Title:
Almost complex structures on homotopy complex projective spaces
Abstract: A manifold is called almost complex if its tangent
bundle is the real reduction of a complex bundle. Since the existence
of an almost complex structure on a given manifold is not homotopy
invariant, we consider this existence question on "homotopy complex
projective spaces," manifolds with the oriented homotopy type of
complex projective space. We will discuss what is required to approach
this problem in general, as well as affirmative results on the
existence and classification of almost complex structures on homotopy
complex projective spaces of dimension 2n for 4 ≤ n ≤ 6.