A matrix $A\in M\_{4}(\mathbb{C})$ is called a simple tensor if $A=B\otimes C$ for two $2\times 2$ matrices $B,C$.


Assume that $X$ is a Hausdorff topological space.Assume that $f:X\to M\_{4}(\mathbb{C})$ is a continuous map such that $f(x)$ is a simple tensor, for every $x\in X$.



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> Are there continous maps $g,h:X\to M\_{2}(\mathbb{C})$ with $f=g\otimes h$?
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Is a pointwise " simple tensor" valued continuous map a tensor product of two continuous maps?
The answer is 'no'. For example, let
$$
X = \{\ B\otimes C\ | \det(B)\det(C) = 1\ \}\subset \mathrm{SL}(4,\mathbb{C}).
$$
Then $X$ is isomorphic to the group $\mathrm{SO}(4,\mathbb{C})$ and hence $\pi\_1(X)\simeq \mathbb{Z}\_2$, while the set
$$
G = \{\ (B, C)\ |\ \det(B)\det(C) = 1\ \}\subset 
\mathrm{GL}(2,\mathbb{C})\times\mathrm{GL}(2,\mathbb{C})
$$
is a connected Lie group that satisfies $\pi\_1(G)\simeq \mathbb{Z}$, with generator given by the subgroup
$$
\{\ (B, B^{-1})\ |\ B\in T\ \}
$$
where $T\subset\mathbb{GL}(2,\mathbb{C})$ is a circle subgroup that generates $\pi\_1\bigl(\mathbb{GL}(2,\mathbb{C})\bigr)\simeq \mathbb{Z}$.


Now, the map $\tau:\mathrm{GL}(2,\mathbb{C})\times\mathrm{GL}(2,\mathbb{C})\to \mathrm{GL}(4,\mathbb{C})$ given by $\tau(B,C) = B\otimes C$ is a group homomorphism, and the preimage under $\tau$ of the space $X$ is the two-component subgroup
$$
G^+ = \{\ (B, C)\ |\ \det(B)\det(C) = \pm1\ \}\subset 
\mathrm{GL}(2,\mathbb{C})\times\mathrm{GL}(2,\mathbb{C}).
$$
(This is because of the identity $\det(B\otimes C) = \det(B)^2\det(C)^2$.)
Note that $\pi\_1(G^+) = \pi\_1(G) = \mathbb{Z}\_2$.


The map $\tau:G\to X$ given by $\tau(B,C) = B\otimes C$ is a group homomorphism. If there were a continuous map $\sigma = (g,h):X\to G^+$ that satisfied $\tau\circ\sigma = \mathrm{id}\_X$, this would induce a homomorphism $\sigma\_\*:\pi\_1(X)\to\pi\_1(G^+)$ that satisfied
$$
\tau\_\*\circ\sigma\_\* = \mathrm{id}:\pi\_1(X)\to\pi\_1(X)\simeq\mathbb{Z}\_2.
$$
However, then $\sigma\_\*$ would be a nontrival homomorphism $\mathbb{Z}\_2\to\mathbb{Z}$, and this does not exist.


