Suppose $X$ is a compact connected Hausdorff space, that $E\to X$ is a rank $n$ topological complex vector bundle over $X$ and, furthermore, that $E\subset \Theta^{2n}(X)$ is a subbundle of the trivial bundle of rank $2n$. Let $E^{\perp}\to X$ be the orthogonal complement vector bundle with respect to the natural inner product in the trivial bundle, so that $E\oplus E^\perp=\Theta^{2n}(X)$. Denote by $H\to \mathbb{C}P^1$ the tautological bundle over $\mathbb{C}P^1$, and by $p_1:X\times\mathbb{C}P^1\to X$ and $p_2:X\times\mathbb{C}P^1$ the canonical projections.
Now consider the vector bundle over $X\times \mathbb{C}P^1$ given by $F=p_1^*E\otimes p_2^*H\oplus p_1^*E^{\perp}\otimes p_2^*\overline{H}$. I believe this vector bundle is, a priori without further asumptions on $E$, non-trivial, however, with the following argument, it seems to be trivial. I would be very happy if someone found the flaw(s) in the proof.
Thanks in advance!
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Proof: Let $\{U_{\alpha}\}_{\alpha=1}^{N}$ be a good open cover of $X$ together with trivializing frame fields $\{s_{\alpha}\}_{\alpha=1}^{N}$ for $E$ and $\{t_{\alpha}\}_{\alpha=1}^{N}$ for $E^{\perp}$ (each $s_{\alpha}=[s_{\alpha,1},...,s_{\alpha,n}]$ is seen as a row vector of linearly independent sections rendering $E|_{U_{\alpha}}\cong U_\alpha\times \mathbb{C}^n$ and similarly for $E^\perp$). The transition functions $\{g_{\alpha\beta}\}$ for $E$ are given by the relations $s_{\beta}=s_{\alpha}\cdot g_{\alpha\beta}$, where $g_{\alpha\beta}:U_{\alpha}\cap U_{\beta}\to \mbox{GL}(n;\mathbb{C})$ and $\cdot$ denotes matrix multiplication, defined over the intersections. Similarly, let $\{g_{\alpha\beta}^{\perp}:U_{\alpha}\cap U_{\beta}\to \mbox{GL}(n;\mathbb{C})\}$ denote the transition functions for $E^\perp$ with respect to the vector bundle atlas defined by the $t_\alpha$'s.
If we view $\mathbb{C}P^1\cong S^2\cong\mathbb{C}\cup\{\infty\}$, we can trivialize $H$ over disks $D_{+}$ and $D_{-}$, with the transition function on the equator being the degree $1$ map $g_{-+}:S^1\ni z\mapsto z\in S^1$. Let $\{e_{+},e_{-}\}$ denote the associated trivializing sections.
With this in mind, we can form an open covering $\{U_{\alpha,+},U_{\alpha,-}\}_{\alpha=1}^{N}$ of $X\times\mathbb{C}P^1$, with $U_{\alpha,\pm}=X\times D_{\pm}$, together with trivializing frame fields $\{S_{\alpha+},S_{\alpha-}\}_{\alpha=1}^{N}$ having the form $S_{\alpha,\pm}=[p_1^{*}s_{\alpha}\otimes p_2^{*}e_{\pm},p_1^{*}t_{\alpha}\otimes p_2^{*}\overline{e}_{\pm}]$, such that the transition maps for the bundle $F$ have the form:
(i) over $U_{\alpha,-}\cap U_{\beta,+}$ the transition function $g_{\alpha-,\beta+}=\text{diag} (zg_{\alpha\beta}, \bar{z}g^{\perp}_{\alpha\beta})$;
(ii) over $U_{\alpha,\pm}\cap U_{\beta,\pm}$, the transition function $g_{\alpha\pm,\beta\pm}=\text{diag} (g_{\alpha\beta}, g^{\perp}_{\alpha\beta})$.
Now the transition function defined in (i) is homotopic, by path connectedness of $\mbox{GL}(2n;\mathbb{C})$, to the transition function $g_{\alpha-,\beta+}=\text{diag} (zg_{\alpha\beta} \cdot\bar{z}g^{\perp}_{\alpha\beta},I_n)$, which in turn is homotopic to $g_{\alpha-,\beta+}=\text{diag} (g_{\alpha\beta}\cdot g^{\perp}_{\alpha\beta},I_n)$, and, finally, homotopic to $g_{\alpha-,\beta+}=\text{diag} (g_{\alpha\beta}, g^{\perp}_{\alpha\beta})$.
This means that the transition functions obtained are homotopic to those one would obtain for $p_{1}^*(E\oplus E^\perp)$. As a consequence, we can build a vector bundle over $[0,1]\times (X\times\mathbb{C}P^1)$, such that the restriction to $\{0\}\times (X\times\mathbb{C}P^1)$ is isomorphic to $F$ and the restriction to $\{1\}\times (X\times\mathbb{C}P^1)$ is isomorphic to $p_1^{*}(E\oplus E^\perp)=p_1^*(\Theta^{2n}(X))=\Theta^{2n}(X\times\mathbb{C}P^1)$. By homotopy invariance of vector bundles, $F\cong \Theta^{2n}(X\times\mathbb{C}P^1)$.
After a discussion with a colleague, we realized that the problem is in the homotopy of transition functions.
First of all, we can take the $z,\bar{z}$ appearing in the transition maps to take values in $S^1$, namely replace them by $z/|z|$ and $\bar{z}/|z|$, by taking an Hermitian inner product in the tautological bundle $H$ (just take the one from $H\subset \Theta^2(S^2)$).
The path I had in mind to get rid of $z$ and $\bar{z}$ appearing in the transition functions was the concatenation of two paths. The first is the following
$g_{\alpha-,\beta+}(t)=\left[\begin{array}{cc} \frac{z}{|z|}g_{\alpha\beta} & 0\\ 0 & I_n \end{array} \right] \left[\begin{array}{cc}\cos\left(\frac{\pi (1-t)}{2}\right)I_n & -\sin\left(\frac{\pi (1-t)}{2}\right)I_n\\ \sin\left(\frac{\pi (1-t)}{2}\right)I_n & \cos\left(\frac{\pi (1-t)}{2}\right)I_n \end{array}\right]\left[\begin{array}{cc} \frac{\bar{z}}{|z|}g_{\alpha\beta}^\perp & 0\\ 0 & I_n \end{array} \right]\left[\begin{array}{cc}\cos\left(\frac{\pi (1-t)}{2}\right)I_n & \sin\left(\frac{\pi (1-t)}{2}\right)I_n\\ -\sin\left(\frac{\pi (1-t)}{2}\right)I_n & \cos\left(\frac{\pi (1-t)}{2}\right)I_n \end{array}\right]$,
which performs an homotopy between $\text{diag}(\frac{z}{|z|}g_{\alpha\beta} ,\frac{\bar{z}}{|z|}g_{\alpha\beta}^\perp)$ ($t=0$) and $\text{diag}(\frac{z}{|z|}g_{\alpha\beta}\cdot \frac{\bar{z}}{|z|}g_{\alpha\beta}^\perp,I_n)=\text{diag}(g_{\alpha\beta}\cdot g_{\alpha\beta}^\perp,I_n)$ ($t=1$). Afterwards, concatenate with the path
$g_{\alpha-,\beta+}(t)=\left[\begin{array}{cc} g_{\alpha\beta} & 0\\ 0 & I_n \end{array} \right] \left[\begin{array}{cc}\cos\left(\frac{\pi t}{2}\right)I_n & -\sin\left(\frac{\pi t}{2}\right)I_n\\ \sin\left(\frac{\pi t}{2}\right)I_n & \cos\left(\frac{\pi t}{2}\right)I_n \end{array}\right]\left[\begin{array}{cc} g_{\alpha\beta}^\perp & 0\\ 0 & I_n \end{array} \right]\left[\begin{array}{cc}\cos\left(\frac{\pi t}{2}\right)I_n & \sin\left(\frac{\pi t}{2}\right)I_n\\ -\sin\left(\frac{\pi t}{2}\right)I_n & \cos\left(\frac{\pi t}{2}\right)I_n \end{array}\right]$
performs an homotopy between $\text{diag}(g_{\alpha\beta}\cdot g_{\alpha\beta}^\perp,I_n)$ ($t=0$) and $\text{diag}(g_{\alpha\beta} , g_{\alpha\beta}^\perp)$ ($t=1$). In the whole process we keep the other transition functions $g_{\alpha\pm,\beta,\pm}$ constant.
The problem is that the resulting collection of functions from the overlaps in the open cover $\{U_{\alpha+}, U_{\alpha -}\}_{\alpha=1}^{N}$ does not give rise, in general, to a $1$-parameter family of cocycles because the cocycle condition fails. Hence, we cannot appeal to the homotopy invariance of topological vector bundles -- there is no path of vector bundles. To see this, consider, for example, the first path. Let $\alpha,\beta,\gamma\in \{1,...,N\}$, then
$g_{\alpha-,\beta +}(1)g_{\beta +,\gamma +}(1)=\text{diag}(g_{\alpha\beta}g_{\alpha\beta}^\perp g_{\beta\gamma}, g_{\beta\gamma}^\perp)\neq \text{diag}(g_{\alpha,\gamma}g_{\alpha,\gamma}^\perp,I_n)=g_{\alpha-,\gamma+}(1).$
There is, however, a trivial situation in which this construction can work. It is when there is only one open set in the cover of $X$, i.e., $E$ and $E^\perp$ are trivial. In that case we recover the usual clutching construction on the sphere and the cocycle is well defined.
Another possible situation where this works is if we the bundle $E^\perp$ is replaced by a trivial bundle $\Theta^n(X)$. In that case, the result we are looking at is:
$p_1^*E\otimes p_2^*H\oplus p_1^*\Theta^n(X)\otimes p_2^*\overline{H}\cong p_1^*(E\oplus \Theta^n(X)).$
To get this result, however, the path of transition functions we wrote above is not enough. Again, this is because we do not get a valid cocycle. To have a cocycle, we take (note that now $g_{\alpha\beta}^\perp$ gets replaced by $I_n$)
$g_{\alpha-,\beta+}(t)=\left[\begin{array}{cc} \frac{z}{|z|}g_{\alpha\beta} & 0\\ 0 & I_n \end{array} \right] R_t\left[\begin{array}{cc} \frac{\bar{z}}{|z|}I_n & 0\\ 0 & I_n \end{array} \right]R_{-t}$,
$g_{\alpha-,\beta-}(t)=\text{diag}(g_{\alpha\beta}, I_n)$,
$g_{\alpha+,\beta+}(t)=R_t\left[\begin{array}{cc} \frac{z}{|z|}I_n & 0\\ 0 & I_n \end{array} \right]R_{-t}\left[\begin{array}{cc} g_{\alpha\beta} & 0\\ 0 & I_n \end{array}\right]R_t\left[\begin{array}{cc} \frac{\bar{z}}{|z|}I_n & 0\\ 0 & I_n \end{array} \right]R_{-t},$
where $R_t=\left[\begin{array}{cc}\cos\left(\frac{\pi (1-t)}{2}\right)I_n & -\sin\left(\frac{\pi (1-t)}{2}\right)I_n\\ \sin\left(\frac{\pi (1-t)}{2}\right)I_n & \cos\left(\frac{\pi (1-t)}{2}\right)I_n \end{array}\right]$ and $t\in[0,1]$. Then one can check that, indeed, for $\alpha,\beta,\gamma\in \{1,...,N\}$, we have the cocycle conditions, defined over the appropriate triple overlaps,
$g_{\alpha+,\beta+}g_{\beta+,\gamma+}=g_{\alpha+,\gamma+},$
$g_{\alpha-,\beta-}g_{\beta-,\gamma-}=g_{\alpha-,\gamma-},$
$g_{\alpha-,\beta+}g_{\beta+,\gamma+}=g_{\alpha-,\gamma+},$
$g_{\alpha-,\beta-}g_{\beta-,\gamma+}=g_{\alpha+,\gamma+},$
needed to define a vector bundle over $[0,1]\times (X\times S^2)$. Observe that the resulting bundle restricted to $\{0\}\times X\times S^2$ is $p_1^*E\otimes p_2^*H\oplus p_1^*\Theta^n(X)\otimes p_2^*\overline{H}$ and the one restricted to $\{1\}\times X\times S^2$ is $p_1^*(E\oplus \Theta^n(X))$. By homotopy invariance of topological vector bundles, the result follows.