As it is well known, every oriented vector bundle $E$ of rank $2k+1$ over a manifold $M$ of dimension $2k+1$ has a nowhere vanishing section. Thus we may decompose $E=F\oplus\varepsilon^1$.
Is $F$ uniquely determined (up to isomorphism)? This question can be rephrased to homotopy theory: if $M \to BSO(2k+1)$ is the classifying map of $E$, how many lifts of this map to $M \to BSO(2k)$ exists?
Best Answer
Not necessarily. For example, consider the space $S^2\times S^1$ and the vector bundle $T(S^2\times S^1)$. We have $T(S^2\times S^1) \cong \pi_1^*TS^2\oplus\pi_2^*TS^1 \cong \pi_1^*TS^2\oplus\varepsilon^1$ where $\pi_i$ are the natural projections. On the other hand, we have $T(S^2\times S^1) \cong \varepsilon^3 \cong \varepsilon^2\oplus\varepsilon^1$. Note that $\pi_1^*TS^2 \not\cong \varepsilon^2$ as the former has non-zero Euler class.