Classifying non-splitting fiber preserving actions on prism manifolds

Finite actions on the Klein four-orbifold and prism manifolds

“…Thus (y, j) is in the normalizer of the group (i, 1), (j, 1), (1, e 2πi b ) in S 3 × S 3 , and thus σ(y, j) induces an isometry σ(y, j) on M (b, 2). It was shown in [3] that σ(y, j) has order 6, and thus we obtain a fiber-preserving Z 6 -action on M (b, 2).…”

“…Actions can fail to preserve a Heegaard Klein bottle when d = 2 and the induced action on Σ(2, 2, 2) is either Z 3 , Z 6 , Dih(Z 3 ) or Dih(Z 6 ). We consider these actions, and show that the orbifold quotients fiber over the following 2-orbifolds: Σ (2,3,3), T h , Σ(2, 3, 4), T v , or O h , all of which are covered by Σ (2,2,2). For the standard actions, Z 3 , Z 6 , Dih(Z 3 ) and Dih(Z 6 ) on M (b, 2), we compute the fundamental groups of the quotient orbifolds.…”

“…Notice that xya(2) = 2 and a fixes vertices 3, 8, 9 on S 2 . Because the edges 2, 14 and 3, 14 are identified in Σ(2, 2, 2), a loop [2,9,8,3] is a fixed set under a. As a result,…”

“…In this section, we consider the finite group actions on a prism manifold M (b, d) which preserve a longitudinal fibering, but do not leave any Heegaard Klein bottle invariant. Actions which leave a Heegaard Klein bottle invariant are said to split, and these were classified in [3]. We consider the orbifold quotients for these actions, and show that they fiber over certain of the orbifolds mentioned in Section 1.…”

“…Hence the group i, j, y 2 = j, y ≃ T * . This implies the quotient space In both [5] and [6], M is a tetrahedral manifold if and only if M fibers over Σ (2,3,3), π 1 (M ) ≃ T * × Z b and g.c.d. {b, 6} = 1.…”