On the homology of $B{\mathrm{\Gamma }}_n^{\mathbb{C}}$ and its application to complex structures on open manifolds

Since the 1970s, it has been known that any open connected manifold of dimension 2, 4 or 6 admits a complex analytic structure whenever its tangent bundle admits a complex linear structure. For half a century, this has been conjectured to hold true for manifolds of any dimension. In this paper, we extend the result to manifolds of dimension 8. To prove the result new ${\mathrm{\Gamma }}_n^{\mathbb{C}}$-structures on ${\mathbb{C}\mathbb{P}}^n$ are constructed. As a consequence we derive a theorem concerning the homology of Haefligers classifying space, $B{\mathrm{\Gamma }}_n^{\mathbb{C}}$. The result then follows from obstruction theory.