopic to identity. Let $\delta(T)$ be a homotopy of
$\delta.$ Then $\exists~{}\sigma(T)\in\mbox{\rm Sp}_{2m}(R[T],(T))$ such that

$\delta(T)V=V\sigma(T)~{}\mbox{and}~{}\sigma(T)^{-1}(\delta(T)\perp
I_{2m-2n})\in\mbox{\rm ESp}_{2m}(R[T],(T)).$

Moreover , if $\sigma(1)=\sigma$, then we have $\delta V=V\sigma$ and
$\sigma^{-1}(\delta\perp I_{2m-2n})\in\mbox{\rm ESp}_{2m}(R).$

${{\bf Proof:}}$ Define,

$\displaystyle J=$ $\displaystyle\\{s\in
R~{}|~{}\delta(T)_{s}V_{s}=V_{s}\sigma(T)~{}\mbox{for
~{}some}~{}\sigma(T)\in\mbox{\rm Sp}_{2m}(R_{s}[T],(T))~{}$
$\displaystyle\mbox{with}~{}\sigma(T)^{-1}(\delta(T)_{s}\perp
I_{2m-2n})\in\mbox{\rm ESp}_{2m}(R_{s}[T])\\}.$

$\bf{Claim:}$ $J$ is an ideal.

For $s\in J,\lambda\in R$, clearly $\lambda s\in J$. So we need to prove that
if $s_{1},s_{2}\in J$ then $s_{1}+s_{2}\in J.$ Since $s_{1},s_{2}\in J$, we
have $(s_{1}+s_{2})s_{1}$, $(s_{1}+s_{2})s_{2}\in J.$ We rename
$R_{s_{1}+s_{2}}$ by $R$, now it suffices to show that

$\displaystyle\delta(T)V$
$\displaystyle=V\sigma(T)~{}\mbox{for}~{}\mbox{some}~{}\sigma(T)\in\mbox{\rm
Sp}_{2m}(R[T],(T))$ $\displaystyle\mbox{with}~{}\sigma(T)^{-1}(\delta(T)\perp
I_{2m-2n})\in\mbox{\rm ESp}_{2m}(R[T],(T))~{}\mbox{provided
that}~{}s_{1}+s_{2}=1~{}\mbox{ and}$ (5)
$\delta(T)_{s_{1}}V_{s_{1}}=V_{s_{1}}\sigma_{1}(T)~{}\mbox{with}~{}\sigma_{1}(T)^{-1}(\delta(T)_{s_{1}}\perp
I_{2m-2n})\in\mbox{\rm ESp}_{2m}(R_{s_{1}}[T],(T)),$ (6)
$\delta(T)_{s_{2}}V_{s_{2}}=V_{s_{2}}\sigma_{2}(T)~{}\mbox{with}~{}\sigma_{2}(T)^{-1}(\delta(T)_{s_{2}}\perp
I_{2m-2n})\in\mbox{\rm ESp}_{2m}(R_{s_{2}}[T],(T)).$

Let

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sigma_{1}(T)(\delta(T)_{s_{1}}\perp
I_{2m-2n})^{-1}=\varepsilon_{1}(T)\in\mbox{\rm ESp}_{2m}(R_{s_{1}}[T],(T)),$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sigma_{2}(T)(\delta(T)_{s_{2}}\perp
I_{2m-2n})^{-1}=\varepsilon_{2}(T)\in\mbox{\rm ESp}_{2m}(R_{s_{2}}[T],(T)).$

Now, $\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\
\end{bmatrix}\varepsilon_{1}(T)_{s_{2}}\varepsilon_{2}(T)_{s_{1}}^{-1}=\begin{bmatrix}V_{s_{1}s_{2}}\\\
0\\\ \end{bmatrix}.$ Let
$\theta(T)=\varepsilon_{1}(T)_{s_{2}}\varepsilon_{2}(T)_{s_{1}}^{-1}.$ By
Quillen’s splitting property, $\mbox{for}~{}b\in(s_{2}^{N}),~{}N>>0$, we have

(7) $\theta(T)=\theta(bT)\\{\theta(bT)^{-1}\theta(T)\\}$

$~{}\mbox{with}~{}\theta(bT)\in\mbox{\rm
ESp}_{2m}(R_{s_{1}}[T]),~{}\mbox{and}~{}\theta(bT)^{-1}\theta(T)\in\mbox{\rm
ESp}_{2m}(R_{s_{2}}[T]).$

Since $\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\
\end{bmatrix}\theta(T)=\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\ \end{bmatrix}$.
We have, $\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\
\end{bmatrix}\theta(bT)=\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\ \end{bmatrix}$
and
$\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\
\end{bmatrix}\theta(bT)^{-1}\theta(T)=\begin{bmatrix}V_{s_{1}s_{2}}\\\ 0\\\
\end{bmatrix}.$

Define, $\theta(bT)=\eta_{1}(T)^{-1},~{}\theta(bT)^{-1}\theta(T)=\eta_{2}(T)$.
Thus we have $V_{i}\eta_{i}(T)=V_{i}$ with $\eta_{i}(0)=\textit{Id}$ and
$\eta_{i}(T)\in\mbox{\rm ESp}_{2m}(R_{s_{i}}[T])$ for $i=1,2.$

In view of equation 7, we gets

(8)
$(\eta_{1}(T)\varepsilon_{1}(T))_{s_{2}}=(\eta_{2}(T)\varepsilon_{2}(T))_{s_{1}}.$

Now, by equations 5 and 6,

$\delta(T)_{s_{1}}V_{s_{1}}=V_{s_{1}}\eta_{1}(T)\sigma_{1}(T)$
$\delta(T)_{s_{2}}V_{s_{2}}=V_{s_{2}}\eta_{2}(T)\sigma_{2}(T).$

In view of equation 8, we have
$(\eta_{1}(T)\sigma_{1}(T))_{s_{2}}=(\eta_{2}(T)\sigma_{2}(T))_{s_{1}}$. Since
$s_{1}+s_{2}=1$, $\exists~{}\sigma(T)\in\mbox{\rm Sp}_{2m}(R[T])$ such that
$\sigma(T)_{s_{1}}=\eta_{1}(T)\sigma_{1}(T)$ and
$\sigma(T)_{s_{2}}=\eta_{2}(T)\sigma_{2}(T)$ with
$\sigma(T)_{s_{i}}^{-1}(\delta(T)_{s_{i}}\perp I_{2m-2n})\in\mbox{\rm
ESp}_{2m}(R_{s_{i}}[T])$ for $i=1,2.$ Since $s_{1}$ and $s_{2}$ are comaximal,
by theorem 3.10,

$\displaystyle\delta(T)V$
$\displaystyle=V\sigma(T)~{}\mbox{for}~{}\mbox{some}~{}\sigma(T)\in\mbox{\rm
Sp}_{2m}(R[T],(T))~{}$
$\displaystyle\mbox{with}~{}\sigma(T)^{-1}(\delta(T)\perp
I_{2m-2n})\in\mbox{\rm ESp}_{2m}(R[T],(T)).$

This proves that $J$ is an ideal.

In view of lemma 3.12, for every maximal ideal $\mathfrak{m}$ of $R$, we have

$\delta(T)_{\mathfrak{m}}V_{\mathfrak{m}}=V_{\mathfrak{m}}\sigma^{{}^{\prime}}(T)~{}\mbox{with}~{}\sigma^{{}^{\prime}}(T)^{-1}(\delta(T)_{\mathfrak{m}}\perp
I_{2m-2n})\in\mbox{\rm ESp}_{2m}(R_{\mathfrak{m}}[T],(T)).$

Thus there exists $s\in R\setminus\mathfrak{m}$, such that

$\delta(T)_{s}V_{s}=V_{s}\sigma^{{}^{\prime}}(T)~{}\mbox{with}~{}\sigma^{{}^{\prime}}(T)^{-1}(\delta(T)_{s}\perp
I_{2m-2n})\in\mbox{\rm ESp}_{2m}(R_{s}[T],(T)).$

Therefore $J\nsubseteq\mathfrak{m}$, for any maximal ideal $\mathfrak{m}$ of
$R$ i.e. $1\in J.$ Thus $\exists~{}\sigma(T)\in\mbox{\rm Sp}_{2m}(R[T],(T))$
such that

$\delta(T)V=V\sigma(T)~{}\mbox{with}~{}\sigma(T)^{-1}(\delta(T)\perp
I_{2m-2n})\in\mbox{\rm ESp}_{2m}(R[T],(T)).$

Now put $T=1$, and take $\sigma(1)=\sigma$ to get the desired result.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

######  Corollary 3.14.

$($[10, Theorem 2.19]$)$ Let $m\geq 2$ and $\alpha,\beta\in\mbox{\rm
Sp}_{2m}(R).$ Let either $\alpha$ or $\beta$ be symplectic homotopic to
identity. Then $\alpha\beta=\beta\alpha\varepsilon,~{}~{}\mbox{for~{}
some}~{}\varepsilon\in\mbox{\rm ESp}_{2m}(R).$

${{\bf Proof:}}$ Let us assume that $\alpha$ is homotopic to identity, so
there exists $\delta(T)\in\mbox{\rm Sp}_{2m}(R[T])$ such that
$\delta(0)=\textit{Id}~{}\mbox{and}~{}\delta(1)=\alpha.$ By theorem 3.13,
there exists $\varepsilon(T)\in\mbox{\rm ESp}_{2m}(R[T])$ with
$\varepsilon(0)=\textit{Id}$ such that

$\delta(T)\beta=\beta\delta(T)\varepsilon(T).$

Put $T=1$ to get the desired result.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

######  Corollary 3.15.

Let $\delta\in\mbox{\rm Sp}_{2n}(R)$ and $V\in\mbox{\rm Sp}\mbox{\rm
Um}_{2n,2m}(R)$. Then $\delta V=V\sigma$ for some $\sigma\in\mbox{\rm
Sp}_{2m}(R)$ such that $(\delta^{-1}\perp\sigma)\in\mbox{\rm
ESp}_{2(n+m)}(R).$

${{\bf Proof:}}$ By $($[18, Lemma 1.1]$)$,
$(\delta\perp\delta^{-1})\in\mbox{\rm ESp}_{4n}(R).$ Since every elementary
symplectic matrix is homotopic to identity, thus by theorem 3.13,

$(\delta\perp\delta^{-1})(V\perp I_{2n})=(V\perp
I_{2n})\sigma^{{}^{\prime}},~{}\mbox{with}~{}\sigma^{{}^{\prime}}\in\mbox{\rm
ESp}_{2(n+m)}(R).$

Write $\sigma^{{}^{\prime}}=\begin{bmatrix}\alpha&\beta\\\ \gamma&\zeta\\\
\end{bmatrix}$ where $\alpha\in M_{2m\times 2m}(R),~{}\beta\in M_{2m\times
2n}(R),~{}\gamma\in M_{2n\times 2m}(R),~{}\zeta\in M_{2n\times 2n}(R)$. Thus
we have,

$(\delta V\perp\delta^{-1})=\begin{bmatrix}V\alpha&V\beta\\\ \gamma&\zeta\\\
\end{bmatrix}.$

Upon compairing both sides we gets $\gamma=0$ and $\zeta=\delta^{-1}.$
Therefore

$\begin{bmatrix}\alpha&\beta\\\ 0&\delta^{-1}\\\ \end{bmatrix}\in\mbox{\rm
ESp}_{2(n+m)}(R).$

Now, take $\alpha=\sigma$, so we have $(\delta^{-1}\perp\sigma)\in\mbox{\rm
ESp}_{2(n+m)}(R)$ and $\delta V=V\sigma.$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

##  4\. Generalised Homotopy and Commutativity Principle for Orthogonal Groups

Throughout this section we will assume that $1/2\in R$, where $R$ is a
commutative ring with $1\neq 0.$ In this section, we will deal with orthogonal
matrices of size at least 6.

######  Notation 4.1.

Let $\phi_{1}=\begin{bmatrix}0&1\\\ 1&0\\\
\end{bmatrix},~{}~{}\phi_{n}=\phi_{n-1}\perp\phi_{1};$ for $n>1$.

######  Notation 4.2.

Let $\sigma$ be the permutation of the natural numbers given by
$\sigma(2i)=2i-1$ and $\sigma(2i-1)=2i$.

######  Definition 4.3.

Orthogonal group $\rm{O}_{2m}(R):$ The subgroup of $\mbox{\rm GL}_{2m}(R)$
consisting of all $2m\times 2m$ matrices $\\{\alpha\in\mbox{\rm
GL}_{2m}(R)~{}\mid\alpha^{t}\phi_{m}\alpha=\phi_{m}\\}$.

######  Definition 4.4.

Elementary orthogonal group $\mbox{\rm EO}_{2m}(R):$ We define for $1\leq
i\neq j\leq 2m,~{}z\in R,$

$oe_{ij}(z)=I_{2m}+zE_{ij}-zE_{\sigma(j)\sigma(i)},~{}~{}~{}\textit{if}~{}i\neq\sigma(j).$

It is easy to verify that all these matrices belong to $\rm{O}_{2m}(R)$. We
call them the elementary orthogonal matrices over $R$. The subgroup generated
by them is called the elementary orthogonal group and is denoted by $\mbox{\rm
EO}_{2m}(R)$.

######  Definition 4.5.

$\mbox{OUm}_{2n,2m}(R)=\\{V\in\mbox{\rm
Um}_{2n,2m}(R)|V\phi_{m}V^{t}=\phi_{n}\\}.$

Let $P$ be a finitely generated projective $R$-module. The module $P\oplus
P^{\ast}$ carries a natural quadratic form $q$ defined by $q(x+f)=f(x)$ for
$x\in P$ and $f\in P^{\ast}.$ The associated bilinear form is given by
$B_{q}(x_{1}+f_{1},x_{2}+f_{2})=f_{1}(x_{2})+f_{2}(x_{1}),x_{1},x_{2}\in
P,f_{1},f_{2}\in P^{\ast}.$ It is easy to see that $q$ is non-singular. The
quadratic space $(P\oplus P^{\ast},q)$ will be called hyperspace of $P.$ The
hyperbolic space of a free $R$-module of rank 1 is called a hyperplane.

######  Definition 4.6.

An orthogonal pair of elements $(w_{1},w_{2})$ is said to be a hyperbolic pair
if $q(w_{1})=1,q(w_{2})=-1.$

######  Remark 4.7.

Hyperbolic plane is generated as an $R$-module by a hyperbolic pair.

######  Lemma 4.8.

Let $R$ be a local ring with $2R=R$ and $V\in\rm{OUm}_{2n,2m}(R),m\geq
n+2,n\geq 1.$ Then $V$ is completable to an elementary orthogonal matrix.

${{\bf Proof:}}$ We will prove it by induction on $n,m.$ Let us assume that
$n=1.$ In view of $($[12, Theorem 7.1 (ii)]$)$ and $($[1, Lemma 2.7]$),$ there
exists $\varepsilon\in\mbox{\rm EO}_{2m}(R)$ such that

$V\varepsilon=\begin{bmatrix}1&0&\cdots&0\\\ 0&1&\cdots&0\\\ \end{bmatrix}.$

Thus $V$ is completable to an elementary orthogonal matrix.

Now assume that $n>1$. We have $m>3.$ In view of $($[12, Theorem 7.1 (ii)]$),$
there exists $\varepsilon_{1}\in\mbox{\rm EO}_{2m}(R)$ such that

$V\varepsilon_{1}=\begin{bmatrix}1&0&\cdots&0\\\ 0&1&\cdots&0\\\ \hskip
36.135pt&V^{\prime}&\hskip 36.135pt\\\
\end{bmatrix}~{}\mbox{for~{}some}~{}V^{\prime}\in\rm{OUm}_{2(n-1),2m}(R).$

Since $V\varepsilon_{1}\in\mbox{OUm}_{2n,2m}(R),$
$(V\varepsilon_{1})\phi_{m}(V\varepsilon_{1})^{t}=\phi_{n}.$ Therefore upon
comparing the coefficients on the both side of the equation, one gets
$V^{\prime}=(0,V^{\prime\prime})$ for some
$V^{\prime\prime}\in\rm{OUm}_{2(n-1),2(m-1)}(R).$ Now, we get the desired
result by induction hypothesis.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

Following the steps of the proof of Proposition 3.12, one gets the following
result :

######  Proposition 4.9.

Let $R$ be a local ring and $V\in\rm{OUm}_{2n,2m}(R)$ for $m\geq n+2,n\geq 2.$
Let $\delta\in\rm{SO}_{2n}(R)$ be orthogonal homotopic to identity. Let
$\delta(T)$ be a homotopy of $\delta.$ Then there exists,
$\sigma(T)\in\mbox{SO}_{2m}(R[T])$ with $\sigma(0)=\textit{Id}$ and
$\sigma(T)^{-1}(\delta(T)\perp I_{2m-2n})\in\mbox{\rm EO}_{2m}(R[T])$ such
that

$\delta(T)V=V\sigma(T).$

By making appropriate modifications in the proof of theorem 2.7 and theorem
3.13, one can prove the following result :

######  Theorem 4.10.

Let $R$ be a commutative ring and $V\in\rm{OUm}_{2n,2m}(R)$ with $m\geq
n+2,n\geq 2.$ Let $\delta\in\rm{SO}_{2n}(R)$ be orthogonal homotopic to
identity. Let $\delta(T)$ be a homotopy of $\delta.$ Then
$\exists~{}\sigma(T)\in\mbox{SO}_{2m}(R[T],(T))$ such that

$\delta(T)V=V\sigma(T)~{}\mbox{and}~{}\sigma(T)^{-1}(\delta(T)\perp
I_{2m-2n})\in\mbox{\rm EO}_{2m}(R[T],(T)).$

Moreover , if $\sigma(1)=\sigma$, then we have $\delta V=V\sigma$ and
$\sigma^{-1}(\delta\perp I_{2m-2n})\in\mbox{\rm EO}_{2m}(R).$

Due to the size restrictions in lemma 4.8 one is not able to deduce whether a
similar homotopy and commutativity principle holds in the orthogonal case. We
began this study in [10]. We add a few more observations on this below.

######  Lemma 4.11.

$($L.N. Vaserstein$)$ $($[18, Theorem 3.5]$)$ Let $m\geq 3$ and $R$ be a local
ring, $\frac{1}{2}\in R$. Then ${\rm{O}_{2m}(R)}/{\mbox{\rm
EO}_{2m}(R)}={\rm{O}_{2}(R)}/{\mbox{\rm EO}_{2}(R)}=O_{2}(R).$

######  Observation 4.12.

Every element $\alpha\in\rm{O}_{2}(R)$ is either of the type
$\begin{bmatrix}u&0\\\ 0&u^{-1}\\\ \end{bmatrix}$ or of the type
$\begin{bmatrix}0&u\\\ u^{-1}&0\\\ \end{bmatrix}$ for some $u\in R^{\ast}.$

######  Theorem 4.13.

Let $R$ be a local ring, $m\geq 3$ and $1/2\in R$. Then we have,

$([\rm{O}_{2m}(R[X]),\rm{O}_{2m}(R[X])]\perp I_{2})\subseteq\mbox{\rm
EO}_{2m+2}(R[X]).$

${{\bf Proof:}}$ Let $\alpha(X),\beta(X)\in\rm{O}_{2m}(R[X])$, we need to
prove that $([\alpha(X),\beta(X)]\perp I_{2})\in\mbox{\rm EO}_{2m+2}(R[X]).$
Define,

$\gamma(X,T)=[\alpha(XT)\perp I_{2},\beta(X)\perp I_{2}]$

For every maximal ideal $\mathfrak{m}$ of $R[X]$, we have
$\gamma(X,T)_{\mathfrak{m}}=[(\alpha(XT)\perp
I_{2})_{\mathfrak{m}},(\beta(X)\perp I_{2})_{\mathfrak{m}}].$ In view of lemma
4.11, $(\beta(X)\perp
I_{2})_{\mathfrak{m}}=(I_{2m}\perp\delta(X))\varepsilon(X)$ for some
$\delta(X)\in\rm{O}_{2}(R[X]_{\mathfrak{m}})$ and $\varepsilon(X)\in\mbox{\rm
EO}_{2m+2}(R[X])_{\mathfrak{m}}.$ By observation 4.12, either
$\delta(X)=\begin{bmatrix}u&0\\\ 0&u^{-1}\\\ \end{bmatrix}$
or $\delta(X)=\begin{bmatrix}0&u\\\ u^{-1}&0\\\
\end{bmatrix},~{}\mbox{for~{}some}~{}u\in R[X]_{\mathfrak{m}}^{\ast}.$
Therefore $\gamma(X,T)_{\mathfrak{m}}\in\mbox{\rm
EO}_{2m+2}(R[X]_{\mathfrak{m}}[T]).$

Now, $\gamma(X,0)=[\alpha(0)\perp I_{2},\beta(X)\perp I_{2}].$ Since $R$ is a
local ring, by lemma 4.11, $\alpha(0)\perp
I_{2}=(I_{2m}\perp\delta)\varepsilon_{1}$ for $\delta\in\rm{O}_{2}(R)$ and
$\varepsilon_{1}\in\mbox{\rm EO}_{2m+2}(R).$ By observation 4.12, either
$\delta=\begin{bmatrix}a&0\\\ 0&a^{-1}\\\
\end{bmatrix}~{}or~{}\begin{bmatrix}0&a\\\ a^{-1}&0\\\
\end{bmatrix},~{}\mbox{for~{}some}~{}a\in R^{\ast}.$ Therefore,
$\gamma(X,0)\in\mbox{\rm EO}_{2m+2}(R[X]).$ Now by local-global principle for
othogonal groups $($[17, Theorem 4.2]$)$, we have

$\gamma(X,1)=[\alpha(X)\perp I_{2},\beta(X)\perp
I_{2}]=([\alpha(X),\beta(X)]\perp I_{2})\in\mbox{\rm EO}_{2m+2}(R[X]).$

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

######  Notation 4.14.

We will denote set of all special orthogonal matrices which are special
orthogonally homotopic to identity by $\rm{HSO}_{2m}(R).$

######  Theorem 4.15.

Let $m\geq 2$ and $R$ be a commutative ring, $\frac{1}{2}\in R$. Then,

$[\rm{HSO}_{2m}(R)\perp I_{2},\rm{O}_{2m}(R)\perp I_{2}]\subseteq\mbox{\rm
EO}_{2m+2}(R).$

${{\bf Proof:}}$ Let $\alpha\in\rm{HSO}_{2m}(R),\beta\in\rm{O}_{2m}(R)$, we
need to prove that $[\alpha\perp I_{2},\beta\perp I_{2}]\in\mbox{\rm
EO}_{2m+2}(R).$ Let $\alpha(T)$ be a homotopy of $\alpha$ and define,

$\gamma(T)=[\alpha(T)\perp I_{2},\beta\perp I_{2}].$

Clearly, $\gamma(0)=Id.$ For every maximal ideal $\mathfrak{m}$ of $R$, we
have

$\gamma(T)_{\mathfrak{m}}=[(\alpha(T)\perp I_{2})_{\mathfrak{m}},(\beta\perp
I_{2})_{\mathfrak{m}}].$

In view of lemma 4.11, $(\beta\perp
I_{2})_{\mathfrak{m}}=(I_{2m}\perp\delta)\varepsilon$ for some
$\delta\in\rm{O}_{2}(R_{\mathfrak{m}})$ and $\varepsilon\in\mbox{\rm
EO}_{2m+2}(R_{\mathfrak{m}}).$ By observation 4.12, either
$\delta=\begin{bmatrix}a&0\\\ 0&a^{-1}\\\
\end{bmatrix}~{}or~{}\begin{bmatrix}0&a\\\ a^{-1}&0\\\
\end{bmatrix},~{}\mbox{for~{}some}~{}a\in R_{\mathfrak{m}}^{\ast}.$ Therefore,
$\gamma(T)_{\mathfrak{m}}\in\mbox{\rm EO}_{2m}(R_{\mathfrak{m}}[T]).$ In view
of local-global principle for orthogonal groups $($[17, Theorem 4.2]$)$, we
have $\gamma(T)\in\mbox{\rm EO}_{2m+2}(R[T]).$ Therefore,
$\gamma(1)=[\alpha\perp I_{2},\beta\perp I_{2}]\in\mbox{\rm EO}_{2m+2}(R).$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\Box$

Acknowledgement: The first author thanks Professor L.N. Vaserstein for some
valuable insights. The second author is thankful to Inspire Faculty Fellowship
(DST/INSPIRE/04/2021/002849) and Startup Research Grant (SRG/2022/000056) for
their support. The second author also thanks IIT Mandi for their Seed Grant.

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Ravi A. Rao, School of Mathematical Sciences, Narsi Monji Institute of
Management Sciences (NMIMS), V.L. Mehta Road, Vile Parle West, Mumbai 400056,
India

E-mail: Ravi A. Rao<EMAIL_ADDRESS>
Sampat Sharma, School of Mathematical and Statistical Sciences, IIT Mandi,
Mandi 175005 (H.P), India

E-mail: Sampat Sharma<EMAIL_ADDRESS><EMAIL_ADDRESS>