Decomposition of Topological Azumaya Algebras with Orthogonal Involution

Arcila-Maya]Niny Arcila-Maya

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

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

Let $\Az{A}$ be a topological Azumaya algebra of degree $mn$ with an orthogonal involution over a CW complex $X$ of dimension less than or equal to $\min\{m,n\}$. We give conditions for the positive integers $m$ and $n$ so that $\Az{A}$ can be decomposed as the tensor product of topological Azumaya algebras of degrees $m$ and $n$ with orthogonal involutions.

§ INTRODUCTION

The concept  of a central simple algebra over a field was generalized by Azumaya <cit.> and Auslander–Goldman <cit.> to the notion of an Azumaya algebra over a commutative ring, and Grothendieck <cit.> defined this concept in the context of a locally ringed topos, that is, in a category of sheaves equipped with a designated choice of sheaf of rings with enough points, $\cO$, having local rings as stalks. One recovers the case of Auslander–Goldman when the category is $(\Spec R)_{\et}$ of étale sheaves over the spectrum of the ring $R$. That is, an Azumaya algebra of degree $n$ is a sheaf of $\cO$ algebras that is locally isomorphic to $\M(n,\cO)$. In the topological case, a topological Azumaya algebra of degree $n$ is defined by taking the sheaf $\cO$ to be the sheaf of continuous functions with value $\CC$, that is, it is a bundle of associative and unital complex algebras over a topological space that is locally isomorphic to the matrix algebra $\fM{n}$, <cit.>.

The endomorphism bundle of a complex vector bundle of rank $n$ is a topological Azumaya algebra of degree $n$ over $X$.

Let $G$ be a group acting on the left on $\fM{n}$ via algebra automorphisms. Let $X$ be a topological space with a free right $G-$action. Suppose the projection $p:X\to X/G$ is a principal $G-$bundle. Then the fiber bundle with fiber $\fM{n}$ associated to $p$, $\fM{n} \to X\times^{G}\fM{n} \to X/G$ is a topological Azumaya algebra of degree $n$ over $X/G$, where $G$ acts on $X\times\fM{n}$ by $(x,M)\cdot g\coloneqq \bigl(x\cdot g,g\cdot M\bigr)$ for all $x\in X$ and $M\in \fM{n}$.

In the theory of central simple algebras over a field $k$, a theorem of Wedderburn states that any central simple algebra $A/k$ has the form $\M(n,D)$, where $D/k$ is a division algebra that is unique up to isomorphism, <cit.>. This theorem reduces the classification problem for finite-dimensional central simple algebras over $k$ to the classification problem for finite-dimensional central division algebras over $k$. Two finite-dimensional central simple algebras $A\iso\M(m,D)$ and $B\iso\M(n,E)$ are Brauer equivalent if their division algebras $D$ and $E$ are isomorphic. The Brauer group of $k$, $\Br(k)$, is the group of equivalence classes of finite-dimensional central simple algebras over $k$ modulo Brauer equivalence, with the tensor product of algebras, <cit.>.

Given two topological Azumaya algebras $\cA$ and $\cA'$ over $X$ of degrees $m$ and $n$, respectively, we can define the tensor product $\cA\tensor \cA'$ by performing the tensor product of (Kronecker product of) matrices $\fM{m}\tensor_{\CC}\fM{n}$ fiberwise. We say two topological Azumaya algebras $\cA$ and $\cA'$ are Brauer equivalent if there exist complex vector bundles $\cV$ and $\cV'$ such that $\cA \tensor \End(\cV) \iso \cA' \tensor \End(\cV')$ as bundles of $\CC$-algebras. The (topological) Brauer group of $X$, $\Br(X)$, is the set of isomorphism classes of topological Azumaya algebras over $X$ modulo Brauer equivalence with the tensor product operation.  The identity of $\Br(X)$ is the class $[\End(\cV)]$ for all complex vector bundles $\cV$, and $[\cA]^{-1}=[\cA^{\text{op}}]$ for all classes $[\cA] \in \Br(X)$ since $\cA\tensor \cA^{\text{op}}\iso \End(\cA)$. We say a topological Azumaya $\cA$ is Brauer trivial if its class $[\cA] \in \Br(X)$ is equal to the identity.

If $D/k$ is a finite-dimensional central division algebra, then $\dim_{k}(D)$ is a square, <cit.>. The degree of $D/k$ is defined by $\displaystyle{\sqrt{\dim_{k}(D)}}$. The theory of central division algebras is equipped with a structure theorem stating that every finite-dimensional central division algebra $D/k$ can be broken up into pieces corresponding to the prime factorization of its degree $\deg(D)=p_{1}^{n_{1}}\cdots p_{r}^{n_{r}}$, i.e. $D$ is isomorphic to $D_{1}\tensor_{k} D_{2} \tensor_{k} \cdots \tensor_{k} D_{r}$ where each $D_{i}$ is a division algebra of degree $p_{i}^{n_{i}}$, and the decomposition is unique up to isomorphism, <cit.>. Saltman asked in <cit.> whether the analogue to the prime decomposition theorem for central simple algebras over a field holds for Azumaya algebras over a commutative ring. Using algebraic topology and topological Azumaya algebras, Antieau–Williams showed that, in general, there is no prime decomposition for these algebras, <cit.>. The author provided conditions for a positive integer $n$ and a topological space $X$ such that a topological Azumaya algebra of degree $n$ on $X$ has a non-unique tensor product decomposition, <cit.>.

Let $A/k$ be a finite-dimensional central simple algebra. An involution on $A$ is an anti-automorphism $\tau:A\to A$ satisfying $(\tau\circ \tau)(a)=a$ for all $a\in A$. A central simple algebra with an involution is denoted by $(A,\tau)$. It can be checked that the center $k$ is preserved under $\tau$. The restriction of $\tau$ to $k$ is therefore an automorphism which is either the identity or of order $2$. Involutions that leave the center invariant are called involutions of the first kind. Involutions whose restriction to $k$ is an automorphism of order $2$ are called involutions of the second kind, <cit.>. A quaternion algebra over a field $k$ is a central simple $k-$algebra of degree $2$. Any quaternion algebra has a symplectic involution called the quaternion conjugation, <cit.>.  Let $(A_{1},\tau_{1}), \dots, (A_{n},\tau_{n})$ be central simple $k-$algebras with involution of the first kind. Then $\tau_{1}\tensor\cdots \tensor\tau_{n}$ is an involution of the first kind on $A_{1}\tensor_{k}\cdots \tensor_{k} A_{n}$, <cit.>.
Knus–Parimala–Srinivas gave a necessary and sufficient decomposability condition for a central simple algebra with an involution on a tensor product of two quaternion algebras, <cit.>. Merkur'ev showed that every central simple algebra with involution is Brauer equivalent to a tensor product of quaternion algebras, <cit.>. However, Amitsur–Rowen–Tignol provided examples of division algebras with involutions of degree $8$ which do not decompose into tensor products of quaternion algebras, <cit.>.

Knus–Parimala–Srinivas generalized the notion of a central simple algebras with an involution to Azumaya algebras over schemes, <cit.>. Saltman presented a classification of involutions of Azumaya algebras over commutative rings into kinds, <cit.>. In this paper we declare an involution on a topological Azumaya algebra to be an involution of the first kind or an involution of the second kind depending on whether or not there is a group action on the base space, respectively. All the involutions we discuss here are involutions of the first kind over, which are classified as orthogonal and symplectic involutions.

Let $X$ be a connected topological space, and let $\cA$ be a topological Azumaya algebra of degree $n$ over $X$. 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. $\tau(ab)=\tau(b)\tau(a)$ holds on sections.
In this case, $(\cA,\tau)$ is called a topological Azumaya algebra with involution.

Let $X$ be a connected topological space, and let $(\cA,\tau)$ be a topological Azumaya algebra with involution over $X$. The involution $\tau$ is said to be orthogonal (symplectic) if the restriction $\tau|_{\cA^{-1}(x)}:\cA^{-1}(x)\to\cA^{-1}(x)$ is an orthogonal (a symplectic) involution of complex algebras for all $x\in X$, in the sense of the definition given in Subsection <ref>.

The goal of this paper is to provide conditions on a positive integer $n$ and a topological space $X$ such that a topological Azumaya algebra of degree $n$ over $X$ with an orthogonal involution has a tensor product decomposition of topological Azumaya algebras with orthogonal involution. The main result of this paper is the following theorem:

Let $m$ and $n$ be positive integers such that $m$ is even, and $n$ is odd. Let $X$ be a CW complex such that $\dim(X)\leq d$ where $d\coloneqq\min\{m,n\}$.

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

From <cit.> there is a bijective correspondence

\begin{align*}\label{bcTAAwithInvo}
\begin{Bmatrix}
\text{Isomorphism classes of degree-$n$}\\
\text{topological Azumaya algebras}\\
\text{over $X$ with an involution}\\
\text{locally isomorphic to $\tau$}
\end{Bmatrix}
\leftrightarrow
\begin{Bmatrix}
\text{Isomorphism classes of principal}\\
\text{$\Aut\bigl(\fM{n},\tau\bigr)$-bundles}\\
\text{over $X$}
\end{Bmatrix},
\end{align*}

where $\tau$ is an involution of the first kind on $\fM{n}$. Proposition <ref>
implies that topological Azumaya algebras over $X$ with involution are classified by $[X,\B\fPO{n}]$ in the orthogonal case, and $[X,\B\fPSp{n}]$ in the symplectic case, where $\B\fPO{n}$ and $\B\fPSp{n}$ are the classifying spaces of the projective complex orthogonal group of degree $n$ and the projective complex sympletic group of degree $2n$, respectively, see Subsection <ref>.

We prove in Theorem <ref> that a map $X\rightarrow \B\fPO{mn}$ can be lifted to $\B\fPO{m}\times \B\fSO{n}$ when the dimension of $X$ is less than $d+1$. The proof of Theorem <ref> relies in the description of the homomorphisms induced on homotopy groups by the $r$-fold direct sum of matrices $\oplus^{r}:\fO{n}\to\fO{rn}$ in the range $\{0,1,\dots,n-1\}$. We call this set “the stable range for $\fO{n}$”.

This paper is organized as follows. The second section contains preliminaries. The third section presents preliminaries on the effect of direct sum and tensor product operations on homotopy groups of compact Lie groups related to the orthogonal groups . The fourth section is devoted to the proof of Theorem <ref>.

§.§ Acknowledgments

The author would like to express her thankfulness to Ben Williams for proposing this research topic, pointing out relevant references, and having devoted a great deal of time to discuss details of the research with the her. She also gratefully acknowledges Kirsten Wickelgren and Diarmuid Crowley for reading the introduction and providing valuable suggestions that improve its exposition.

§.§ Notation

Throughout this paper, we adopt the notation for the classical Lie groups as in <cit.>. We denote by $\fM{n}$ the set of $n\times n$ complex matrices. We use $\fGL{n}$, $\fO{n}$, $\fSO{n}$, and $\fSp{n}$ to denote the complex general linear group of degree $n$, the complex orthogonal group of degree $n$, the special complex orthogonal group of degree $n$, and the complex symplectic group of degree $2n$, respectively. These groups are closed subgroups of $\fGL{n}$ in the orthogonal cases, and of $\fGL{2n}$ in the symplectic case. In matrix terms these groups are defined as

\begin{align*}
&\fGL{n}=\bigl\{M \in \fM{n}:\; M \text{ is invertible}\bigr\},\\
&\fO{n}=\bigl\{M \in \fM{n}: \; M^{\tr}M=MM^{\tr}=I_{n}\bigr\},\\
&\fSO{n}=\bigl\{M \in \fO{n}: \; \det(M)=1\bigr\},\; \text{ and } \;\\
&\fSp{n}=\bigl\{M \in \fM{2n}: \; M^{\tr}J_{2n} M=J_{2n} \bigr\},
\end{align*}

where $\tr$ denotes transposition, and

\[
\begin{pmatrix}
0 & I_{n}\\
-I_{n} & 0
\end{pmatrix}.
\]

Furthermore, $\fPO{n}$ shall denote the projective complex orthogonal group of degree $n$, and $\fPSp{n}$ shall denote the projective complex sympletic group of degree $2n$.

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 $\pi_{i}(X,x_{0})$.

§ PRELIMINARIES

§.§ Involutions on matrix algebras

Let $A \in \fGL{n}$, we denote by $\Int_{A}:\fM{n}\to\fM{n}$ the inner automorphism induced by $A$, $\Int_{A}(M)=AMA^{-1}$ for all $M\in \fM{n}$.

Observe that transposition, $\tr:\fM{n} \to \fM{n}$, is an involution of the first kind on $\fM{n}$. All automorphisms of $\bigl(\fM{n},\tr\bigr)$ arise by conjugation by an invertible matrix: Let $\tau$ be an arbitrary involution on $\fM{n}$. Since $\tr\circ \,\tau$ is an automorphism of $\bigl(\fM{n},\tr\bigr)$, then $\tr\circ\, \tau=\Int_{A}$ for some $A \in \fGL{n}$. Hence $\tau(M)=A^{-\tr}M^{\tr}A^{\tr}$. Since $\tau$ is an involution, then

\[
\]

Therefore $\Int_{A^{-1}A^{\tr}}=\id_{\fM{n}}$, this is $A^{-1}A^{\tr}=\lambda I_{n}$ for some $\lambda \in \CC$. This last equation implies $A^{\tr}=\lambda A$, and thus $A=\lambda^{2}A$. Hence $\lambda=1$ or $\lambda=-1$, i.e. $A^{\tr}=A$ or $A^{\tr}=-A$.

Consider the subspaces

\begin{align*}
\Bigl(\fM{n},\tau\Bigr)^{+}&=\Bigl\{M\in \fM{n}:\;\;\tau(M)=M\Bigr\} \quad \text{(symmetric elements), and}\\
\Bigl(\fM{n},\tau\Bigr)^{-}&=\Bigl\{M\in \fM{n}:\;\;\tau(M)=-M\Bigr\} \quad \text{(skew symmetric elements).}
\end{align*}

Observe that

\[
\Bigl(\fM{n},\tau\Bigr)^{+}=%
\begin{cases}
A\Bigl(\fM{n},\tr\Bigr)^{+} & \text{if $A^{\tr}=A$},\\
A\Bigl(\fM{n},\tr\Bigr)^{-} & \text{if $A^{\tr}=-A$}.
\end{cases}
\]

where $\dim_{\CC}\Bigl(\fM{n},\tr\Bigr)^{+}=\frac{1}{2}n(n+1)$ and $\dim_{\CC}\Bigl(\fM{n},\tr\Bigr)^{-}=\frac{1}{2}n(n-1)$.

In summary, we obtain the following proposition.

Let $\tau$ be an involution of the first kind on $\fM{n}$.

* The involution $\tau$ on $\fM{n}$ has the form $\tau=\Int_{A}\circ \tr$ for some $A\in \fGL{n}$ such that $A^{\tr}=\pm A$.

* The subspace $\Bigl(\fM{n},\tr\Bigr)^{+}$ has complex dimension $\frac{1}{2}n(n+1)$ if and only if $A^{\tr}=A$.

* The subspace $\Bigl(\fM{n},\tr\Bigr)^{+}$ has complex dimension $\frac{1}{2}n(n-1)$ if and only if $A^{\tr}=-A$.

If $\tau$ is an involution on $\fM{n}$. The type of $\tau$ is said to be orthogonal if $A^{\tr}=A$, and symplectic if $A^{\tr}=-A$. Up to isomorphism, the matrix algebra $\fM{n}$ can carry at most one involution (transposition) if $n$ is odd; and up to two involutions (orthogonal and symplectic) if $n$ is even, <cit.>.

§.§ Non-degenerate bilinear forms

Let a pair $(V,B)$ denote a finite-dimensional complex vector space, $V$, with a non-degenerate bilinear form, $B$. If $\dim_{\CC}V$ is even, $V$ can be given both a skew-symmetric bilinear form and a symmetric bilinear form.  If $\dim_{\CC}V$ is odd, $V$ can be given a symmetric bilinear form. These bilinear forms are unique up to isomorphism, given that $\CC$ is algebraically closed, <cit.>.

The automorphism group of $(V,B)$ is

\[
\Aut(V,B)=\Bigl\{T \in \fGL{n}: \; B(Tv,Tw)=B(v,w) \text{ for all } v,w \in V\Bigr\}.
\]

Let $(\CC^{n}, B)$ and $(\CC^{2n}, B')$ denote the spaces $\CC^{n}$ and $\CC^{2n}$ with the standard symmetric bilinear form and the standard skew-symmetric bilinear form, respectively. Then

\[
\Aut(\CC^{n}, B) \iso \fO{n} \quad \text{ and } \quad \Aut(\CC^{2n}, B') \iso \fSp{n}.
\]

Let $(V,B)$ and $(V,B')$ be finite-dimensional complex vector spaces with non-degenerate bilinear forms. The tensor product $(V, B)\tensor (V',B')$ is is a non-degenerate bilinear form over $\CC$, where $(B \tensor B')(v\tensor v',w\tensor w')=B(v,w)B'(v',w')$ (multiplication in $\CC$), <cit.>.

The tensor product is bifunctorial, so that one obtains induced maps

\begin{equation}\label{tensorAut}
\begin{tikzcd}
\tensor: \Aut(V,B) \times \Aut(V',B') \arrow[r] &\Aut(V \tensor V', B \tensor B').
\end{tikzcd}
\end{equation}

The following tensor product operation is of interest in the subsequent sections. Let $(\CC^{m}, B)$ and $(\CC^{n}, B')$ be the spaces $\CC^{m}$ and $\CC^{n}$ with the standard symmetric bilinear forms.

Then, the tensor product (<ref>) induces a homomorphism

\begin{equation*}
\begin{tikzcd}
\tensor:\fO{m} \times \fO{n} \arrow[r] & \fO{mn}.
\end{tikzcd}
\end{equation*}

The tensor product (<ref>) also induces the following tensor products on Lie groups.

*  Let $(\CC^{2m}, B)$ be the space $\CC^{2m}$ with the standard skew-symmetric bilinear form, and $(\CC^{n}, B')$ be the space $\CC^{n}$ with the standard symmetric bilinear form.

Then, the tensor product (<ref>) induces a homomorphism

\begin{equation*}
\begin{tikzcd}
\tensor:\fSp{m} \times \fO{n}\arrow[r] & \fSp{mn}.
\end{tikzcd}
\end{equation*}

*  Let $(\CC^{2m}, B)$ and $(\CC^{2n}, B')$ be the spaces $\CC^{2m}$ and $\CC^{2n}$ with the standard skew-symmetric bilinear forms.

From the tensor product (<ref>) there is a homomorphism

\begin{equation*}
\begin{tikzcd}
\tensor:\fSp{m} \times \fSp{n} \arrow[r] & G,
\end{tikzcd}
\end{equation*}

where $G\leq \fGL{4mn}$ denotes $\Aut(\CC^{4mn}, B \tensor B')$. Observe that $B\tensor B'$ is not the standard symmetric bilinear form on $\CC^{4mn}$. Let $P \in \fGL{4mn}$  be matrix associated to changing the basis of $\CC^{4mn}$ to an orthonormal basis. Then we have the commutative square below

\begin{equation*}
\begin{tikzcd}[row sep=large]
G \arrow[r,hookrightarrow] \arrow[d,"\Int_{P}"] & \GL(4mn,\CC) \arrow[d,"\Int_{P}"]\\
\Or(4mn,\CC) \arrow[r,hookrightarrow] \arrow[u,leftarrow,"\iso"] & \GL(4mn,\CC). \arrow[u,leftarrow,"\iso"]
\end{tikzcd}
\end{equation*}

Thus, the composite $\Int_{P}\circ \,\tensor:\fSp{m} \times \fSp{n} \to G \to \fO{4mn}$ gives a homomorphism

\begin{equation*}
\begin{tikzcd}
\stensor:\fSp{m} \times \fSp{n}\arrow[r] &  \fO{4mn}.
\end{tikzcd}
\end{equation*}

§ STABILIZATION OF OPERATIONS ON THE COMPLEX ORTHOGONAL GROUP

In what follows, we shall drop $\CC$ from the notations $\fO{n}$ and $\fSO{n}$.

We compute the homotopy groups of $\PO(n)$ and $\PSO(n)$ in low degrees.

When $n=1$, then $\Or(1)=S^{0}$, $\SO(1)=\{1\}$ and $\PO(1)=\PSO(1)=\{1\}$.

When $n=2$, then $\SO(2)=\U_{1}=S^{1}$, $\PO(2)\iso\Or_{2}$ and $\PSO(2)\iso S^{1}$.

Let $n\geq 3$ and $i< n-1$, the first homotopy groups of the orthogonal group are given by Bott periodicity,

\[
\pi_{i}(\Or(n))\iso
\begin{cases}
0 & \text{if $i=2,4,5,6$ (mod 8),}\\
\ZZ/2 & \text{if $i=0,1$ (mod 8),}\\
\ZZ & \text{if $i=3,7$ (mod 8).}
\end{cases}
\]

The special orthogonal group is the connected component of the identity of $\Or(n)$, then $\pi_{0}(\SO(n))$ is trivial. Moreover, for $n\geq 2$ there is a fiber sequence $\SO(n)\hookrightarrow \Or(n) \xrightarrow{\det} \ZZ/2$ where $\ZZ/2$ has the discrete topology, then we can use the long exact sequence associated to it to see that $\pi_{i}(\SO(n))\iso\pi_{i}(\Or(n))$ for $i\geq 1$.

To calculate the homotopy groups of the projective orthogonal group and the projective special orthogonal group in low degrees when $n\geq 3$ consider the diagrams (<ref>) and (<ref>) below for $k\geq 2$.

\begin{equation}\label{cd:Oeven}
\begin{tikzcd}[column sep=large]
\{\pm I_{2k}\} \arrow[equal,r] \arrow[hookrightarrow,d] & \{\pm I_{2k}\}  \arrow[r] \arrow[hookrightarrow,d] & \{1\} \arrow[d] \\
\SO(2k) \arrow[hookrightarrow,r] \arrow[twoheadrightarrow,d] & \Or(2k) \arrow[twoheadrightarrow,r,"\det"] \arrow[twoheadrightarrow,d] & \{\pm 1\} \arrow[equal,d] \\
\PSO(2k) \arrow[r] & \PO(2k) \arrow[r,"\det"] & \{\pm 1\}
\end{tikzcd}
\end{equation}

\begin{equation}\label{cd:Oodd}
\begin{tikzcd}[column sep=large]
\{I_{2k-1}\} \arrow[hookrightarrow,r] \arrow[hookrightarrow,d] & \{\pm I_{2k-1}\}  \arrow[twoheadrightarrow,r] \arrow[hookrightarrow,d] & \{\pm 1\} \arrow[equal,d] \\
\SO(2k-1) \arrow[hookrightarrow,r] \arrow[twoheadrightarrow,d] & \Or(2k-1) \arrow[twoheadrightarrow,r,"\det"] \arrow[twoheadrightarrow,d] & \{\pm 1\} \arrow[d]\\
\PSO(2k-1) \arrow[r] & \PO(2k-1) \arrow[r,"\det"] & \{ 1\}
\end{tikzcd}
\end{equation}

All columns as well as the two top rows of diagrams (<ref>) and (<ref>) are exact. The nine-lemma implies that the bottom rows in (<ref>) and (<ref>) are also exact. Therefore,

\begin{align*}
\pi_{i}(\PO(2k))&\iso \pi_{i}(\PSO(2k)) \iso \pi_{i}(\SO(2k))\quad \text{ for all } i\geq 2, \text{ and }\\
\PO(2k-1)&\iso \PSO(2k-1)\iso \SO(2k-1)
%\pi_{i}(\PO(2k-1))&= \pi_{i}(\PSO(2k-1)) = \pi_{i}(\SO(2k-1)) \quad \text{ for all } i\geq0.
\end{align*}

The only calculation left is the one of the fundamental group of $\PSO(2k)$ for $k\geq2$. Let $n\geq 3$. By definition, the $n$-th spinor group $\Spin(n)$ is the universal double covering group of $\SO(n)$, <cit.>. This group is also the universal covering of $\PSO(n)$, $\Ker(p)\rightarrow \Spin(n) \xrightarrow{p} \PSO(n)$, and

\[
\pi_{1}(\PSO(n))=\Ker(p)=Z(\Spin(n)).
\]

From <cit.> the center of $\Spin(n)$ for $n\geq3$ is given by

\begin{equation}\label{ZSpinn}
\begin{cases}
\ZZ/2 & \text{if $n$ is odd},\\
\ZZ/2\oplus\ZZ/2 & \text{if $n\equiv 0$ (mod 4)},\\
\ZZ/4 & \text{if $n\equiv 2$ (mod 4)}.
\end{cases}
\end{equation}

§.§ First unstable homotopy group of On

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

The first unstable homotopy group of $\Or(n)$ happens in degree $n-1$. Consider the following segment of the long exact sequence

\begin{equation*}\label{segmentles}
\begin{tikzcd}
\pi_{n}(S^{n}) \arrow[r,"\partial"] & \pi_{n-1}(\Or(n)) \arrow[r,"i_{*}"] & \pi_{n-1}(\Or(n+1))  \arrow[r]  & 0.
\end{tikzcd}
\end{equation*}

By exactness of the sequence above, there is a short exact sequence

\begin{equation}\label{ses:piOn}
\begin{tikzcd}
0 \arrow[r] & \Ker i_{*} \arrow[r,hookrightarrow] & \pi_{n-1}(\Or(n)) \arrow[r,"i_{*}"] & \pi_{n-1}(\Or)  \arrow[r]  & 0,
%\pi_{n}(S^{n})/\Ker \partia
\end{tikzcd}
\end{equation}

where $\Or\coloneqq \colim_{n\to \infty} \Or(n)$ and $\pi_{n-1}(\Or)\iso \pi_{n-1}(\Or(n+1))$.

Let $n=3,7$. From <cit.>, $\Ker i_{*}$ is trivial. Thus $\pi_{n-1}(\Or(n))\iso\pi_{n-1}(\Or)$, i.e. $\pi_{n-1}(\Or(n))$ is trivial.

Let $n=2$, then $\pi_{1}(\Or(2))=\pi_{1}(S^{1})=\ZZ$.

Let $n\neq 1, 2, 3, 7$. The short exact sequence (<ref>) splits, <cit.>. Therefore,

\begin{align}\label{pin1On}
\pi_{n-1}(\Or(n)) \iso %
\begin{cases}
\ZZ \oplus \ZZ & \text{if $n\equiv0,4$ (mod 8)}, \\
\ZZ/2 \oplus \ZZ/2& \text{if $n\equiv1$ (mod 8)}, \\
\ZZ \oplus \ZZ/2 & \text{if $n\equiv2$ (mod 8)}, \\
\ZZ/2 & \text{if $n\equiv3,5,7$ (mod 8)},\\
\ZZ & \text{if $n\equiv6$ (mod 8)}.
\end{cases}
\end{align}

Let $G\in \bigl\{\Or(n), \SO(n), \PO(n), \PSO(n)\bigr\}$. Tables <ref>, <ref>, <ref>, and <ref> summarize the previous results.

$G$     $\Or(n)$     $\SO(n)$     $\PO(2k)$     $\PSO(2k)$

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

Connected components of compact Lie groups related to the complex orthogonal group

$G$     $\Or(n)$     $\SO(n)$     $\PO(2k)$     $\PSO(2k)$

$\pi_{1}(G)$     $\ZZ/2$     $\ZZ/2$     $Z(\Spin(2k))$     $Z(\Spin(2k))$

Fundamental group of compact Lie groups related to the complex orthogonal group

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

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

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

$n$ equals to             2     3     4     5     6     7

$\pi_{n-1}(G)$               $\ZZ$     0     $\ZZ\oplus \ZZ$     $\ZZ/2$     $\ZZ$     0

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

$\pi_{n-1}(G)$     $\ZZ \oplus \ZZ$      $\ZZ/2\oplus \ZZ/2$     $\ZZ/2\oplus \ZZ$     $\ZZ/2$     $\ZZ\oplus \ZZ$     $\ZZ/2$     $\ZZ$     $\ZZ/2$

First unstable homotopy group compact Lie groups related to the complex orthogonal group

§.§ Stabilization

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

\begin{equation*}
\s(A)=
\begin{pmatrix}
A & 0\\
0 & I_{n}
\end{pmatrix}.
\end{equation*}

Since the map $\s$ is equal to the composite of consecutive canonical inclusions, it follows that $\s$ is $(m-1)$-connected.

Let $\esta$ denote the homomorphisms the map $\s$ induces on homotopy groups. From now on, we will identify $\pi_{i}(\Or(m))$ with $\pi_{i}(\Or(m+n))$ for all $i<m-1$ through the isomorphism

\begin{equation}
\begin{tikzcd}[row sep=tiny, column sep=normal]
\esta:\pi_{i}(\Or(m)) \arrow[r,rightarrow,"\iso"] & \pi_{i}(\Or(m+n)).
\end{tikzcd}
\end{equation}

The following lemma is a technical result that will be used to prove Lemma <ref> and Lemma <ref>.

Let $G$ be a Lie group and let $G_{0}$ be the component of the identity. If $r:G\rightarrow G$ is conjugation by $P \in G_{0}$, then there is a basepoint preserving homotopy $H$ from $r$  to $\id_{G}$ such that for all $t \in [0,1]$, $H(-,t)$ is a homomorphism.

Since $G$ is path-connected, there exists a path $\alpha$ from $P$ to $I_{m}$ in $G$. Define $H:G\times [0,1] \rightarrow G$ by $H(A,t)=\alpha(t)A\alpha(t)^{-1}$. Observe that $H(-,t):G\to G$, $A\mapsto H(A,t)$ is a homomorphism. Moreover, $H$ is such that  $H(e_{G},t)=e_{G}$, $H(A,0)=r(A)$, and $H(A,1)=A$.

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

\[
\s_{j}(A)=\diag(I_{n},\dots,I_{n},A,I_{n},\dots,I_{n}),
\]

where $A$ is in the $j$-th position. Then the map $\s_{j}$ is pointed homotopic to $\s_{j+1}$ for $j=1,\dots,r-1$.

Let $P_{j}$ be the permutation matrix

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

We consider two cases according to the parity of $n$.

If $n$ even, then $\det(P_{j})=1$. Moreover, $P_{j}$ is such that $\s_{j+1}(A)=P_{j}\s_{j}(A)P_{j}$ for $j=1,\dots,r-1$.

In the case $n$ odd, $\det(P_{j})=-1$. The matrices

\[
\begin{pmatrix}
I_{rn-2} & & \\
& 0 & 1 \\
& 1 & 0
\end{pmatrix}%
\quad \text{and} \quad%
\begin{pmatrix}
0 & 1 &  \\
1 & 0 &  \\
& & I_{rn-2}
\end{pmatrix}
\]

are such that $\det(W_{d}P_{j})=\det(W_{u}P_{j})=1$. Moreover,

\[
\s_{2}(A)=(W_{d}P_{j})\s_{1}(A)(W_{d}P_{j})^{-1},\quad \text{and} \quad \s_{j+1}(A)=(W_{u}P_{j})\s_{j}(A)(W_{u}P_{j})^{-1}
\]

for $j=2,\dots,r-1$. By Lemma <ref> the result follows.

We call the $\s_{j}$ maps stabilization maps. As $\s_{1}$ is equal to $\s:\Or(n) \to \Or(n+(r-1)n)$, it follows that $\s_{j}$ is $(n-1)$-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 $\pi_{i}(\Or(n))$ with $\pi_{i}(\Or(rn))$ for $i<n-1$ through $\esta$.  The identification allows one to introduce a slight abuse of notation, namely to identify $x$ and $\esta(x)$ for $x\in \pi_{i}(\Or(n))$ and $i<n-1$.

§.§ Operations on homotopy groups

We do not write proofs of some results in this section given that they are similar to the proofs of the results in <cit.>.

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

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

If $m<n$ and $i<m-1$, then $\oplus_{*}(x,y)=x+y$ for $x\in \pi_{i}(\Or(m))$ and $y\in\pi_{i}(\Or(n))$.

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

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

If $i<n-1$, then $\oplus_{*}^{r}(x)=rx$ for $x\in \pi_{i}(\Or(n))$.

Let $L, R: \Or(m)\rightarrow \Or(mn)$ be the homomorphisms $L(A)=A\tensor I_{n}$ and $R(A)=I_{n}\tensor A$. There is a basepoint preserving homotopy $H$ from $L$  to $R$ such that for all $t \in [0,1]$, $H(-,t)$ is a homomorphism.

Let $A\in \Or(m)$.

\[
\begin{pmatrix}
a_{11}I_{n}&  \cdots &  a_{1m}I_{n}\\
\vdots&  \ddots &  \vdots\\
a_{m1}I_{n}&  \cdots &  a_{mm}I_{n}
\end{pmatrix}%
\quad \text{and} \quad
\begin{pmatrix}
A&  \cdots &  0\\
\vdots&  \ddots &  \vdots\\
0&  \cdots &  A
\end{pmatrix}=A^{\oplus n}.
\]

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}]
%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_{3},e_{n+3},e_{2n+3},e_{3n+3},\dots, e_{(m-1)n+3},\dots,\\
%		   &e_{n-1},e_{2n-1},e_{3n-1},\dots,e_{mn-1}\\
%		   &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 $L(A)=P_{m,n}R(A)P_{m,n}^{-1}$.

If $\det(P_{m,n})=1$, the result follows from Lemma <ref>. Suppose $\det(P_{m,n})=-1$. Consider $W_{d}$ and $W_{u}$ from the proof of Lemma <ref>.  Observe that $\det(P_{m,n}W_{d})=\det(P_{m,n}W_{u})=\det(W_{d}^{-1}W_{u})=1$, and $R(A)=\s_{1}(A)\s_{2}(A)\cdots \s_{n}(A)$. Then

\begin{align*}
&=P_{m,n}\s_{1}(A)\s_{2}(A)\cdots \s_{n}(A)P_{m,n}^{-1}\\
&=P_{m,n}\Bigl(W_{d} \s_{1}(A) \cdots \s_{n-1}(A) W_{d}^{-1}\Bigr)\Bigl(W_{u}\s_{n}(A)W_{u}^{-1}\Bigr)P_{m,n}^{-1}\\
&=(P_{m,n}W_{d})\s_{1}(A)\cdots\s_{n-1}(A)(P_{m,n}W_{d})^{-1} (P_{m,n}W_{u})\s_{n}(A)(P_{m,n}W_{u})^{-1}.
\end{align*}

Applying Lemma <ref> twice yields the result.

Let $i\in \NN$, the homomorphism $\otimes_{*}:\pi_{i}(\Or(m))\times\pi_{i}(\Or(n)) \to \pi_{i}(\Or(mn))$ is given by

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

If $m<n$ and $i<m-1$, then $\otimes_{*}(x,y)=nx+my$ for $x \in \pi_{i}(\Or(m))$ and $y\in \pi_{i}(\Or(n))$.

Under the hypothesis of Corollary <ref>, the homomorphism

\[
\tensor_{*}:\pi_{m-1}(\Or(m))\times\pi_{m-1}(\Or(n)) \to \pi_{m-1}(\Or(mn))
\]

is given by $\tensor_{*}(x,y)=n\esta(x)+m\esta(y)= n\esta(x)+my$. This can be seen by observing that $\otimes_{*}$ is equal to the sum of the two paths around diagram (<ref>).

\begin{equation}\label{cd:stabtensor}
\begin{tikzcd}[row sep=large,column sep=scriptsize]
&\pi_{m-1}(\Or(m))\times\pi_{m-1}(\Or(n)) \arrow[dr,twoheadrightarrow,"\proj_{2}"] & \\
\pi_{m-1}(\Or(m)) \arrow[ru,twoheadleftarrow,"\proj_{1}"] \arrow[d,twoheadrightarrow,"\esta"] & &\pi_{m-1}(\Or(n)) \arrow[d,rightarrow,"\esta"]\\
\pi_{m-1}(\Or(mn)) & & \pi_{m-1}(\Or(mn)) \arrow[dl,rightarrow,"\times m"] \arrow[u,leftarrow,"\iso"]\\
& \pi_{m-1}(\Or(mn))\times \pi_{m-1}(\Or(mn)) \arrow[ul,leftarrow,"\times n"] \arrow[d,rightarrow,"+"]  &\\
\end{tikzcd}
\end{equation}

Let $i\in \NN$. The homomorphism $\otimes^{r}_{*}:\pi_{i}(\Or(n)) \rightarrow \pi_{i}(\Or(n^{r}))$ is given by

\[
\otimes^{r}_{*}(x)=rn^{r-1}\esta(x) \quad\text{for}\quad x \in \pi_{i}(\Or(n)).
\]

If $i<n-1$, the map $\otimes^{r}_{*}(x)=rn^{r-1}x$ for $x \in \pi_{i}(\Or(n))$.

§.§ Tensor product on the quotient

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

The methods used in <cit.> to establish decomposition of topological Azumaya algebras apply to those whose degrees are relatively prime. For this reason, we study the tensor product

\begin{equation}\label{qO}
\begin{tikzcd}
\otimes:\PO(m)\times \PO(n) \arrow[r] & \PO(mn)
\end{tikzcd}
\end{equation}

in two cases: when $m$ and $n$ are odd, and when $m$ is even and $n$ is odd.

§.§.§ Case $m$ and $n$ odd

Let $m$ and $n$ be positive integers such that $m$ and $n$ are odd. Since $\PO(m)\times\PO(n)=\SO(m)\times\SO(n)$, the tensor product in (<ref>) may be written as

\begin{equation}\label{qOo}
\begin{tikzcd}
\otimes:\SO(m)\times \SO(n) \arrow[r] & \SO(mn),
\end{tikzcd}
\end{equation}

and by Proposition <ref> we have

Let $i\in \NN$. The homomorphism

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

is given by $\otimes_{*}(x,y)=n\esta(x)+m\esta(y)$

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

§.§.§ Case $m$ even and $n$ odd

Let $m$ and $n$ be positive integers such that $m$ is even and $n$ is odd. The tensor product operation $\otimes:\Or(m)\times\SO(n) \rightarrow \Or(mn)$ sends the center of $\Or(m)\times\SO(n)$ to the center of $\Or(mn)$. As a consequence, the operation descends to the quotient

\begin{equation}\label{qOeo}
\begin{tikzcd}
\otimes:\PO(m)\times \SO(n) \arrow[r] & \PO(mn).
\end{tikzcd}
\end{equation}

Let $i>1$. The homomorphism

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

is given by $\otimes_{*}(x,y)=n\esta(x)+m\esta(y)$

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

There is a map of fibrations

\begin{equation}\label{cd:fOeo}
\begin{tikzcd}
Z(\Or(m))\times\{I_{n}\} \arrow[r,hookrightarrow] \arrow[d,"\otimes"] & \Or(m)\times \SO(n) \arrow[r,twoheadrightarrow] \arrow[d,"\otimes"] & \PO(m)\times\SO(n) \arrow[d,"\otimes"]\\
Z(\Or(mn)) \arrow[r,hookrightarrow] & \Or(mn) \arrow[r,twoheadrightarrow] & \PO(mn).
\end{tikzcd}
\end{equation}

From the homomorphism between the long exact sequences associated to the fibrations in diagram (<ref>), we obtain a commutative square

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

for $i>1$. From this diagram and Proposition <ref> we have that for all $i>1$, $\otimes_{*}(x,y)=n\esta(x)+m\esta(y)$ for $x \in \pi_{i}(\PO(m))$ and $y\in \pi_{i}(\SO(n))$.

The homomorphism

\begin{equation*}
\begin{tikzcd}
\otimes_{*}: \pi_{1}(\PO(m))\times\pi_{1}(\SO(n)) \arrow[r] & \pi_{1}(\PO(mn))
\end{tikzcd}
\end{equation*}

is given by the following expressions.

* If $m\equiv0$ (mod $4$) and $n$ is odd, then $\otimes_{*}(x,\beta)=x$ for $x\in \ZZ/2\oplus\ZZ/2$ and $\beta\in \ZZ/2$.

* If $m\equiv2$ (mod $4$) and $n$ is odd, then $\otimes_{*}(\alpha, \beta)=n\alpha$ for $\alpha \in \ZZ/4$ and $\beta \in \ZZ/2$.

From the long exact sequence associated to diagram (<ref>) there is a diagram of exact sequences

\begin{equation}\label{cd:meno}
\begin{tikzcd}
0 \arrow[d] & 0 \arrow[d] \\
\pi_{1}(\Or(m))\times \pi_{1}(\SO(n)) \arrow[d] \arrow[r,"\otimes_{*}"]  & \pi_{1}(\Or(mn)) \arrow[d] \\
\pi_{1}(\PO(m))\times \pi_{1}(\SO(n)) \arrow[d] \arrow[r,"\otimes_{*}"] & \pi_{1}(\PO(mn)) \arrow[d]\\
\pi_{0}Z(\Or(m)) \arrow[d] \arrow[r,equal] & \pi_{0}Z(\Or(mn)) \arrow[d]\\
\pi_{0}(\Or(m)) \arrow[d] \arrow[r,"\otimes_{*}"] & \pi_{0}(\Or(mn)) \arrow[d]\\
\pi_{0}(\PO(m)) \arrow[r,"\otimes_{*}"] & \pi_{0}(\PO(mn)).
\end{tikzcd}
\end{equation}

Since $Z(\Or(m))$ is contained in the connected component of the identity of $\Or(m)$, then the homomorphism $\pi_{0}Z(\Or(m)) \to \pi_{0}(\Or(m))$ is trivial. Hence we have a diagram of short exact sequences

\begin{equation}\label{cd:meno2}
\begin{tikzcd}[column sep=small]
\pi_{1}(\Or(m))\times \pi_{1}(\SO(n)) \arrow[r,hookrightarrow] \arrow[d,"\otimes_{*}"] & \pi_{1}(\PO(m))\times \pi_{1}(\SO(n)) \arrow[r,twoheadrightarrow] \arrow[d,"\otimes_{*}"] & \pi_{0}Z(\Or(m)) \arrow[d,equal]\\
\pi_{1}(\Or(mn)) \arrow[r,hookrightarrow] & \pi_{1}(\PO(mn)) \arrow[r,twoheadrightarrow] & \pi_{0}Z(\Or(mn)).
\end{tikzcd}
\end{equation}

By Corollary <ref>, and the parities of $m$ and $n$, the homomorphism $\tensor_{*}:\pi_{1}(\Or(m))\times \pi_{1}(\SO(n)) \to \pi_{1}(\Or(mn))$ is the projection onto the first coordinate. Diagram (<ref>) becomes

\begin{equation}\label{cd:tppi1}
\begin{tikzcd}
0 \arrow[r] & \ZZ/2\times \ZZ/2 \arrow[r] \arrow[d,"\proj_{1}"] & Z(\Spin(m))\times \ZZ/2 \arrow[r] \arrow[d,"\otimes_{*}"] & \ZZ/2 \arrow[r] \arrow[equal,d] & 0\\
0 \arrow[r] & \ZZ/2 \arrow[r] & Z(\Spin(mn)) \arrow[r] & \ZZ/2 \arrow[r] & 0.
\end{tikzcd}
\end{equation}

Part 1. If $m=4k$, then diagram (<ref>) takes the form

\begin{equation}\label{cd:tppi14k}
\begin{tikzcd}
0 \arrow[r] & \ZZ/2\times \ZZ/2 \arrow[r,"i\times i"] \arrow[d,"\proj_{1}"] & (\ZZ/2\oplus \ZZ/2)\times \ZZ/2 \arrow[r,"\varphi"] \arrow[d,"\otimes_{*}"] \arrow[r,dotted,"s",bend left=35,leftarrow] & \ZZ/2 \arrow[r] \arrow[equal,d] & 0\\
0 \arrow[r] & \ZZ/2 \arrow[r,"i"] & \ZZ/2\oplus \ZZ/2 \arrow[r] & \ZZ/2 \arrow[r] & 0.
\end{tikzcd}
\end{equation}

Let $s: \ZZ/2 \rightarrow (\ZZ/2\oplus \ZZ/2)\times \ZZ/2$ be a section of $\varphi$, this section must satisfy $1\mapsto (0\oplus1,0)$. The composite $\tensor_{*}\circ s$, and $s$ make the short exact sequences in diagram (<ref>) split compatibly. Thus

\begin{equation*}
\begin{tikzcd}
(\ZZ/2\oplus \ZZ/2)\times \ZZ/2 \arrow[r,"\iso"] \arrow[d,"\otimes_{*}"] & \im(i\times i)\oplus \im s \arrow[d,"\proj_{1}\;\oplus\;\id"]\\
\ZZ/2\oplus \ZZ/2 \arrow[r,"\iso"] & \im i\oplus \im(\tensor_{*}\circ s).
\end{tikzcd}
\end{equation*}

Hence $\tensor_{*}(\alpha\oplus\beta,\gamma)=\alpha\oplus\beta$.

Part 2. If $m=4k+2$, then diagram (<ref>) takes the form

\begin{equation}\label{cd:tppi14k2}
\begin{tikzcd}
0 \arrow[r] & \ZZ/2\times \ZZ/2 \arrow[r] \arrow[d,"\proj_{1}"] & \ZZ/4 \times \ZZ/2 \arrow[r] \arrow[d,"\tensor_{*}"] & \ZZ/2 \arrow[r] \arrow[equal,d] & 0\\
0 \arrow[r] & \ZZ/2 \arrow[r] & \ZZ/4 \arrow[r] & \ZZ/2 \arrow[r] & 0.
\end{tikzcd}
\end{equation}

By direct inspection we have $\tensor_{*}(1,0)=n$ and $\tensor_{*}(0,1)=0$. Therefore, $\tensor_{*}(\alpha,\beta)= n\alpha$.

The homomorphism

\begin{equation*}
\begin{tikzcd}
\otimes_{*}: \pi_{0}(\PO(m))\times\pi_{0}(\SO(n)) \arrow[r] & \pi_{0}(\PO(mn))
\end{tikzcd}
\end{equation*}

is a bijection.

§ PROOF OF THEOREM <REF>

We use the following technical lemma to prove Proposition <ref> and Proposition <ref>.

Let $a$ and $b$ be positive integers such that $a$ and $b$ are relatively prime. There exist positive  integers $u$ and $v$ such that $|vb-ua|=1$.

Since $a$ and $b$ are relatively prime they generate the unit ideal in $\ZZ$. That is, the equation $ax+by=1$ has integer solutions $x_{0}, y_{0}$. Given that $ax_{0}+by_{0}=1$ and $a, b$ are positive, then exactly one of $x_{0}, y_{0}$ is negative. By changing the sign of the negative one we obtain $u, v$ both positive integers satisfying $ua-vb=\pm1$.

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

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

Let $m$ and $n$ be positive integers such that $m$ and $n$ are odd, and relatively prime. Let $d$ denote $\min\{m,n\}$. Then there exist positive integers $u$ and $v$ satisfying $|vn-um|=1$, so that there exist a positive integer $N$ and a homomorphism $\Tr: \SO(m)\times \SO(n)\rightarrow \SO(N)$ such that for all $i<d-1$ the homomorphisms induced on homotopy groups

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

are given by $\Tr_{i}(x,y)=ux+vy$

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

Assume without loss of generality $m<n$. Since $m$ and $n$ are relatively prime, there exist positive integers $u$ and $v$ such that $vn-um=\pm1$ by Lemma <ref>. Let $N$ denote $um+vn$, and let $\Tr$ denote the composite

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

Let $i<m-1$. From Corollaries <ref> and <ref> we have that $\Tr_{i}(x,y)=ux+vy$.

Let $m$ and $n$ be positive integers such that $m$ is even, and $n$ is odd. Then there exist positive integers $u$ and $v$ satisfying $\bigl|vn-2um^{2}\bigr|=1$, so that there exist a positive integer $N$ and a homomorphism  $\widetilde{\Tr}: \PO(m)\times \SO(n) \rightarrow \SO(N)$ such that the homomorphisms induced on homotopy groups

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

are given by the following expressions. Let $d$ denote $\min\{m,n\}$.

* If $1<i<d-1$,

\[
\widetilde{\Tr}_{i}(x,y)=2umx+vy \quad\text{for}\quad x \in \pi_{i}(\PO(m)) \quad\text{and}\quad y \in \pi_{i}(\SO(n)).
\]

* If $i=1$, then

\begin{equation*}
\begin{cases}
\widetilde{\Tr}_{1}(\alpha\oplus\beta,\gamma)=z\beta+\gamma & \text{if $m\equiv0$ (mod $4$)},\\
\widetilde{\Tr}_{1}(\delta,\gamma)=z'\delta+\gamma & \text{if $m\equiv2$ (mod $4$)}.
\end{cases}
\end{equation*}

for $z, z', \alpha, \beta, \gamma \in \ZZ/2$ and $\delta \in \ZZ/4$.

Without loss of generality, suppose $m<n$. Since $m$ and $n$ are relatively prime, there exist positive integers $u$ and $v$ such that $vn-2um^{2}=\pm1$ by Lemma <ref>. Let $N$ denote $um^{2}+vn$, and let $\Tr$ denote the composite

\begin{equation*}
\begin{tikzcd}[column sep=large]
\Or(m)\times \SO(n) \arrow[d,"(\tensor^{2}\text{,}\id)"] \\
\SO(m^{2})\times \SO(n) \arrow[d,"(\oplus^{u}\text{,}\oplus^{v})"] \\
\SO(um^{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 $\bigl(I_{m^{2}},I_{n^{2}}\bigr)$ by $(\tensor^{2},\id)$, hence to the identity by the composite $\Tr$ defined above. Hence $\Tr$ factors through $\PO(m)\times \SO(n)$

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

From Proposition <ref>, and Corollaries <ref>, <ref> and <ref> we have that $\Tr_{i}(x,y)=2umx+vy$ for $i<m-1$.

The map of fibrations

\begin{equation*}
\begin{tikzcd}
Z(\Or(m))\times\{I_{n}\} \arrow[r,hookrightarrow] \arrow[d] & \Or(m)\times \SO(n) \arrow[r,twoheadrightarrow] \arrow[d,"\Tr"] & \PO(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}(\Or(m))\times\pi_{i}(\SO(n)) \arrow[r,"\iso"] \arrow[d,"\Tr_{i}"] & \pi_{i}(\PO(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 $\widetilde{\Tr}_{i}(x,y)=\Tr_{i}(x,y)=2umx+vy$ for $1<i<m-1$.

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

\begin{equation*}
\begin{tikzcd}
\pi_{1}(\Or(m))\times\pi_{1}(\SO(n)) \arrow[r,hookrightarrow] \arrow[d,"\Tr_{1}"] & \pi_{1}(\PO(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:TTtilde}
\begin{tikzcd}
\ZZ/2\times \ZZ/2 \arrow[r,hookrightarrow] & Z(\Spin(m))\times \ZZ/2 \arrow[d,"\widetilde{\Tr}_{1}"]\\
& \ZZ/2. \arrow[ul,leftarrow,"\Tr_{1}"]
\end{tikzcd}
\end{equation}

Observe that the equality $vn-2um^{2}=\pm1$ implies $v$ is odd, then $\Tr_{1}(\alpha,\beta)=2um\alpha+v\beta=\beta$, i.e. $\Tr_{1}$ is projection onto the second coordinate.

Subcase i: Suppose $m=4k$ for some $k\in\ZZ$. Diagram (<ref>) takes the form

\begin{equation*}
\begin{tikzcd}
\ZZ/2\times \ZZ/2 \arrow[r,hookrightarrow] & (\ZZ/2\oplus \ZZ/2)\times \ZZ/2 \arrow[d,"\widetilde{\Tr}_{1}"]\\
& \ZZ/2 \arrow[ul,leftarrow,"\Tr_{1}"]
\end{tikzcd}
\end{equation*}

where the horizontal homomorphism is the inclusion $(\alpha,\beta)\mapsto (\alpha\oplus0,\beta)$. Thus $\widetilde{\Tr}_{1}(1\oplus0,0)=0$ and $\widetilde{\Tr}_{1}(\mathbf{0},1)=1$. Let $z$ denote $\widetilde{\Tr}_{1}(0\oplus1,0)$. Hence $\widetilde{\Tr}_{1}(\alpha\oplus\beta,\gamma)=z\beta+\gamma$.

Subcase ii: Suppose $m=4k+2$ for some $k\in\ZZ$. Diagram (<ref>) takes the form

\begin{equation*}
\begin{tikzcd}
\ZZ/2\times \ZZ/2 \arrow[r,hookrightarrow] & \ZZ/4\times \ZZ/2 \arrow[d,"\widetilde{\Tr}_{1}"]\\
& \ZZ/2 \arrow[ul,leftarrow,"\Tr_{1}"]
\end{tikzcd}
\end{equation*}

where the horizontal homomorphism is the inclusion $(\alpha,\beta)\mapsto (\alpha,\beta)$. Thus $\widetilde{\Tr}_{1}(0,1)=1$. Let $z'$ denote $\widetilde{\Tr}_{1}(1,0)$. Hence $\widetilde{\Tr}_{1}(\delta,\gamma)=z'\delta+\gamma$.

Let $J$ denote the map

\begin{equation*}
\begin{tikzcd}[row sep=tiny]
J:\B\PO(m)\times\B\PO(n) \arrow[r,mapsto] & \B\PO(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{Ji}
J_{i}: \pi_{i}(\B\PO(m))\times \pi_{i}(\B\PO(n)) \rightarrow \pi_{i}(\B\PO(mn))\times\pi_{i}(\B\SO(N))
\end{equation}

§.§ A d-connected map where d is the minimum of two odd positive integers

Let $m$ and $n$ be positive integers such that $m$ and $n$ are odd, and relatively prime.. The  map $J$ is $d$-connected where $d$ denotes $\min\{m,n\}$.

Without loss of generality, let $m<n$. Let $i<m$. By Propositions <ref> and <ref>, and Corollary <ref> the homomorphism

\begin{equation*}
J_{i}: \pi_{i}(\B\SO(m))\times \pi_{i}(\B\SO(n)) \rightarrow \pi_{i}(\B\SO(mn))\times\pi_{i}(\B\SO(N))
\end{equation*}

is represented by the matrix

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

Observe that this matrix is invertible for all $i<m$ because by Proposition <ref> its determinant satisfies $nv-um=\pm 1$.

Let $i=m$. Propositions <ref> and <ref>, and Corollary <ref> show that the homomorphism $\Tr_{m-1}:\pi_{m-1}(\SO(m))\times\pi_{m-1}(\SO(n)) \to \pi_{m-1}(\SO(N))$ is given by $\Tr_{m-1}(x,y)=\esta(u\esta(x))+\esta(v\esta(y))=u\esta(x)+vy$, where $vn-um=\pm1$. This can be seen by observing that $\Tr_{m-1}$ is equal to the sum of the two paths around diagram (<ref>).

\begin{equation}\label{cd:stabTodd}
\begin{tikzcd}[row sep=large,column sep=tiny]
&\pi_{m-1}(\SO(m))\times\pi_{m-1}(\SO(n))  \arrow[dr,twoheadrightarrow,"\proj_{2}"] & \\
\pi_{m-1}(\SO(m)) \arrow[ru,twoheadleftarrow,"\proj_{1}"] \arrow[d,twoheadrightarrow,"\esta"] & &\pi_{m-1}(\SO(n)) \arrow[d,rightarrow,"\esta"]\\
\pi_{m-1}(\SO(um)) & & \pi_{m-1}(\SO(vn)) \arrow[dl,rightarrow,"\times v"] \arrow[u,leftarrow,"\iso"]\\
& \pi_{m-1}(\SO(um))\times \pi_{m-1}(\SO(vn)) \arrow[ul,leftarrow,"\times u"] \arrow[d,rightarrow,"\esta\times\esta"]  &\\
&\pi_{m-1}(\SO(N))\times\pi_{m-1}(\SO(N)) \arrow[d,rightarrow,"+"] \arrow[u,leftarrow,"\iso"]& \\
\end{tikzcd}
\end{equation}

Hence $J_{m}$ is given by

\begin{equation*}
\begin{tikzcd}[row sep=tiny]
J_{m}: \pi_{m}(\B\SO(m))\times \pi_{m}(\B\SO(n)) \arrow[r]& \pi_{m}(\B\SO(mn))\times\pi_{m}(\B\Or_{N})\\
(x,y) \arrow[r,mapsto] & \bigl(n\esta_{1}(x)+my,u\esta_{2}(x)+vy\bigr)
\end{tikzcd}
\end{equation*}

\begin{equation*}
\begin{tikzcd}
\esta_{1}:\pi_{m-1}(\SO(m)) \arrow[r,twoheadrightarrow] & \pi_{m-1}(\SO(mn))
\end{tikzcd}
\end{equation*}

\begin{equation*}
\begin{tikzcd}
\esta_{2}:\pi_{m-1}(\SO(m))\arrow[r,twoheadrightarrow] &\pi_{m-1}(\SO(um))
\end{tikzcd}
\end{equation*}

are epimorphisms.

If $m=1$ or $m\equiv 3,5,7$ (mod 8), then $J_{m}$ has trivial target.

Let $m\neq1$ and $m\equiv 1$ (mod 8), then

\[
J_{m}: (\ZZ/2\oplus\ZZ/2)\times \ZZ/2 \to \ZZ/2\times\ZZ/2
\]

is given by

\begin{align*}
J_{m}(x\oplus y,z)&=\bigl(\esta_{1}(x\oplus y)+z,u\esta_{2}(x\oplus y)+vz\bigr)
\end{align*}

Observe that $J_{m}$ factors as

\begin{equation*}
\begin{tikzcd}[column sep=huge]
(\ZZ/2\oplus\ZZ/2)\times \ZZ/2 \arrow[r,"(\esta\text{,}\id)"] & \ZZ/2\times\ZZ/2 \arrow[r,"\begin{pmatrix} n \quad m \\ u \quad v \end{pmatrix}"] & \ZZ/2\times\ZZ/2.
\end{tikzcd}
\end{equation*}

Hence $J_{m}$ is an epimorphism.

§.§ A d-connected map where d is the minimum of two positive integers of opposite parity

Let $m$ and $n$ be positive integers such that $m$ is even, and $n$ is odd. Let $d$ denote $\min\{m,n\}$. The homomorphism $J_{i}$ is an isomorphism for all $i<d$.

Without loss of generality, suppose $m<n$.

\begin{equation}\label{Jeo}
J_{i}: \pi_{i}(\B\PO(m))\times \pi_{i}(\B\SO(n)) \rightarrow \pi_{i}(\B\PO(mn))\times\pi_{i}(\B\SO(N))
\end{equation}

Let $i<m$. Given that the homotopy groups of the spaces involved are zero in degrees $i\equiv 3, 5, 6, 7$ (mod 8), it suffices to prove that $J_{i}$ is an isomorphism for $i=1,2$, and for $i\equiv 0,1,2,4$ (mod 8) with $i>2$.

Let $i=1$. By Proposition <ref> the homomorphism $J_{1}: \ZZ/2 \rightarrow \ZZ/2$ is the identity.

Let $i=2$. The homomorphism (<ref>) takes the form

\begin{equation*}
\begin{tikzcd}
J_{2}: Z(\Spin(m)) \times \ZZ/2 \arrow[r] & Z(\Spin(mn)) \times \ZZ/2.
\end{tikzcd}
\end{equation*}

Case i: Let $m=4k$ for some $k\in \ZZ$. From Propositions <ref> and <ref>, $J_{2}$ is given by

\begin{equation*}
\begin{tikzcd}[row sep=tiny]
J_{2}: (\ZZ/2\oplus \ZZ/2)\times \ZZ/2 \arrow[r] & (\ZZ/2\oplus \ZZ/2) \times \ZZ/2\\
(\alpha\oplus\beta,\gamma) \arrow[r,mapsto] & \Bigl(\alpha\oplus\beta,z\beta+\gamma\Bigr),
\end{tikzcd}
\end{equation*}

i.e. $J_{2}$ is represented by the invertible matrix

\[
\begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0 \\
0 & z & 1
\end{pmatrix} \quad \text{where} \quad z\in \ZZ/2.
\]

Hence $J_{2}$ is an isomorphism.

Case ii: Let $m=4k+2$ for some $k\in \ZZ$. From Proposition  <ref>, $f_{\tensor}(\alpha,\beta)=n\alpha$. Since $vn-2um^{2}=\pm 1$, then $f_{\tensor}(\alpha,\beta)=\alpha$. By Proposition <ref>, $J_{2}$ is given by

\begin{equation*}
\begin{tikzcd}[row sep=tiny]
J_{2}: \ZZ/4\times \ZZ/2 \arrow[r] & \ZZ/4 \times \ZZ/2\\
(\alpha,\beta) \arrow[r,mapsto] & \Bigl(\alpha,z'\alpha+\beta\Bigr),
\end{tikzcd}
\end{equation*}

i.e. $J_{2}$ is represented by the invertible matrix

\[
\begin{pmatrix}
1 & 0 \\
z' & 1
\end{pmatrix} \quad \text{where} \quad z'\in \ZZ/2.
\]

Hence $J_{2}$ is an isomorphism.

Let $i>2$. The homomorphism (<ref>) takes the form $J_{i}: \ZZ\times\ZZ \rightarrow \ZZ\times\ZZ$ if $i\equiv0,4$ (mod 8), and $J_{i}: \ZZ/2\times \ZZ/2 \rightarrow \ZZ/2\times \ZZ/2$ if $i\equiv1,2$ (mod 8).

Note that both homomorphisms are represented by the matrix

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

for $i<m$. By Proposition <ref>, the determinant of this matrix is $vn-2um^{2}=\pm 1$. This proves $J_{i}$ is an isomorphism.

\begin{equation}\label{cd:stabTevenodd}
\begin{tikzcd}[row sep=large,column sep=tiny]
&\pi_{m-1}(\Or(m))\times\pi_{m-1}(\SO(n))  \arrow[dr,rightarrow,"\proj_{2}"] & \\
\pi_{m-1}(\Or(m)) \arrow[ru,leftarrow,"\proj_{1}"] \arrow[d,twoheadrightarrow,"\esta"] & &\pi_{m-1}(\SO(n)) \arrow[dd,rightarrow,equal]\\
\pi_{m-1}(\Or(m^{2})) \arrow[d,rightarrow,"\times (2m)"] & &  \\
\pi_{m-1}(\Or(m^{2})) \arrow[d,rightarrow,"\esta"] & & \pi_{m-1}(\SO(n)) \arrow[d,rightarrow,"\esta"] \\
\pi_{m-1}(\Or(um^{2})) \arrow[u,leftarrow,"\iso"] & &  \pi_{m-1}(\SO(vn)) \arrow[dl,rightarrow,"\times v"] \arrow[u,leftarrow,"\iso"] \\
& \pi_{m-1}(\Or(um^{2}))\times\pi_{m-1}(\SO(vn)) \arrow[d,rightarrow,"\esta\times \esta"] \arrow[ul,leftarrow,"\times u"] & \\
&\pi_{m-1}(\SO(N))\times\pi_{m-1}(\SO(N)) \arrow[d,rightarrow,"+"] \arrow[u,leftarrow,"\iso"]& \\
\end{tikzcd}
\end{equation}

Let $m$ and $n$ be positive integers such that $m$ is even, and $n$ is odd. Let $d$ denote $\min\{m,n\}$. The induced homomorphism

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

is an epimorphism.

Suppose $m<n$. Propositions <ref> and <ref>, and Corollaries <ref> and <ref> show that $\Tr_{m}:\pi_{m}(\Or(m))\times\pi_{m}(\SO(n)) \to \pi_{m}(\SO(N))$ is given by

\[
\Tr_{m-1}(x,y)=\esta(u\esta(2m\esta(x)))+\esta(v\esta(y))=2um\esta(x)+vy,
\]

where $vn-2um^{2}=\pm1$. This can be seen by observing that $\Tr_{m-1}$ is equal to the sum of the two paths around diagram (<ref>).

Hence (<ref>) is given by

\begin{equation*}
\begin{tikzcd}[row sep=tiny]
J_{m}: \pi_{m}(\B\PO(m))\times \pi_{m}(\B\SO(n)) \arrow[r]& \pi_{m}(\B\SO(mn))\times\pi_{m}(\B\Or_{N})\\
(x,y) \arrow[r,mapsto] & \bigl(n\esta_{1}(x)+my,2um\esta_{2}(x)+vy\bigr)
\end{tikzcd}
\end{equation*}

\begin{equation*}
\begin{tikzcd}
\esta_{1}:\pi_{m-1}(\Or(m)) \arrow[r,twoheadrightarrow] & \pi_{m-1}(\Or(mn))
\end{tikzcd}
\end{equation*}

\begin{equation*}
\begin{tikzcd}
\esta_{2}:\pi_{m-1}(\Or(m))\arrow[r,twoheadrightarrow] &\pi_{m-1}(\Or(m^{2}))
\end{tikzcd}
\end{equation*}

are epimorphisms.

Observe that for $m=1$ or $m\equiv 3, 5, 6, 7$ (mod 8), the homomorphism $J_{m}$ has trivial target.

Let $m=2$. Then $J_{2}: \ZZ\times \ZZ/2 \to \ZZ/4\times\ZZ/2$ is given by

\[
\]

Thus $J_{2}$ is an epimorphism.

Let $i>2$ and $m\equiv 0,4$ (mod 8). Then $J_{m}: (\ZZ\oplus \ZZ)\times \ZZ \to \ZZ\times\ZZ$  is given by

\[
J_{m}(x\oplus y,z)=\bigl(n\esta_{1}(x\oplus y)+mz,2um\esta_{2}(x\oplus y)+vz\bigr).
\]

Note that $J_{m}$ factors as

\begin{equation*}
\begin{tikzcd}[column sep=huge]
(\ZZ\oplus\ZZ)\times \ZZ \arrow[r,"(\esta\text{,}\id)"] & \ZZ\times\ZZ \arrow[r,"\begin{pmatrix} n \;\;\;\; 2um \\ m \quad\;\;\; v \end{pmatrix}"] & \ZZ\times\ZZ.
\end{tikzcd}
\end{equation*}

Hence $J_{m}$ is an epimorphism.

Let $i>2$ and $m\equiv 2$ (mod 8). Then $J_{m}: (\ZZ/2\oplus \ZZ)\times \ZZ/2 \to \ZZ/2\times\ZZ/2$  is given by

\[
J_{m}(x\oplus y,z)=\bigl(\esta_{1}(x\oplus y),z\bigr).
\]

Then $J_{m}$ is an epimorphism.

If we suppose $n<m$. In the same manner it can be proved that

\begin{equation*}
\begin{tikzcd}[row sep=tiny]
J_{n}: \pi_{n}(\B\PO(m))\times \pi_{n}(\B\SO(n)) \arrow[r]& \pi_{n}(\B\SO(mn))\times\pi_{n}(\B\Or_{N})\\
(x,y) \arrow[r,mapsto] & \bigl(nx+m\esta_{1}(y),2umx+v\esta_{2}(y)\bigr)
\end{tikzcd}
\end{equation*}

is an epimorphism, where

\begin{equation*}
\begin{tikzcd}
\esta_{1}:\pi_{n-1}(\SO(n)) \arrow[r,twoheadrightarrow] & \pi_{n-1}(\SO(mn))
\end{tikzcd}
\end{equation*}

\begin{equation*}
\begin{tikzcd}
\esta_{2}:\pi_{n-1}(\SO(n))\arrow[r,twoheadrightarrow] &\pi_{n-1}(\SO(vn)).
\end{tikzcd}
\end{equation*}

From Propositions <ref>, and <ref> we obtain Corollary <ref>.

Let $m$ and $n$ be positive integers such that $m$ is even, and $n$ is odd. Let $d$ denote $\min\{m,n\}$. The map $J$ is $d$-connected.

§.§ Factorization through f:BPOm x BSOn–>BPOmn

Diagrammatically speaking, we want to find a map

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

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

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

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

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

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

\begin{equation*}
\begin{tikzcd}
J_{\#}:\bigl[X, \B\PO(m)\times\B\SO(n)\bigr] \arrow[r] & \bigl[X,\B\PO(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 $\cA_{m}\times\cA_{n}:X \rightarrow \B\PO(m)\times\B\SO(n)$ such that $J\circ (\cA_{m}\times\cA_{n})\simeq s\circ \cA$.

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

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