## Abstract for Class VII Paper

The purpose of this note is to obtain a restriction on the fundamantel
groups of non-elliptic complex surfaces in Kodaira's classification. The
main result is that if M is such a surface and N is a Riemannian
manifold of constant negative curvature, tehn the image of the fundamental
group of M in the fundamental group of N under any homomorphism is either
cyclic or trivial.

As a corollary we find that a the fundamental group of a
compact manifold of constant negative curvature
and dimension at least three is not isomorphic to that of any
compact complex surface. We also find that a compact manifold
of constant negative curvature and real dimension four has no complex structure.

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Last modified by jac at 15:51 on 12/27/1997.