It is not difficult to see that there are non-trivial stably trivial bundles of rank $2$ for a closed surface $\Sigma$ of genus $\neq 1$ using that $T \Sigma$ is non-trivial, but what happens in the genus $1$ case? Are there stably trivial real vector bundle over $T^2$ which are non-trivial?
In case the answer is affirmative, I'm actually interested in the following situation. Suppose we have an orientable real vector bundle $E \rightarrow T^2$ of rank $2$ such that $E \oplus \epsilon^2$ is trivial, where $\epsilon^2 \rightarrow T^2$ is the trivial real vector bundle of rank $2$. Can we conclude in this case that $E$ is also trivial?
Thank you in advance.
Best Answer
I solved the question, I'll leave the answer for reference.
The answer is yes for the first question and no for the second. Consider the tangent bundle of $T^2 \times S^2$, which is trivial. $T^2 \times S^2$ contains tori of nonzero self-intersection number. Pick one such torus, say $Y$, and let $E \rightarrow Y$ be the restriction of the tangent bundle of $T^2 \times S^2$ to $Y$. This is a rank $4$ trivial vector bundle over a torus, and it decomposes as $E = TY \oplus NY$, where $TY$ is the tangent bundle and $NY$ the normal bundle. Since a torus is parallelizable, $TY$ is trivial, so we are in the situation of the second question above. However, $NY$ is not trivial, since its Euler class coincides with its self-intersection number, which is nonzero.