Lattice of Finite Group Actions on Prism Manifolds

Affiliation(s)

Department of Mathematics and Computer Science, Saint Louis University, St. Louis, USA.

Department of Mathematics, King’s College, Wilkes-Barre, USA.

Department of Mathematics and Computer Science, Saint Louis University, St. Louis, USA.

Department of Mathematics, King’s College, Wilkes-Barre, USA.

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.

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.

Cite this paper

J. Kalliongis and R. Ohashi, "Lattice of Finite Group Actions on Prism Manifolds,"*Advances in Pure Mathematics*, Vol. 2 No. 3, 2012, pp. 149-168. doi: 10.4236/apm.2012.23022.

J. Kalliongis and R. Ohashi, "Lattice of Finite Group Actions on Prism Manifolds,"

References

[1] J. Kalliongis and A. Miller, “Orientation Reversing Ac- tions on Lens Spaces and Gaussian Integers,” Journal of Pure and Applied Algebra, Vol. 212, No. 3, 2008, pp. 652-667. doi:10.1016/j.jpaa.2007.06.022

[2] R. Stanley, “Enumerative Combinatorics Volume 1,” Wads- worth & Brooks/Cole, New York, 1986.

[3] R. Ohashi, “The Isometry Groups on Prism Manifolds, Dissertation,” Saint Louis University, Saint Louis, 2005.

[4] J. Kalliongis and R. Ohashi, “Finite Group Actions on Prism Manifolds Which Preserve a Heegaard Klein Bot- tle,” Kobe Journal of Math, Vol. 28, No. 1, 2011, pp. 69- 89.

[5] B. A. Davey and H. A. Priestley, “Introduction to Lattices and Order,” Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1990.

[1] J. Kalliongis and A. Miller, “Orientation Reversing Ac- tions on Lens Spaces and Gaussian Integers,” Journal of Pure and Applied Algebra, Vol. 212, No. 3, 2008, pp. 652-667. doi:10.1016/j.jpaa.2007.06.022

[2] R. Stanley, “Enumerative Combinatorics Volume 1,” Wads- worth & Brooks/Cole, New York, 1986.

[3] R. Ohashi, “The Isometry Groups on Prism Manifolds, Dissertation,” Saint Louis University, Saint Louis, 2005.

[4] J. Kalliongis and R. Ohashi, “Finite Group Actions on Prism Manifolds Which Preserve a Heegaard Klein Bot- tle,” Kobe Journal of Math, Vol. 28, No. 1, 2011, pp. 69- 89.

[5] B. A. Davey and H. A. Priestley, “Introduction to Lattices and Order,” Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1990.