Decomposition of Symplectic Vector Bundles and Azumaya Algebras

Arcila-Maya]Niny Arcila-Maya

Department of Mathematics, Duke University, Durham NC, 27708-0320 USA

[2020]55P99 (Primary), 55Q52, 55S45, 16H05, 16W10 (Secondary)

Let $X$ be a connected CW complex. Let $\Az{V}$ be a symplectic vector bundle of rank $2mn$ over $X$, and let $\Az{A}$ be a topological Azumaya algebra of degree $2mn$ with a symplectic involution over a CW complex $X$. We give conditions for the positive integers $m$ and $n$, and the dimension of $X$ so that $\Az{V}$ can be decomposed as the tensor product of a symplectic vector bundle of rank $2m$ and an orthogonal vector bundle of rank $n$; and so that $\Az{A}$ can be decomposed as the tensor product of topological Azumaya algebras of degrees $2m$ and $n$ with involutions of the first kind.

§ INTRODUCTION

It is a result of Merkur'ev that every central simple algebra with involution is Brauer equivalent to a tensor product of quaternion algebras, <cit.>. Knus–Parimala–Srinivas gave a necessary and sufficient decomposability condition for an involution on a tensor product of two quaternion algebras, <cit.>.
Nevertheless, there exist examples of division algebras of degree $8$ with involutions of the first kind, in the sense that these involutions act as the identity on the base, which do not decompose into tensor products of quaternion algebras

Let $n$ be a positive integer, and let $X$ be a connected topological space. A topological Azumaya algebra of degree $n$ over $X$ is a fiber bundle $\cA$ over $X$ that is locally isomorphic as $\CC-$algebras to the matrix algebra $\M(n,\CC)$.  An involution on $\cA$  is a morphism of fiber bundles $\tau:\cA \rightarrow \cA$ such that $\tau\circ\tau=\id_{\cA}$, and when restricted to fibers it is an anti-automorphism of complex algebras, i.e. $\tau|_{\cA^{-1}(x)}$ is linear and $\tau(ab)=\tau(b)\tau(a)$ for all $a, b \in \cA^{-1}(x)$ and for all $x\in X$. In this case, $(\cA,\tau)$ is called a topological Azumaya algebra with involution. The involution $\tau$ is said to be symplectic (orthogonal) if the restriction $\tau|_{\cA^{-1}(x)}:\cA^{-1}(x)\to\cA^{-1}(x)$ is a symplectic (an orthogonal) involution of complex algebras for all $x\in X$.

Although on the level of Brauer groups, there is a tensor product decomposition of topological Azumaya algebras with symplectic involution, in general this decomposition does not hold. We prove in Proposition <ref> that topological Azumaya algebras with symplectic involution may not decompose as the tensor product of topological Azumaya algebras with involutions of the first kind. Additionally, we provide conditions on a positive integer $n$ and a topological space $X$ such that a topological Azumaya algebra of degree $2n$ over $X$ with a symplectic involution decomposes as the tensor product of topological Azumaya algebras with involution. We prove the following theorem:

Let $X$ be a CW complex such that $\dim(X)\leq 7$. Let $m$ and $n$ be positive integers such that $m>1$, $n>7$, and $n$ is odd.

If $\cA$ is a topological Azumaya algebra of degree $2mn$ over $X$ with a symplectic involution, then there exist topological Azumaya algebras $\cA_{2m}$ and $\cA_{n}$ of degrees $2m$ and $n$, respectively, such that $\cA_{2m}$ has a symplectic involution, $\cA_{n}$ has an orthogonal involution and is Brauer-trivial, and $\cA\iso \cA_{2m}\tensor\cA_{n}$.

For the particular case of topological Azumaya algebras of the form $\End(\cV)$ for $\cV$ a symplectic vector bundle of rank $2n$, we obtain the decomposition result for the dimension of the underlying space in a larger range:

Let $m$ and $n$ be positive integers such that $n$ is odd. Let $X$ be a CW complex such that $\dim(X)\leq n$. If $\cV$ is a symplectic bundle of rank $2mn$ over $X$, then there exist a symplectic bundle $\cV_{2m}$ of rank $2m$, and an orthogonal bundle $\cV_{n}$ of rank $n$ such that $\cV\iso \cV_{2m}\tensor\cV_{n}$.

We prove in Theorem <ref> that a map $X\rightarrow \B\fPSp{mn}$ can be lifted to $\B\fPSp{m}\times \B\fSO{n}$ when the dimension of $X$ is less than $7$. The proof of Theorem <ref> relies in the description of the homomorphisms induced on homotopy groups by the $r$-fold direct sum of symplectic matrices $\splus^{r}:\fSp{m}\to\fSp{rm}$, and by the tensor product operations $\tensor:\fSp{m} \times \fO{n} \to \fSp{mn}$ and $\stensor:\fSp{m} \times \fSp{n} \to \fO{4mn}$, defined in <cit.>, in the range $\{0,1,\dots,4m+2\}$. We call this set “the stable range for $\fSp{n}$”. More concretely we prove the following result

Let $i<4m+2$ and $i<n-1$, and let $\tensor:\fSp{m} \times \fO{n} \to \fSp{mn}$ be the tensor product. The induced homomorphism $\tensor_{*}:\pi_{i}(\Sp(m))\times\pi_{i}(\Or(n)) \to \pi_{i}(\Sp(mn))$
is given by

\[
\otimes_{*}(x,y)=nx+2my \quad \text{for} \quad x \in \pi_{i}(\Sp(m)) \quad \text{and} \quad y\in \pi_{i}(\Or(n)).
\]

This paper is a sequel to <cit.>, where we determined conditions for the decomposition of topological Azumaya algebras with orthogonal involutions over finite CW complexes.

§.§ Acknowledgments

The author extends gratitude to Ben Williams for suggesting the research topic,  investing substantial time in detailed research discussions, and for thoroughly reviewing the manuscript. Additionally, heartfelt appreciation is extended to Kirsten Wickelgren for offering valuable suggestions that enhance its overall exposition of this manuscript.

§.§ Notation

Let $\FF$ be a ring with identity. We denote by $\M(n,\FF)$ the group of squares matrices of order $n$ with entries in $\FF$, and by $\GL(n,\FF)=\M(n,\FF)^{\times}$.

When $\FF$ is the field of the complex (real) numbers, then the complex (real) special linear group of degree $n$, the complex (real) orthogonal group of degree $n$, and the complex (real) special orthogonal group of degree $n$ shall be denoted by $\SL(n,\FF)$, $\Or(n,\FF)$, and $\SO(n,\FF)$, respectively. The complex unitary group of degree $n$, the compact symplectic group of degree $n$, and the complex symplectic group of degree $2n$ shall be denoted by $\U(n,\CC)$, $\U(n,\HH)$, and $\fSp{n}$, respectively.

The projective special complex group of degree$n$, and the projective complex sympletic group of degree $2n$ shall be denoted by $\fPSL{n}\coloneqq \fSL{n}/Z(\fSL{n})$, and $\fPSp{n}\coloneqq \fSp{n}/Z(\fSp{n})$, respectively, where $Z(-)$ is the center.

All topological spaces will have the homotopy type of a CW complex. We fix basepoints for connected topological spaces, and for topological groups we take the identities as basepoints. We write $\pi_{i}(X)$ in place of $\pi_{i}(X,x_{0})$.

§ STABILIZATION OF OPERATIONS ON THE COMPLEX SYMPLECTIC GROUP

In what follows, we will drop $\CC$ from $\fGL{n}$, $\fO{n}$ and $\fSp{n}$. We recall the homotopy groups of $\Sp(n)$ in low degrees, and compute the homotopy groups of $\PSp(n)$ in low degrees.

When $n=1$, then $\Sp(1)=\SL(2,\CC)$ and $\PSp(1)=\PSL(2,\CC)$. Let $n\geq1$ and $i< 4n+2$, the first homotopy groups of the symplectic group are given by Bott periodicity, <cit.>.

\begin{align*}
\pi_{i}(\Sp(n))&\iso
\begin{cases}
0 & \text{if $i=0,1,2,6$ (mod 8),}\\
\ZZ/2 & \text{if $i=4,5$ (mod 8),}\\
\ZZ & \text{if $i=3,7$ (mod 8)}.
\end{cases}\\
\pi_{4n}(\Sp(n))\iso\pi_{4n+1}(\Sp(n))&\iso
\begin{cases}
\ZZ/2 & \text{if $n$ is odd,}\\
0 & \text{if $n$ is even}.
\end{cases}
\end{align*}

The quotient map $\Sp(n) \rightarrow \PSp(n)$ is a universal cover with fiber $\ZZ/2$, then $\pi_{1}(\PSp(n))\iso \ZZ/2$. The symplectic group $\Sp
(n)$ is a subgroup of $\SL(2n,\CC)$. For $n>0$ there is a fibration $\{\pm I_{2n}\}\hookrightarrow \Sp(n) \rightarrow \PSp(n)$, then we can use the long exact sequence associated to it to see that $\pi_{i}(\PSp(n))\iso\pi_{i}(\Sp(n))$ for $i\geq 2$.

§.§ First unstable homotopy group of Sp(n)

Let $A\in \Sp(n)$, we write $A$ as a block matrix

\[
\begin{pmatrix}
A_{11} & A_{12}\\
A_{21} & A_{22}
\end{pmatrix}
\]

where the $A_{ij}$'s are $n\times n$ matrices for $i,j=1,2$, and they satisfy the following conditions: $A_{11}^{\tr}A_{21}$ are $A_{12}^{\tr}A_{22}$ are symmetric, and $A_{11}^{\tr}A_{22}-A_{21}^{\tr}A_{12}=I_{n}$.

The standard inclusion of the symplectic group $i:\Sp(n) \hookrightarrow \Sp(n+1)$ is defined as

\begin{equation*}
\begin{pmatrix}
&  &  & 0 &  &  &  & 0\\
& A_{11} &  & \vdots &  & A_{12} &  & \vdots \\
&  &  & 0 &  &  &  & 0\\
0 & \cdots & 0 & 1 & 0 & \cdots & 0 & 0 \\
&  &  & 0 &  &  &  & 0\\
& A_{21} &  & \vdots &  & A_{22} &  & \vdots \\
&  &  & 0 &  &  &  & 0\\
0 & \cdots & 0  & 0 &  0 & \cdots & 0  & 1
\end{pmatrix}.
\end{equation*}

The standard inclusion of the symplectic group is $(4n+2)$-connected, hence it induces an isomorphism on homotopy groups in degrees less than $4n+2$ and an epimorphism in degree $4n+2$. This can be seen by observing the long exact sequence for the fibration $\Sp(n) \hookrightarrow \Sp(n+1) \rightarrow \Sp(n+1)/\Sp(n)\simeq S^{4n+3}$.

The first unstable homotopy group of $\Sp(n)$ happens in degree $4n+2$. From <cit.>

\begin{align*}
\pi_{4n+2}(\Sp(n))\iso
\begin{cases}
\ZZ/(2n+1)! & \text{if $n$ is even,}\\
\ZZ/(2n+1)!\cdot2 & \text{if $n$ is odd}.
\end{cases}
\end{align*}

$i>1$ and $i$ (mod 8)     0     1     2     3     4     5     6     7

$\pi_{i}(G)$     0     0     0     $\ZZ$     $\ZZ/2$     $\ZZ/2$     0     $\ZZ$

Homotopy groups of compact Lie groups related to the complex symplectic group for $i=2,\dots,4n-1$

§.§ Stabilization

Direct sum of symplectic matrices

The direct sum of symplectic matrices $\splus:\Sp(m)\times \Sp(n) \to \Sp(m+n)$ is given by

\[
A\splus B=
\left(\begin{NiceArray}{cc:cc}
A_{11} &  & A_{12} & \\
& B_{11} &  & B_{12}\\
\hdottedline
A_{21} &  & A_{22} & \\
& B_{21} &  & B_{22}
\end{NiceArray}\right).
\]

$r$-fold direct sum of symplectic matrices

The $r$-fold direct sum of matrices $\splus^{r}:\Sp(n) \to \Sp(rn)$ is given by

\[
A^{\splus r}=
%A_{11} &  & A_{12} & \\
% & B_{11} &  & B_{12}\\
%A_{21} &  & A_{22} & \\
% & B_{21} &  & B_{22}
%A_{11} & 0 & A_{12} & 0\\
%0 & B_{11} & 0 & B_{12}\\
%A_{21} & 0 & A_{22} & 0\\
%0 & B_{21} & 0 & B_{22}
\left(\begin{NiceArray}{cc}
A_{11}^{\oplus r} & A_{12}^{\oplus r} \\
A_{21}^{\oplus r} & A_{22}^{\oplus r}
\end{NiceArray}\right).
\]

Let $m,n\in \NN$. Define the map $\s:\Sp(m) \rightarrow \Sp(m+n)$ by

\begin{equation*}
\s(A)=
\left(\begin{NiceArray}{cc:cc}
A_{11} &  & A_{12} & \\
& I_{n} &  & I_{n} \\
\hdottedline
A_{21} &  & A_{22} & \\
& I_{n}  &  & I_{n}
\end{NiceArray}\right).
\end{equation*}

Since the map $\s$ is equal to the composite of consecutive canonical inclusions, it follows that $\s$ is $(4m+2)$-connected.

Let $\esta$ denote the homomorphisms the map $\s$ induces on homotopy groups. From now on, we will identify $\pi_{i}(\Sp(m))$ with $\pi_{i}(\Sp(m+n))$ for all $i<4m+2$ through the isomorphism

\begin{equation}
\begin{tikzcd}
\esta:\pi_{i}(\Sp(m)) \arrow[r,"\iso"] & \pi_{i}(\Sp(m+n)).
\end{tikzcd}
\end{equation}

Let $n,r \in \NN$. For all $j=1,\dots,r$ define $\s_{j}:\Sp(n) \rightarrow \Sp(rn)$ by

\[
\s_{j}(A)=
\begin{pmatrix}
\s_{j}(A_{11}) & \s_{j}(A_{12})\\
\s_{j}(A_{21}) & \s_{j}(A_{22})
\end{pmatrix}.
\]

where $\s_{j}(A_{kl})=\diag(I_{n},\dots,I_{n},A_{kl},I_{n},\dots,I_{n})$,
where $A_{kl}$ is in the $j$-th position for $k,l=1,2$. Then the map $\s_{j}$ is pointed homotopic to $\s_{j+1}$ for $j=1,\dots,r-1$.

Let $P$ be the permutation matrix

\[
\begin{pmatrix}
I_{(j-1)n} & & &  \\
& 0 & I_{n} & \\
& I_{n} & 0 & \\
& & & I_{(r-j-1)n}
\end{pmatrix}.
\]

Observe that $\det(P_{j})=1$, and $\s_{j+1}(A_{kl})=P\s_{j}(A_{kl})P$ for $j=1,\dots,r-1$ and $k,l=1,2$. Then

\[
\begin{pmatrix}
P & 0\\
0 & P
\end{pmatrix}
\begin{pmatrix}
\s_{j}(A_{11}) & \s_{j}(A_{12})\\
\s_{j}(A_{21}) & \s_{j}(A_{22})
\end{pmatrix}
\begin{pmatrix}
P & 0\\
0 & P
\end{pmatrix}%
\begin{pmatrix}
\s_{j+1}(A_{11}) & \s_{j+1}(A_{12})\\
\s_{j+1}(A_{21}) & \s_{j+1}(A_{22})
\end{pmatrix}.
\]

Since $\diag(P,P) \in \Sp(n)$ and $\Sp(n)$ is path-connected, the result follows from <cit.>.

We call the $\s_{j}$ maps stabilization maps. As $\s_{1}$ is equal to $\s:\Sp(n) \to \Sp(n+(r-1)n)$, it follows that $\s_{j}$ is $(4n+2)$-connected for all $j=1,\dots,r$. From Lemma <ref> the homomorphisms induced on homotopy groups by the stabilization maps are equal, hence $\esta$ also denotes $\pi_{i}(\s_{1})=\cdots=\pi_{i}(\s_{r})$.Thus we identify $\pi_{i}(\Sp(n))$ with $\pi_{i}(\Sp(rn))$ for $i<4n+2$ through $\esta$.  The identification allows one to introduce a slight abuse of notation, namely to identify $x$ and $\esta(x)$ for $x\in \pi_{i}(\Sp(n))$ and $i<4n+2$.

§.§ Operations on homotopy groups
We do not write proofs for some results in this section as they are analogous to the proofs in <cit.>.

Let $i\in \NN$. The homomorphism $\splus_{*}: \pi_{i}(\Sp(m))\times \pi_{i}(\Sp(n)) \rightarrow \pi_{i}(\Sp(m+n))$ is equal to

\[
\splus_{*}(x,y)=\esta(x)+\esta(y) \quad \text{for} \quad x\in \pi_{i}(\Sp(m)) \quad \text{and} \quad y\in\pi_{i}(\Sp(n)).
\]

If $m<n$ and $i<4m+2$, then $\splus_{*}(x,y)=x+y$ for $x\in \pi_{i}(\Sp(m))$ and $y\in\pi_{i}(\Sp(n))$.

Let $i\in \NN$. The homomorphism $\splus^{r}_{*}:\pi_{i}(\Sp(n))\rightarrow \pi_{i}(\Sp(rn))$ is equal to

\[
\splus^{r}_{*}(x)=r\esta(x)  \quad \text{for} \quad x\in \pi_{i}(\Sp(n)).
\]

If $i<4n+2$, then $\splus_{*}^{r}(x)=rx$ for $x\in \pi_{i}(\Sp(n))$.

We want to understand the effect on homotopy groups of the tensor product operations $\tensor:\fSp{m} \times \fO{n} \to \fSp{mn}$ and $\stensor:\fSp{m} \times \fSp{n} \to \fO{4mn}$, defined in <cit.>.

§.§.§ Effect of tensor:Sp(m)xO(n) –> Sp(mn) on homotopy groups in the stable range of Sp(m)

Let $L$ and $R$ denote the restrictions of $\tensor$ to $\Sp(m) \times \{I_{n}\}$ and $\{I_{2m}\} \times  \Or(n)$, respectively. By the mixed-product property of the tensor product of matrices, if $(A,B) \in \Sp(m) \times \Or(n)$ then $A\tensor B=(A\tensor I_{n})(I_{2m}\tensor B)$. Hence, $A\tensor B=L(A)R(B)$.

Let $L:\Sp(m)\rightarrow \Sp(mn)$ be the homomorphism $L(A)=A\tensor I_{n}$. There is a basepoint preserving homotopy $H$ from $L$ to the $n$-fold direct sum map of symplectic matrices $\splus^{n}:\Sp(m) \to \Sp(mn)$ such that for all $t \in [0,1]$, $H(-,t)$ is a homomorphism.

Let $A\in \Sp(m)$,

\[
\begin{pmatrix}
A_{11}\tensor I_{n} & A_{12}\tensor I_{n}\\
A_{21}\tensor I_{n} & A_{22}\tensor I_{n}\\
\end{pmatrix}
%a_{11}I_{n}&  \cdots &  a_{1,2m}I_{n}\\
%\vdots&  \ddots &  \vdots\\
%a_{2m,1}I_{n}&  \cdots &  a_{2m,2m}I_{n}
%\quad \text{and} \quad
%A&  \cdots &  0\\
%\vdots&  \ddots &  \vdots\\
%0&  \cdots &  A
%a_{11}I_{n} & \cdots & a_{1m}I_{n} & a_{1,m+1}I_{n} &  \cdots & a_{1,2m}I_{n}\\
%\vdots & \ddots & \vdots & \vdots & \ddots &  \vdots\\
%a_{m1}I_{n}&  \cdots & a_{mm}I_{n} & a_{m,m+1}I_{n} &  \cdots & a_{m,2m}I_{n}\\
%a_{m+1,1}I_{n}&  \cdots & a_{m+1,m}I_{n} & a_{m+1,m+1}I_{n} &  \cdots & a_{m+1,2m}I_{n}\\
%\vdots & \ddots & \vdots & \vdots & \ddots &  \vdots\\
%a_{2m,1}I_{n}&  \cdots & a_{2m,m}I_{n} & a_{2m,m+1}I_{n} &  \cdots & a_{2m,2m}I_{n}\\
\quad \text{and} \quad %
A^{\splus n}=
\begin{pmatrix}
A_{11}^{\oplus n} & A_{12}^{\oplus n}\\
A_{21}^{\oplus n} & A_{22}^{\oplus n}
\end{pmatrix}.
\]

Let $P_{m,n}$ be the permutation matrix

\begin{align*}
P_{m,n}=[&e_{1},e_{n+1},e_{2n+1},\dots, e_{(m-1)n+1},%
e_{2},e_{n+2},e_{2n+2},\dots, e_{(m-1)n+2},\\%
e_{n},e_{2n},e_{3n},\dots, e_{(m-1)n},e_{mn}]
\end{align*}

where $e_{i}$ is the $i$-th standard basis vector of $\CC^{mn}$ written as a column vector. Observe that $A_{ij}\tensor I_{n}=P_{m,n}A_{ij}^{\oplus n}P_{m,n}^{-1}$ for $i,j=1,2$. This implies

\[
\begin{pmatrix}
P_{m,n} & 0\\
0 & P_{m,n}
\end{pmatrix}%
A^{\splus n}%
\begin{pmatrix}
P_{m,n}^{-1} & 0\\
0 & P_{m,n}^{-1}
\end{pmatrix}.
\]

The result follows from <cit.>.

Doubling homomorphism: The 2-fold direct sum $\oplus^{2}:\Or(n) \to \Or(2n)$ is such that $\im \oplus^{2} \subset \Sp(n)$. Let $d$ denote this homomorphism $d:\Or(n)\to \Sp(n)$

\[
\begin{pmatrix}
A & 0\\
0 & A
\end{pmatrix}.
\]

Let $R:\Or(n)\rightarrow \Sp(mn)$ be the homomorphism $R(A)=I_{2m}\tensor A$.  Then $R_{*}(x)=m\esta(d_{*}(x))$ for all $x \in \pi_{i}(\Or(n))$and $i\in \NN$.

For all $j=1,\dots,m$ define $\sigma_{j}:\Or(2n)\to \Or(2mn)$ by $\sigma_{j}(M)=\diag(I_{2n},\dots,I_{2n},M,I_{2n},\dots,I_{2n})$ where $M$ is the $j-$th position. Observe that $R$ can be decomposed as a composite:

\begin{equation*}
\begin{tikzcd}
\Or(n) \arrow[d,"d",swap] & A \arrow[d,mapsto] \\
\Or(2n) \arrow[d,"\Delta",swap] & d(A) \arrow[d,mapsto] \\
(\Or(2n))^{\times m} \arrow[d,"\sigma_{1}\times \cdots \times\sigma_{m}",swap] & \Bigl(d(A),\dots,d(A)\Bigr) \arrow[d,mapsto] \\
(\Or(2mn))^{\times m} \arrow[d,"\m",swap]  & \Bigl(\sigma_{1}d(A),\dots,\sigma_{m}d(A)\Bigr) \arrow[d,mapsto] \\
\Sp(mn) & \sigma_{1}d(A)\cdot\cdots\cdot\sigma_{m}d(A).
\end{tikzcd}
\end{equation*}

By <cit.>, all maps $\sigma_{j}$ are homotopic to $\sigma_{1}$, and thus, $R_{*}(x)=m\esta(d_{*}(x))$.

Let $i\in\NN$. The homomorphism $\tensor_{*}:\pi_{i}(\Sp(m))\times\pi_{i}(\Or(n))\to\pi_{i}(\Sp(mn))$ is equal to

\[
\tensor_{*}(x,y)=n\esta(x) + m\esta(d_{*}(y)) \quad \text{for} \quad x\in \pi_{i}(\Sp(m)) \quad \text{and} \quad y \in \pi_{i}(\Or(n)).
\]

Let $x\in \pi_{i}(\Sp(m))$ and $y \in \pi_{i}(\Or(n))$. By Lemmas <ref> and <ref>, and the mixed-product property of the tensor product $\tensor_{*}(x,y)=L_{*}(x)+R_{*}(y)=\splus^{n}_{*}(x)+\oplus^{m}_{*}(d_{*}(y))=n\esta(x)+m\esta(d_{*}(y))$ for all $i\in \NN$.

In view of Proposition <ref>, we focus our attention on determining the effect of the doubling homomorphism $d$ on homotopy groups in the stable range.

Let $i<n-1$. The homomorphism $d_{*}:\pi_{i}(\Or(n)) \longrightarrow \pi_{i}(\Sp(n))$

is an isomorphism onto $2\pi_{i}(\Sp(n))$ if $i\equiv 3,7$ (mod 8). In all other cases, either the source or the target is trivial.

Let $c:\Or(n,\RR)\hookrightarrow \fU{n}$ and $q:\fGL{n} \to \GL(n,\HH)$ be the inclusions. Since the conjugation on $\HH$ extends that on $\CC$, the inclusion $q$ restricts to $q:\fU{n} \to \U(n,\HH)$.

The homomorphism of $\RR$-algebras $c': \M(n,\HH) \to  \fM{2n}$ given by

\[
c'(A + jB)=%
\begin{pmatrix}
A & -\overline{B}\\
B & \overline{A}
\end{pmatrix}
\]

is injective, and its restriction to $\U(n,\HH)$ is an isomorphism, $c': \U(n,\HH) \longrightarrow \fU{2n}\cap \fSp{n}$  <cit.>. Thus we arrive at the following homomorphism

\begin{equation*}
\begin{tikzcd}[column sep=large, row sep=0.5pt]
\fU{n} \arrow[r,"q"] & \U(n,\HH) \arrow[r,"c'"] & \fU{2n}\cap\fSp{n} \arrow[r,hook,"\simeq"] \arrow[l,leftarrow,"\iso"]& \fSp{n}\\
A \arrow[rrr,mapsto] & & & \begin{pmatrix}
A & 0\\
0 & \overline{A}
\end{pmatrix}.
\end{tikzcd}
\end{equation*}

Hence, we can factor $d$ as follows

\begin{equation*}
\begin{tikzcd}[row sep=large]
\fO{n} \arrow[rr,"d"]  &  & \fSp{n} \\
\Or(n,\RR) \arrow[r,"c"] \arrow[u,hook,"\simeq"] & \fU{n} \arrow[r,"q\circ c'"] & \fSp{n}\cap \fU{2n}. \arrow[u,hook,"\simeq"]
\end{tikzcd}
\end{equation*}

The effect of $d_{*}$ on homotopy groups in the stable range for $\fO{n}$ is identified with the effect of $q_{*}\circ c_{*}$.
From <cit.>, <cit.>, and <cit.> the result follows.

Let $i<4m+2$ and $i<n-1$, the homomorphism $\tensor_{*}:\pi_{i}(\Sp(m))\times\pi_{i}(\Or(n)) \to \pi_{i}(\Sp(mn))$
is given by

\[
\otimes_{*}(x,y)=nx+2my \quad \text{for} \quad x \in \pi_{i}(\Sp(m)) \quad \text{and} \quad y\in \pi_{i}(\Or(n)).
\]

Corollary <ref>, <cit.>, and Propositions <ref> and <ref> yield the result.

§.§.§ Effect of stensor:Sp(m)xSp(n) –> O(4mn) on homotopy groups in the stable range of Sp(m)

Recall that $\stensor:\Sp(m) \times \Sp(n) \to \Or(4mn)$ is equal to the composite

\begin{equation*}
\begin{tikzcd}
\Sp(m) \times \Sp(n) \arrow[r,"\tensor"] & G \arrow[r,"\Int_{P}"] & \Or(4mn),
\end{tikzcd}
\end{equation*}

where $G\leq \GL(4mn)$ and $P\in \GL(4mn)$ is a basis change matrix, <cit.>. Let $L$ and $R$ denote the restrictions of $\stensor$ to $\Sp(m) \times \{I_{2n}\}$ and $\{I_{2m}\} \times  \Sp(n)$, respectively.

Let $c'$ and $c''$ denote the inclusions

\begin{equation*}
\begin{tikzcd}
c':\fSp{n} \arrow[r,hook] & \fGL{2n} &\text{and}& c'':\fO{n} \arrow[r,hook] & \fGL{2n}.
\end{tikzcd}
\end{equation*}

Let $m\leq n$ and $i<4m+2$, the homomorphism $\bigl(c''_{*}\circ\stensor_{*}\bigr):\pi_{i}(\Sp(m))\times\pi_{i}(\Sp(n)) \to \pi_{i}(\GL(4mn))$ is given by

\[
\bigl(c''_{*}\circ\stensor_{*}\bigr)(x,y)=2nc'_{*}(x)+2mc'_{*}(y)  \quad \text{for} \quad x \in \pi_{i}(\Sp(m)) \quad \text{and} \quad y \in \pi_{i}(\Sp(n)).
\]

By the mixed-product property of the tensor product of matrices, $A\stensor B=L(A)R(B)$.

Consider the commutative diagrams

\begin{equation*}
\begin{tikzcd}[row sep=large]
\Sp(m) \arrow[r,"c'"] \arrow[d,"L"] &  \GL(2m) \arrow[d,"L"] && \Sp(n) \arrow[r,"c'"] \arrow[d,"R"] &  \GL(2n) \arrow[d,"R"]\\
G \arrow[r,hookrightarrow] \arrow[d,"\Int_{P}"] &  \GL(4mn) \arrow[d,"\Int_{P}"] & \text{ and } & G \arrow[r,hookrightarrow] \arrow[d,"\Int_{P}"] &  \GL(4mn) \arrow[d,"\Int_{P}"]\\
\Or(4mn) \arrow[r,"c''"] \arrow[u,leftarrow,"\iso"] & \GL(4mn) \arrow[u,leftarrow,"\iso"] && \Or(4mn) \arrow[r,"c''"] \arrow[u,leftarrow,"\iso"] & \GL(4mn). \arrow[u,leftarrow,"\iso"]
\end{tikzcd}
%\begin{tikzcd}[row sep=large]
%\Sp(m) \arrow[r,"c'"] \arrow[d,"L"] &  \GL(2m) \arrow[d,"L"]\\
%G \arrow[r,hookrightarrow] \arrow[d,"\Int_{P}"] &  \GL(4mn) \arrow[d,"\Int_{P}"]\\
%\Or(4mn) \arrow[r,"c"] \arrow[u,leftarrow,"\iso"] & \GL(4mn) \arrow[u,leftarrow,"\iso"]
\end{equation*}

From these we obtain the commutative squares below.

\begin{equation}\label{cd:cL}
%\begin{tikzcd}[row sep=large]
%\pi_{i}(\Sp(m)) \arrow[r,"c'_{*}"] \arrow[d,"L_{*}"] &  \pi_{i}(\GL(2m)) \arrow[d,"L_{*}"] && \pi_{i}(\Sp(n)) \arrow[r,"c'_{*}"] \arrow[d,"R_{*}"] &  \pi_{i}(\GL(2n)) \arrow[d,"R_{*}"]\\
%\pi_{i}(\Or(4mn)) \arrow[r,"c_{*}"] & \pi_{i}(\GL(4mn)) && \pi_{i}(\Or(4mn)) \arrow[r,"c_{*}"] & \pi_{i}(\GL(4mn)).
\begin{tikzcd}[row sep=large]
\pi_{i}(\Sp(m)) \arrow[r,"c'_{*}"] \arrow[d,"L_{*}"] &  \pi_{i}(\GL(2m)) \arrow[d,"L_{*}"]\\
\pi_{i}(\Or(4mn)) \arrow[r,"c''_{*}"] & \pi_{i}(\GL(4mn))
\end{tikzcd}
\end{equation}

\begin{equation}\label{cd:cR}
\begin{tikzcd}[row sep=large]
\pi_{i}(\Sp(n)) \arrow[r,"c'_{*}"] \arrow[d,"R_{*}"] &  \pi_{i}(\GL(2n)) \arrow[d,"R_{*}"]\\
\pi_{i}(\Or(4mn)) \arrow[r,"c''_{*}"] & \pi_{i}(\GL(4mn))
\end{tikzcd}
\end{equation}

Observe that $R:\GL(2n)\to \GL(4mn)$ is equal to $(2n)$-fold direct sum, then $R_{*}:\pi_{i}(\GL(2n))\to \pi_{i}(\GL(4mn))$ is given by $R_{*}(y)=2m\esta(y)$ for $y\in \pi_{i}(\GL(2n))$, <cit.>. Moreover, $R,L:\GL(2m)\to \GL(4mn)$ are pointed homotopic <cit.>, then $L_{*}=R_{*}:\pi_{i}(\GL(2m))\to \pi_{i}(\GL(4mn))$, and $L_{*}(x)=2n\esta(x)$ for $x\in \pi_{i}(\GL(2m))$.

Therefore, by the commutativity of squares (<ref>) and (<ref>), we have that $L_{*}:\pi_{i}(\Sp(m))\to \pi_{i}(\Or(4mn))$ and $R_{*}:\pi_{i}(\Sp(n))\to \pi_{i}(\Or(4mn))$ satisfy the following equalities

\[
(c''_{*}\circ L_{*})(x)=2n\esta(c'_{*}(x)) \quad \text{and} \quad (c''_{*}\circ R_{*})(y)=2m\esta(c'_{*}(y)).
\]

Thus, $(c''_{*}\circ L_{*})(x)=2nc'_{*}(x)$ and $(c''_{*}\circ R_{*})(y)=2mc'_{*}(y)$ for $x \in \pi_{i}(\Sp(m))$, $y\in \pi_{i}(\Sp(n))$ and $i<4m+2$.

Let $i$ be in the stable range for $\fSp{n}$. Then the induced homomorphism $c'_{*}:\pi_{i}(\fSp{n}) \to \pi_{i}(\fGL{n})$

is an isomorphism if $i\equiv 3$ (mod 8) and is an isomorphism onto $2\pi_{i}(\fGL{2n})$ if $i\equiv 7$ (mod 8). In all other cases, it is trivial because the source is torsion and the target is torsion-free.

Let $m\leq n$ and $i<4m+2$, the homomorphism

\begin{equation*}
\begin{tikzcd}
\stensor_{*}:\pi_{i}(\Sp(m))\times\pi_{i}(\Sp(n))  \arrow[r] & \pi_{i}(\Or(4mn))
\end{tikzcd}
\end{equation*}

is given by

\[
\stensor_{*}(x,y)=%
\begin{cases}
nx+my & \text{if $i\equiv 3$ (mod 8)},\\
4(nx+my) & \text{if $i\equiv 7$ (mod 8)},\\
0 & \text{otherwise},\\
\end{cases}
\]

for $x \in \pi_{i}(\Sp(m))$ and $y \in \pi_{i}(\SO(n))$.

Let $i<4m+2$ and $i\equiv 3,7$ (mod 8), then by Proposition <ref>, diagrams (<ref>) and (<ref>) have the form

\begin{equation*}
\begin{tikzcd}[row sep=large]
\pi_{i}(\Sp(m)) \arrow[r,"c'_{*}"] \arrow[d,"L_{*}"] &  \pi_{i}(\GL(2m)) \arrow[d,"\times(2n)"]\\
\pi_{i}(\Or(4mn)) \arrow[r,"c''_{*}"] & \pi_{i}(\GL(4mn))
\end{tikzcd}
\end{equation*}

\begin{equation*}
\begin{tikzcd}[row sep=large]
\pi_{i}(\Sp(n)) \arrow[r,"c'_{*}"] \arrow[d,"R_{*}"] &  \pi_{i}(\GL(2n)) \arrow[d,"\times(2m)"]\\
\pi_{i}(\Or(4mn)) \arrow[r,"c''_{*}"] & \pi_{i}(\GL(4mn)).
\end{tikzcd}
\end{equation*}

By <cit.> and Theorem <ref>,  if $i\equiv 3$ (mod 8), then $c'_{*}$ is an isomorphism and $c''_{*}$ is an isomorphism onto $2\pi_{i}(\fGL{4mn})$. Therefore, $2L_{*}(x)=2nx$ and $2R_{*}(y)=2my$, i.e. $L_{*}(x)=nx$ and $R_{*}(x)=my$.

By <cit.> and Theorem <ref>, if $i\equiv 7$ (mod 8), then $c'_{*}$ is an isomorphism onto $2\pi_{i}(\fGL{2l})$ for $l=m,n$ and $c''_{*}$ is an isomorphism. Therefore, $L_{*}(x)=4nx$ and $R_{*}(y)=4my$.

In all other cases, either the source or the target is trivial.

Let $i<4m+2$, the homomorphism

\begin{equation*}
\begin{tikzcd}
\stensor^{2}_{*}:\pi_{i}(\Sp(m)) \arrow[r] & \pi_{i}(\Or(4m^{2}))
\end{tikzcd}
\end{equation*}

is given by

\[
\stensor^{2}_{*}(x)=%
\begin{cases}
2mx & \text{if $i\equiv 3$ (mod 8)},\\
8mx & \text{if $i\equiv 7$ (mod 8)},\\
0 & \text{otherwise},\\
\end{cases}
\]

for $x \in \pi_{i}(\Sp(m))$.

§.§ Tensor product on the quotient

We want to describe the effect of the tensor product operation on the homotopy groups of the projective complex symplectic group.

The methods we use in <cit.> and <cit.> to establish the decomposition of topological Azumaya algebras apply to those whose degrees are relatively prime. For this reason, we only study the tensor product (<ref>) where $n$ is odd. The tensor product operation $\tensor:\Sp(m)\times\SO(n) \to \Sp(mn)$ sends $Z(\Sp(m))\times\{I_{n}\}$ to $Z(\Sp(mn))$. As a consequence, the operation descends to the quotient

\begin{equation}\label{qSp}
\begin{tikzcd}
\tensor:\PSp(m)\times\SO(n) \arrow[r] & \PSp(mn).
\end{tikzcd}
\end{equation}

Let $i<4m+1$. The homomorphism

\begin{equation*}
\begin{tikzcd}
\tensor_{*}:\pi_{i}(\PSp(m))\times\pi_{i}(\SO(n)) \arrow[r] & \pi_{i}(\PSp(mn))
\end{tikzcd}
\end{equation*}

is given by $\tensor_{*}(x,y)=nx+2my$ for $x\in \pi_{i}(\PSp(m))$, $y \in \pi_{i}(\SO(n))$.

There exits a map of fibrations

\begin{equation}\label{mfSp}
\begin{tikzcd}
Z(\Sp(m))\times\{I_{n}\} \arrow[r,hookrightarrow] \arrow[d,"\tensor"] & \Sp(m)\times\SO(n) \arrow[r,twoheadrightarrow] \arrow[d,"\tensor"] & \PSp(m)\times\SO(n) \arrow[d,"\tensor"]\\
Z(\Sp(mn)) \arrow[r,hookrightarrow] & \Sp(mn) \arrow[r,twoheadrightarrow] & \PSp(mn).
\end{tikzcd}
\end{equation}

Then there exits a homomorphism between the long exact sequences associated to the fibrations in diagram (<ref>). For $i>1$ we obtain a commutative square

\begin{equation*}
\begin{tikzcd}
\pi_{i}(\Sp(m))\times\pi_{i}(\SO(n))\arrow[r,"\iso"] \arrow[d,"\otimes_{*}"] & \pi_{i}(\PSp(m))\times\pi_{i}(\SO(n)) \arrow[d,"\otimes_{*}"]\\
\pi_{i}(\Sp(mn)) \arrow[r,"\iso"] & \pi_{i}(\PSp(mn)).
\end{tikzcd}
\end{equation*}

From this diagram and Proposition <ref> we have that for all $1<i<4m+1$, $\tensor_{*}(x,y)=nx+2my$ where $x\in \pi_{i}(\PSp(n))$ and $y \in \pi_{i}(\SO(n))$.

If $i=0,1$, then $\tensor_{*}$ is trivial.

§ PROOF OF THEOREM <REF>

Let $m$ and $n$ be positive integers such that $n$ is odd. By applying the classifying-space functor to the homomorphism (<ref>) we obtain a map

\begin{equation*}
\begin{tikzcd}
f_{\tensor}: \B\PSp(m)\times\B\SO(n) \arrow[r] & \B\PSp(mn).
\end{tikzcd}
\end{equation*}

Let $m$ and $n$ be positive integers such that $n$ is odd. Let $d$ denote $\min\{4m+2,n-1\}$. There exists a homomorphism $\widetilde{\Tr}:\PSp(m)\times \SO(n) \rightarrow \SO(N)$, for some positive integer $N$, such that for all $i<d$ the homomorphisms induced on homotopy groups

\begin{equation*}
\begin{tikzcd}
\widetilde{\Tr}_{i}:\pi_{i}(\PSp(m))\times \pi_{i}(\SO(n)) \arrow[r] & \pi_{i}(\SO(N))
\end{tikzcd}
\end{equation*}

are given by the following expressions, where $u$ and $v$ are some positive integers, independent of $i$, for which $\bigl|vn-4um^{2}\bigr|=1$.

* If $i=1$, then $\widetilde{\Tr}_{1}(x,y)=zx+y$ for $x,y \in \ZZ/2$ and some $z\in \ZZ/2$.

* If $1<i<d$, then

\[
\widetilde{\Tr}_{i}(x,y)=%
\begin{cases}
vy & \text{if $i\equiv 0, 1$ (mod 8)},\\
2umx+vy & \text{if $i\equiv 3$ (mod 8)},\\
8umx+vy & \text{if $i\equiv 7$ (mod 8)},\\
0 & \text{otherwise},\\
\end{cases}
\]

for $x \in \pi_{i}(\PSp(m))$ and $y \in \pi_{i}(\SO(n))$.

Without loss of generality suppose $4m+2<n-1$. There exist positive integers $u$ and $v$ such that $vn-4um^{2}=\pm1$.
Let $N$ denote $4um^{2}+vn$, and let $\Tr$ denote the composite

\begin{equation*}
\begin{tikzcd}[column sep=large]
\Sp(m)\times \SO(n) \arrow[d,"(\stensor^{2}\text{,}\id)"]\\
\SO(4m^{2})\times \SO(n) \arrow[d,"(\oplus^{u}\text{,}\oplus^{v})"]\\
\SO(4um^{2})\times \SO(vn) \arrow[d,"\oplus"]\\
\SO(N).
\end{tikzcd}
\end{equation*}

Note that the elements $\bigl(\pm I_{m}, I_{n}\bigr)$ are sent to $\bigl(I_{4m^{2}},I_{n}\bigr)$ by $(\stensor^{2},\id)$, hence to the identity by the composite $\Tr$ defined above. Hence $\Tr$ factors through $\PSp(m)\times \SO(n)$

\begin{equation*}
\begin{tikzcd}[column sep=huge]
\Sp(m)\times \SO(n) \arrow[rd,bend left=20,"\Tr"] \arrow[d,twoheadrightarrow]  &  \\
\PSp(m)\times \SO(n) & \SO(N). \arrow[l,leftarrow,"\widetilde{\Tr}"]
\end{tikzcd}
\end{equation*}

From Corollaries <ref> and <ref> we have that for $i<4m+2$

\[
\Tr_{i}(x,y)=%
\begin{cases}
2umx+vy & \text{if $i\equiv 3$ (mod 8)},\\
8umx+vy & \text{if $i\equiv 7$ (mod 8)},\\
vy & \text{otherwise}.\\
\end{cases}
\]

The map of fibrations

\begin{equation*}
\begin{tikzcd}
Z(\Sp(m))\times \{I_{n}\} \arrow[r,hookrightarrow] \arrow[d] & \Sp(m)\times \SO(n) \arrow[r,twoheadrightarrow] \arrow[d,"\Tr"] & \PSp(m)\times \SO(n) \arrow[d,"\widetilde{\Tr}"]\\
\{I_{N}\} \arrow[r] & \SO(N) \arrow[r,equal] & \SO(N)
\end{tikzcd}
\end{equation*}

induces a commutative diagram

\begin{equation*}%\label{iabove1}
\begin{tikzcd}
\pi_{i}(\Sp(m))\times\pi_{i}(\SO(n)) \arrow[r,"\iso"] \arrow[d,"\Tr_{i}"] & \pi_{i}(\PSp(m))\times\pi_{i}(\SO(n)) \arrow[d,"\widetilde{\Tr}_{i}"]\\
\pi_{i}(\SO(N)) \arrow[r,equal] & \pi_{i}(\SO(N))
\end{tikzcd}
\end{equation*}

for $i>1$. Then $\widetilde{\Tr}_{i}(x,y)=\Tr_{i}(x,y)$ for $1<i<4m+2$.

For $i=1$, the map of fibrations induces the commutative diagram below

\begin{equation*}
\begin{tikzcd}
\pi_{1}(\Sp(m))\times\pi_{1}(\SO(n)) \arrow[r,hookrightarrow] \arrow[d,"\Tr_{1}"] & \pi_{1}(\PSp(m))\times\pi_{1}(\SO(n)) \arrow[d,"\widetilde{\Tr}_{1}"]\\
\pi_{1}(\SO(N)) \arrow[r,equal] & \pi_{1}(\SO(N)),
\end{tikzcd}
\end{equation*}

which takes the form

\begin{equation}\label{cd:TTtildeOSp}
\begin{tikzcd}
0\times \ZZ/2 \arrow[r,hookrightarrow] &  \ZZ/2 \times\ZZ/2  \arrow[d,"\widetilde{\Tr}_{1}"]\\
& \ZZ/2. \arrow[ul,leftarrow,"\Tr_{1}"]
\end{tikzcd}
\end{equation}

Observe that the equality $vn-4um^{2}=\pm1$ implies $v$ is odd, then $\Tr_{1}(0,\beta)=\beta$, i.e. $\Tr_{1}$ is the identity. Thus, $\widetilde{\Tr}_{1}(0,1)=1$. Let $z$ denote $\widetilde{\Tr}_{1}(1,0)$. Then $\widetilde{\Tr}_{1}(x,y)=zx+y$.

§.§ A d-connected map where d=min(4m+3,n)

Let $J$ denote the map

\begin{equation}\label{JSpO}
\begin{tikzcd}[row sep=tiny]
J:\B\PSp(m)\times\B\SO(n) \arrow[r,mapsto] & \B\PSp(mn)\times\B\SO(N)\\
(x,y) \arrow[r,mapsto] & \Bigl(f_{\tensor}(x,y)\text{,}\B\widetilde{\Tr}(x,y)\Bigr).
\end{tikzcd}
\end{equation}

Let $J_{i}$ denote the homomorphism induced on homotopy groups by $J$.

\begin{equation}\label{JiOSp}
\begin{tikzcd}
J_{i}: \pi_{i}(\B\PSp(m))\times \pi_{i}(\B\SO(n)) \arrow[r] & \pi_{i}(\B\PSp(mn))\times\pi_{i}(\B\SO(N))
\end{tikzcd}
\end{equation}

Let $m$ and $n$ be positive integers such that $n$ is odd. Let $d$ denote $\min\{4m+3,n\}$ and let $i<d$, then the homomorphism $J_{i}$ is an isomorphism if $i>0$ and $i\not \equiv 0$ (mod 8).

Without loss of generality we can assume $4m+3<n$.

Let $0<i<4m+1$ and $i\not\equiv 0$ (mod 8). Observe that $J_{i}$ is trivial for $i=0,1$ and, $i\equiv 3,7$ (mod 8) with $i>2$. Then we study the cases $i=2$ and, $i\equiv 1,2,4,5,6$ (mod 8) with $i>2$.

Let $i=2$. By Propositions <ref> and <ref> the homomorphism $J_{2}:\ZZ/2\times\ZZ/2 \to \ZZ/2 \times\ZZ/2$ is represented by the invertible matrix

\[
\begin{pmatrix}
1 & 0\\
z & 1
\end{pmatrix}.
\]

Let $2<i<4m+1$ and $i\equiv4$ (mod 8). By Propositions <ref> and <ref> the homomorphism $J_{i}:\ZZ\times \ZZ \to \ZZ \times\ZZ$ is represented by the invertible matrix

\[
\begin{pmatrix}
n & 2m\\
2um & v
\end{pmatrix}.
\]

Let $2<i<4m+1$ and $i\equiv1,2$ (mod 8). By Propositions <ref> and <ref> the homomorphism $J_{i}:0\times \ZZ/2 \to 0\times \ZZ/2$ is given by $J_{i}(0,y)=(2my,vy)=(0,y)$.

Let $2<i<4m+1$ and $i\equiv5,6$ (mod 8). By Propositions <ref> and <ref> the homomorphism $J_{i}:\ZZ/2\times 0 \to \ZZ/2\times 0$ is given by $J_{i}(x,0)=(nx,2umx)=(x,0)$.

Let $m$ be even, then $4m+1\equiv 1$ and $4m+2\equiv 2$ (mod 8). Thus, $J_{4m+1}$ and $J_{4m+2}$ take the form $0\times \ZZ/2 \to 0\times \ZZ/2$. As in the case $i\equiv 1,2$, we have $J_{4m+1}(0,y)=J_{4m+2}(0,y)=(0,y)$.

Let $m$ be odd, then $4m+1\equiv 5$ and $4m+2\equiv 6$ (mod 8). Thus, $J_{4m+1}$ and $J_{4m+2}$ take the form $\ZZ/2\times 0 \to \ZZ/2\times 0$. As in the case $i\equiv 5,6$, we have $J_{4m+1}(x,0)=J_{4m+2}(x,0)=(x,0)$.

As corollary of Propositions <ref> we obtain the following.

Let $m$ and $n$ be positive integers such that $m>1$, $n>7$, and $n$ is odd. Then the map $J$ is $7$-connected.

§.§ Factorization through f:BPSp2m x BSOn –> BPSp2mn

Theorem <ref> is a Corollary of Theorem <ref>.

Let $X$ be a CW complex such that $\dim(X)\leq 7$. Let $m$ and $n$ be positive integers such that $m>1$, $n>7$, and $n$ is odd. Every map $\cA:X \rightarrow \B\PSp(mn)$ can be lifted to $\B\PSp(m)\times \B\SO(n)$ along the map $f_{\tensor}$.

Diagrammatically speaking, we want to find a map

\begin{equation*}
\begin{tikzcd}
\cA_{2m}\times\cA_{n}:X \arrow[r] & \B\PSp(m)\times \B\SO(n)
\end{tikzcd}
\end{equation*}

such that diagram (<ref>) commutes up to homotopy

\begin{equation}\label{cd:lpOSp}
\begin{tikzcd}[execute at begin picture={\useasboundingbox (-4.5,-1) rectangle (4.5,1);},row sep=large,column sep=huge]
& \B\PSp(m)\times \B\SO(n) \arrow[d,"f_{\tensor}"] \\
X \arrow[r,"\cA"] \arrow[ur,dotted,"\cA_{2m}\times\cA_{n}",bend left=20]  & \B\PSp(mn).
\end{tikzcd}
\end{equation}

Without loss of generality supposse $4m+2<n-1$. Corollary <ref> yields a map $J: \B\PSp(m)\times \B\SO(n) \rightarrow \B\SO(N)$ where $N$ is some positive integer so that $N\gg n-1>4m+2$. Observe that $f_{\tensor}$ factors through $\B\PSp(mn)\times \B\SO(N)$, so we can write $f_{\tensor}$ as the composite of $J$ and the projection $\proj_{1}$ shown in diagram (<ref>).

\begin{equation}\label{cd:JprojOSp}
\begin{tikzcd}[row sep=large,column sep=huge]
\B\PSp(m)\times \B\SO(n) \arrow[r,"J"] \arrow[d,"f_{\tensor}"] & \B\PSp(mn)\times \B\SO(N) \\
\B\PSp(mn) \arrow[ur,leftarrow,"\proj_{1}"]&
\end{tikzcd}
\end{equation}

Since $J$ is $7$-connected and $\dim(X)\leq 7$, then by Whitehead's theorem

\begin{equation*}
\begin{tikzcd}
J_{\#}:\bigl[X, \B\PSp(m)\times \B\SO(n)\bigr] \arrow[r] &  \bigl[X,\B\PSp(mn)\times \B\SO(N)\bigr]
\end{tikzcd}
\end{equation*}

is a surjection, <cit.>.

Let $s$ denote a section of $\proj_{1}$. The surjectivity of $J_{\#}$ implies $s\circ \cA$ has a preimage $\cA_{2m}\times\cA_{n}:X \rightarrow \B\PSp(2)\times \B\SO(n)$ such that $J\circ (\cA_{2m}\times\cA_{n})\simeq s\circ \cA$.

Commutativity of diagram (<ref>) follows from commutativity of diagram (<ref>). Thus, the result follows.

Let $m$ and $n$ be positive integers such that $n$ is odd.

* If $4m+4<n$ and $4m+4<4mn$, then $f_{\tensor}: \B\PSp(m)\times\B\SO(n) \to \B\PSp(mn)$ does not have a section.

* If $n\in \{3,5,7\}$, $n<4m+3$ and $n<4mn$, then then $f_{\tensor}: \B\PSp(m)\times\B\SO(n) \to \B\PSp(mn)$ does not have a section.

Suppose $f_{\tensor}$ has a section, namely $\sigma$. By Proposition <ref>, the homomorphism induced on homotopy groups by $f_{\tensor}$ is given by $(x,y)\mapsto n\esta(x)+m\esta(d_{*}(y))$ for all $x\in \pi_{i}(\B\PSp(m))$, $y\in\pi_{i}(\B\SO(n))$, and $i \in \NN$.

Since $\sigma$ is a section of $f_{\tensor}$, the image of $(f_{\tensor})_{i}\circ \sigma_{i}$ is equal to $\pi_{i}(\B\PSp(mn))$ for all $i\geq 1$.

* From Table <ref> and <cit.>, $\pi_{4m+4}(\B\SO(n))\iso\pi_{4m+4}(\B\PSp(mn))\iso \ZZ$. From <cit.>, $\pi_{4m+4}(\B\PSp(m))\iso\ZZ/2$. Therefore, the image of $(f_{\tensor})_{4m+4}:\pi_{4m+4}(\B\PSp(m))\times\pi_{4m+4}(\B\SO(n))\to\pi_{4m+4}(\B\PSp(mn))$ is $m\ZZ$ which implies that the image of $(f_{\tensor})_{4m+4} \,\circ\, \sigma_{4m+4}$ is contained in $m\ZZ$. However, this image is equal to the integers.

* Let $C$ denote the set $\bigl\{(8,3), (12,5), (16,7)\bigr\}$. From Table <ref>, $\pi_{i}(\B\PSp(m))\iso\pi_{i}(\B\PSp(mn))\iso \ZZ$  for all $(i,m) \in C$, and from <cit.> $\pi_{i}(\B\SO(n))$ is torsion for all $(i,m) \in C$. Therefore, the image of $(f_{\tensor})_{i}:\pi_{i}(\B\PO(m))\times\pi_{i}(\B\SO(n))\to\pi_{i}(\B\PO(mn))$ is $n\ZZ$ for all $(i,m) \in C$. In a similar way as in (1), this leads to a contradiction.

Let $m$ and $n$ be positive integers such that $n$ is odd. If $\cA$ is a topological Azumaya algebra of degree $2mn$ with a symplectic involution over a finite CW complex of dimension higher than $\min\{4m+3,n\}$, then $\cA$ may not decompose as the tensor product of topological Azumaya algebras of degrees $2m$ and $n$ with symplectic involutions.

cambia de linea.

* Let $4m+4<n$ and $4m+4<4mn$. Let $\cS$ be a topological Azumaya algebra of degree $2mn$ with a symplectic involution on the unit $(4m+4)-$sphere, $S^{4m+4}$, such that $\cS$ generates $\pi_{4m+4}(\B\PSp(mn))$. An argument similar to the one in Proposition <ref> part (1) can be used to prove the statement.

* If $n\in \{3,5,7\}$, and $16<4m+3<4mn$, we can show the statement follows by considering a topological Azumaya algebra of degree $2mn$ with a symplectic involution on the unit $i-$sphere that generates $\pi_{i}(\B\PSp(mn))$ for $(i,m) \in C$.

§ DECOMPOSITION OF SYMPLECTIC VECTOR BUNDLES

In this section, we show that symplectic vector bundles decompose as a tensor product of symplectic and orthogonal vector bundles in the stable range of $\fSO{n}$.

The tensor product operation $\tensor:\Sp(m) \times \SO(n) \to \Sp(mn)$ induces a map at the level of classifying spaces $f'_{\tensor}:\B\Sp(m) \times \B\SO(n) \to \B\Sp(mn)$. Let $F$ denote the homotopy fiber of $f'_{\tensor}$. From Proposition <ref>, we observe that $\pi_{i}(F)\iso \pi_{i}(\B\SO(n))$ for all $i<4m+3$ and $i<n$.

Therefore, the Moore-Postnikov tower of $f'_{\tensor}$ takes the form shown in diagram (<ref>) where $1<i<n-1$ and $i\equiv 3$ (mod $8$).

\begin{equation}\label{MPofBSp}
\begin{tikzcd}[row sep=small]
\B\Sp(m)\times\B\SO(n) \arrow[d] &\\
Y[n-1] \arrow[d] \arrow[r,"k_{n-1}"] & \K\Bigl(\pi_{n}(\B\SO(n)),n+1\Bigr)\\
Y[n-2] \arrow[d] \arrow[r,"k_{n-1}"] & \K\Bigl(\pi_{n-1}(\B\SO(n)),n\Bigr)\\
\vdots \arrow[d] & \\
Y[i+6] \arrow[d]  \arrow[r,"k_{i+6}"] & \K(\ZZ/2,i+8)\\
Y[i+5] \arrow[d]  \arrow[r,"k_{i+5}"] & \K(\ZZ/2,i+7)\\
Y[i+4] \arrow[d]  \arrow[r,"k_{i+4}"] & \K(\ZZ,i+6) \\
Y[i] \arrow[d] \arrow[r,"k_{i}"] & \K(\ZZ,i+2) \\
\vdots \arrow[d] & \\
%Y[9] \arrow[d]  \arrow[r,"k_{9}"] & \K(\ZZ/2,11)\\
%Y[8] \arrow[d]  \arrow[r,"k_{8}"] & \K(\ZZ/2,10)\\
%Y[7] \arrow[d]  \arrow[r,"k_{7}"] & \K(\ZZ,9) \\
%Y[3] \arrow[d] \arrow[r,"k_{3}"] & \K(\ZZ,5) \\
\B\Sp(mn) \arrow[r,"k_{1}"] & \K(\ZZ/2,3)
\end{tikzcd}
\end{equation}

The quaternionic Grassmannian $G_{m}(\HH^{\infty})$ is a model for the classifying space of $\Sp(mn)$ <cit.>. Moreover, from <cit.> and the universal coefficients theorem for cohomology $\Hy^{*}(G_{m}(\HH^{\infty});A)\iso A[\alpha_{1},\dots,\alpha_{m}]$ with $|\alpha_{i}|=4i$ for all $i=1,\dots,m$ and $A = \ZZ, \ZZ/2$ .

Let $\sk_{i}\B\Sp(mn)$ be the $i-$th skeleton of $\B\Sp(mn)$. We have that $\Hy^{i}(\sk_{n}\B\Sp(mn)) \iso \Hy^{i}(\B\Sp(mn))$ for $i<n$, and $\Hy^{i}(\sk_{n}\B\Sp(mn)) \iso 0$ for $i>n$.

Since $\Hy^{i}(G_{m}(\HH^{\infty});A)\iso 0$ for $i\not\equiv 0$ (mod $4$), we have that $\Hy^{i}(\sk_{n}\B\Sp(mn);A)\iso 0$ for $i<n$,  $i\not\equiv 0$ (mod $4$) as well for $i\geq n+1$. Since $n$ is odd, it follows that $\Hy^{i}(\sk_{n}\B\Sp(mn);A)\iso 0$ for $i\not\equiv 0$ (mod $4$). Since $i+2, i+6, i+7$, and $i+8$ are not congruent to zero modulo $4$ for $i\equiv 3$ (mod $8$), there is a lifting, $\xi$, of the inclusion $\sk_{n}\B\Sp(mn) \hookrightarrow \B\Sp(mn)$ along the Moore-Postnikov tower of $f'_{\tensor}$.

Let $\cV: X\to \B\Sp(mn)$ be a symplectic bundle of rank $2mn$ and $\dim(X)\leq n$. By CW approximation, $\cV$ factors through the inclusion $\sk_{n}\B\Sp(mn) \hookrightarrow \B\Sp(mn)$. Thus, there exists a not necessarily unique lifting of $\cV$ along $f'_{\tensor}$ that makes diagram (<ref>) commute up to homotopy, namely the composite of the dotted arrow and $\xi$. The result follows.

\begin{equation}\label{liftingV}
\begin{tikzcd}[row sep=normal]
& & \B\Sp(m)\times\B\SO(n) \arrow[dd]\\
& \sk_{n}\B\Sp(mn)  \arrow[dr,hookrightarrow] \arrow[ur,"\xi",bend left=15] & \\
X \arrow[rr,"\cV"] \arrow[ru,dotted] &  & \B\Sp(mn)
\end{tikzcd}
\end{equation}

author=Arcila-Maya, Niny,
title=Decomposition of topological Azumaya algebras,
journal=Canadian Mathematical Bulletin,

author=Arcila-Maya, Niny,
title=Decomposition of topological Azumaya algebras with orthogonal
publisher=arXiv. Accepted for publication after corrections at the Journal
of Pure and Applied Algebra.,

author=Amitsur, S. A.,
author=Rowen, L. H.,
author=Tignol, J. P.,
title=Division algebras of degree 4 and 8 with involution,
journal=Bulletin (New Series) of the American Mathematical Society,
pages=691 693,

author=Hatcher, Allen,
title=Algebraic topology,
publisher=Cambridge University Press, Cambridge,
ISBN=0-521-79160-X; 0-521-79540-0,

author=Hatcher, Allen,
title=Vector bundles & K-theory,
url=https://pi.math.cornell.edu/ hatcher/VBKT/VB.pdf,

author=Knus, Max-Albert,
author=Parimala, R.,
author=Srinivas, V.,
title=Involutions on rank 16 central simple algebras,
journal=Journal of the Indian Math. Soc.,

author=Merkur'ev, A. S.,
title=On the norm residue symbol of degree 2,
journal=Dokl. Akad. Nauk SSSR,

author=Mathematical Society of Japan, Corporate,
author=Itô, Kiyosi,
title=Encyclopedic dictionary of mathematics (2nd ed.),
publisher=MIT Press,
address=Cambridge, MA, USA,

author=Mimura, Mamoru,
author=Toda, Hiroshi,
title=Homotopy groups of symplectic groups,
journal=Journal of Mathematics of Kyoto University,
pages=251 273,

author=Mimura, Mamoru,
author=Toda, Hirosi,
title=Topology of Lie groups. I, II,
series=Translations of Mathematical Monographs,
publisher=American Mathematical Society, Providence, RI,
note=Translated from the 1978 Japanese edition by the authors,

author=Spanier, Edwin H.,
title=Algebraic Topology,
publisher=Springer New York,
address=New York, NY,