Cohomology-free diffeomorphisms of low-dimension tori (vol 18, pg 985, 1998)

A C-infinity diffeomorphism p of a manifold M is cohomologically rigid if for each smooth function f on M there is a constant f(0) so that the cohomological equation h - h circle phi = f - f(0) has a smooth solution h. We prove that all of the eigenvalues of the mapping on H-1 (T-n, R) induced by 4 cohomologically rigid diffeomorphism phi of the torus T-n are roots of unity if n < 4. The same is true for n=4 provided that phi preserves orientation. We do not know whether it is true when n=4 and phi reverses orientation or when n > 4. The purpose of this erratum is to give a proof for Proposition 1.8 in [31 for the tori T-n, n < 4. It is also true for n=4 provided that the diffeomorphism preserves orientation. This result plays a fundamental role in [3]. All of the objects we consider here are C-infinity. A diffeomorphism phi of a manifold M is cohomologically rigid (CR) if every function f on M is cohomologous to a constant, i.e. there exist a constant f(0) and a smooth function h on M so that h-h circle phi=f-f(0). In [3] it is referred to as cohomology-free diffeomorphism. It was proved in [3, Proposition 1.2] that a CR diffeomorphism preserves a smooth volume form, is uniquely ergodic and thus minimal.