Stable Complex Structures On Real Vector Bundles Over Connected Sums Of Quaternionic Projective Spaces

For integers $n\ge 2$ and integers $\mathrm{\ell },m\ge 0$, not both zero, let $M_{\mathrm{\ell },m}^{4n}=\mathrm{\ell }{\mathbb{H}\mathbb{P}}^n\mathrm{\#}m\overline{{\mathbb{H}\mathbb{P}}^n}$ denote the connected sum of $\mathrm{\ell }$ copies of the quaternionic projective space ${\mathbb{H}\mathbb{P}}^n$ and $m$ copies of ${\mathbb{H}\mathbb{P}}^n$ endowed with the opposite orientation. By analysing the image of the complexification map $c\text{\hspace{0.17em}⁣}:\widetilde{KO}(M_{\mathrm{\ell },m}^{4n})\to \tilde{K}(M_{\mathrm{\ell },m}^{4n})$, we characterise, in terms of Pontryagin classes, the real vector bundles over $M_{\mathrm{\ell },m}^{4n}$ admitting a stable complex structure, and deduce that $M_{\mathrm{\ell },m}^{4n}$ is stably almost complex if and only if $n=2$ and $\mathrm{\ell }-m$ is even. Consequently no $M_{\mathrm{\ell },m}^{4n}$ with $n\ge 3$ admits an almost complex structure. Sato and Suzuki asserted the non-existence of almost complex structures on $M_{\mathrm{\ell },m}^{4n}$ in 1974 for $3\le n\le 10$, except possibly when $n=3$ and $\mathrm{\ell }=3m+1$; we establish it for all $n\ge 3$, and in the stronger form of stable almost complex structures.