Vol. 97, No. 1, 1981

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Topological proof of the G-signature theorem for G finite

Patrick M. Gilmer

Vol. 97 (1981), No. 1, 105–114

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.

Mathematical Subject Classification 2000
Primary: 57S17
Secondary: 58G10
Received: 18 October 1977
Published: 1 November 1981
Patrick M. Gilmer
Department of Mathematics
Louisiana State University
376 Lockett Hall
Baton Rouge LA 70803
United States