Mahan Mj

A manifold admits a good complexification  in the sense of Totaro
 if it can be realised as the set of real
points of an affine variety such that the inclusion of the real locus
is a homotopy equivalence. Conjecturally only manifolds admitting
a Riemannian metric of non-negative curvature admit good complexifications.

We produce restrictions on fundamental groups of manifolds admitting
good complexifications in the sense of Totaro by proving the following Cheeger-Gromoll type splitting theorem:
 Let M be a closed manifold admitting a good complexification. Then $M$ admits a finite-sheeted regular covering $M_1$
such that $M_1$ admits a fiber bundle structure with base $(S^1)^k$ and
fiber $N$,  where $N$ admits a good complexification and has zero  virtual first Betti number.
We give several applications to manifolds of dimension at most 5.