Abstract
The set of finite group actions (up to equivalence) which operate on a prism manifold M, preserve a Heegaard Klein bottle and have a fixed orbifold quotient type, form a partially ordered set. We describe the partial ordering of these actions by relating them to certain sets of ordered pairs of integers. There are seven possible orbifold quotient types, and for any fixed quotient type we show that the partially ordered set is isomorphic to a union of distributive lattices of a certain type. We give necessary and sufficent conditions, for these partially ordered sets to be isomorphic and to be a union of Boolean algebras.
Highlights
The set of finite group actions which operate on a prism manifold M, preserve a Heegaard Klein bottle and have a fixed orbifold quotient type, form a partially ordered set
There are seven possible orbifold quotient types, and for any fixed quotient type we show that the partially ordered set is isomorphic to a union of distributive lattices of a certain type
This paper examines the partially ordered sets consisting of equivalence classes of finite group actions acting on prism manifolds and having a fixed orbifold quotient type
Summary
This paper examines the partially ordered sets consisting of equivalence classes of finite group actions acting on prism manifolds and having a fixed orbifold quotient type. For a fixed quotient type, we show that the partially ordered set is a union of distributive lattices of a certain type. Finite group actions on prism manifolds were studied in [3]. A G-action is primitive if it does not contain a nontrivial normal subgroup which acts freely. These actions determine minimal elements in the partially ordered sets.
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