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

Abstract

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 $Γ_n^\mathbb{C}$-structures on $\mathbb{CP}^n$ are constructed. As a consequence we derive a theorem concerning the homology of Haefligers classifying space, $BΓ_n^\mathbb{C}$. The result then follows from obstruction theory.