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Model solvmanifolds for lefschetz and nielsen theories
Abstract
In this paper we construct a class of
solvmanifolds and certain (diagonal type) self maps on them. These
solvmanifolds and their maps serve firstly as rich source of examples. Secondly
they serve as models for Nielsen theory in the sense that any map f : S → S of
an arbitrary compact solvmanifold S, has the same Lefschetz and Nielsen
theory (ordinary
and periodic) as a “diagonal” map f′ on a “model solvmanifold” S′.
Thus Nielsen theory calculations never get more complicated than they do on
these model maps
and spaces.
The models also have the property (unlike arbitrary
solvmanifolds) that their diagonal maps are characterized by a single simply
purely matrix theoretical condition which is merely necessary for arbitrary
solvmanifolds. As a result, the models S′often exhibit many more
maps than the original solvmanifolds S form which they are derived.
Moreover,
it
is
often on these “extra” maps are where the more interesting Nielsen theory
occurs.
Mathematics Subject
Classification (2000): Primary: 37C25, 14M17, 55M20; Secondary: 03C65
Quaestiones Mathematicae 25 (2002), 483-501
solvmanifolds and certain (diagonal type) self maps on them. These
solvmanifolds and their maps serve firstly as rich source of examples. Secondly
they serve as models for Nielsen theory in the sense that any map f : S → S of
an arbitrary compact solvmanifold S, has the same Lefschetz and Nielsen
theory (ordinary
and periodic) as a “diagonal” map f′ on a “model solvmanifold” S′.
Thus Nielsen theory calculations never get more complicated than they do on
these model maps
and spaces.
The models also have the property (unlike arbitrary
solvmanifolds) that their diagonal maps are characterized by a single simply
purely matrix theoretical condition which is merely necessary for arbitrary
solvmanifolds. As a result, the models S′often exhibit many more
maps than the original solvmanifolds S form which they are derived.
Moreover,
it
is
often on these “extra” maps are where the more interesting Nielsen theory
occurs.
Mathematics Subject
Classification (2000): Primary: 37C25, 14M17, 55M20; Secondary: 03C65
Quaestiones Mathematicae 25 (2002), 483-501
