# The classification of isometry groups of a Seifert fibered space N which is double covered by a lens space L.

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Ryo Ohashi, SLU

What Topology Seminar Fri, Nov 19, 2004 from 11:00 AM to 11:50 AM RH 316 vCal iCal

Abstract: Let W be a twisted I-bundle over a Klein bottle fibered trivially whose base space is a Mobius band and V be a (p,q)-fibered solid torus.  Since the boundaries of V and  W  are tori, we may form a quotient space N by identifying the boundary tori.  Let  f : bd(V ) -> bd(W) denote the glueing map.  One can choose f to be a fiber preserving map.     Next, we let  N = V \cup_{f} W be such the quotient space.  As the attaching preserves the fiber, N is also a (p,q)-fibered Seifert fibered space, which is known as a prism manifold.  We can observe that the base space of N is topologically a real projective plane that contains at most one cone point of order p.In the talk, we will carefully observe that N is double covered by a symmetric lens space L, which gives us a geometric structure on N.  It turns out that L and Isom(N) are completely determined by the initial assignment of a fibration type on V.  We will observe how a fibration type affects topological structures on N and L as well as Isom(N).  Moreover, this allows us to compute the isometry group of N denoted by Isom(N).The technique to compute Isom(N) depends on the liftability criteria, that is, if  h : L -> N is a covering map and if g \in Homeo(N), then there is \bar{g}: L \rightarrow L such that h \circ \bar{g} = g \circ h if and only if g_* \circ h_* (\pi_1(L)) is a subgroup of  h_*(\pi_1(L)).Unfortunately, the liftability does not work in certain cases. Thus, Isom(N) was not able to classify completely in my last talk.  In these cases, a "pullback" method will be used.  In other words, we will use the fact that there is a subgroup in S^3 \oplus S^3 and an epimorphism \hat{\rho} such that image of the subgroup under \hat{\rho} is Isom(N).  Further, the key ingredient to determine Isom(N) is as follows: If G_1 and G_2 are any finite non-cyclic subgroups of Isom_+(S^3) = SO(3) such that G_1 \cong G_2 and acting on S^3 freely, then the two groups are a difference of some conjugate.  This will give us the complete classification of Isom(N).

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