A note on stable complex structures on real vector bundles over manifolds (Q2344322)

Let \(M\) be an \(n\)-dimensional closed oriented smooth manifold with \(n\equiv 0 \mod 8\) and let \(\xi\) be a real vector bundle over \(M\) admitting a stable complex structure over the \((n-1)\)-skeleton \(M^{n-1}\) of \(M\). That means that there exists a complex vector bundle \(\eta\) over \(M^{n-1}\) such that \(r(\tilde{\eta})=i^*(\tilde{\xi})\) where \(\tilde{x}\) denotes the stable class of \(x\), \(r\) the realification homomorphism and \(i\) the inclusion of \(M^{n-1}\). In this paper the necessary and sufficient condition for the existence of a stable complex structure on \(\xi\) is formulated in terms of the characteristic classes of \(\xi\) and \(M\), which is stated as follows: {\parindent=6mm \begin{itemize} \item(a) \(Sq^2\rho_2H^{n-2}(M; \mathbb{Z})\neq0 \;\;\text{or}\) \item (b) \(Sq^2\rho_2H^{n-2}(M; \mathbb{Z})=0, \;\;\langle \mathrm{ch}(\tilde{c}(\tilde{\xi}))\cdot \hat{\mathfrak{A}}(M), [M] \rangle \equiv 0 \;\text{mod} \;2\). \end{itemize}} In order to prove this, the author first makes the observation that the condition (a) above holds true if and only if \[ \text{Im} \{p^* : \widetilde{KO}(S^n)\to \widetilde{KO}(M)\} \subset \text{Im} \{r : \tilde{K}(M)\to \widetilde{KO}(M)\} \] where \(p : M \to M/M^{n-1}=S^n\) denotes the canonical projection. In fact, this observation serves as a keystone in the proof.