The G-Signature Theorem
was originally proved by Atiyah and Singer as a corollary of their general index
theorem for elliptic operators. Subsequently Ossa gave a proof for G finite and the fix
point set orientable. His methods are mainly topological. However, he uses the theory
of elliptic operators to show the g-signature of a fix point free diffeomorphism of
finite order is zero. Janich and Ossa gave a short completely topological proof of the
theorem for involutions. In part one, we give a complete proof for semi-free
actions and simultaneously a proof for general actions modulo the theorem for
fix point free actions. In essence our argument here is similar to that of
Ossa. However it is shorter and conceptually simpler. Also we derive the
formula in a natural way as opposed to verifying it. In part two, we prove
a theorem which we use in part one to prove the result for fix point free
actions. I wish to thank my advisor Professor E. Thomas for much help and
encouragement.
