Stably Equivalent and Isomorphism Class of Real Line Bundles

Question

Let us denote by $\text{Vect}_{1}(X)$ the set of isomorphism classes of real line bundles of a compact nice manifold $X$. It is well known that the reduced $K$-group $\tilde{K}\mathcal{O}(X)$ can be identified with the set of stable equivalence classes in $\text{Vect}(X)$, subsequently denoted by $\text{Vect}(X)/{\sim_{\text{st}}}$, and that the first Stiefel-Whitney class on line bundles $w_{1}:\text{Vect}_{1}(X)\to H^{1}(X,\mathbb{Z}_{2})$ defines an isomorphism. My question is that in view of this, if the natural projection $$\text{Vect}_{1}(X)\twoheadrightarrow\text{Vect}_{1}(X)/{\sim_{\text{st}}},$$ is a bijection. If no, there must exit two non isomorphic line bundles $E\nsim F$ such that $E\sim_{\text{st}} F$. By the properties of $w_{1}$, we obtain $w_{1}(E)=w_{1}(F)$ which contradicts that the map $w_{1}:\text{Vect}_{1}(X)\to H^{1}(X,\mathbb{Z}_{2})$ is a bijection. Is this argumentation correct? It seems strange to me that the isomorphism classes of line bundles coincide with the stably equivalent ones.

Answer 1

Your argument is correct. That is, two real line bundles are stably isomorphic if and only if they are isomorphic. As you've noted, this boils down to the fact that real line bundles are determined up to isomorphism by a single stable characteristic class, the first Stiefel-Whitney class $w_1$. Analogously, two complex line bundles are stably isomorphic if and only if they are isomorphic, again because complex line bundles are determined up to isomorphism by a single stable characteristic class, the first Chern class $c_1$.