Fixed points of surface diffeomorphisms

Boju Jiang and Jianhan Guo

Vol. 160 (1993), No. 1, 67–89

DOI: 10.2140/pjm.1993.160.67

Abstract

We give a complete proof of the following theorem which was conjectured by Jakob Nielsen for closed oriented surfaces. Theorem 1 Let f : M → M be a homeomorphism of a compact surface. When M is closed, then f is isotopic to a diffeomorphism with N(f) fixed points, where N(f) is its Nielsen number. When M has boundary, N(f) should be replaced by the relative Nielsen number N(f;M,∂M) defined by Schirmer.

Another result is the inequality |L(f) − χ(M)|≤ N(f) − χ(M) when χ(M) < 0, where L(f) is the Lefschetz number and χ(M) is the Euler characteristic.

Mathematical Subject Classification 2000

Primary: 57N05

Secondary: 57M20, 57R50

Milestones

Received: 17 October 1990

Revised: 3 March 1992

Published: 1 September 1993