Closed $3$-manifolds with no periodic maps
Transactions of the American Mathematical Society
Examples of closed, orientable, aspherical 3-manifolds are constructed on which every action of a finite group is trivial. 1. Introduction. In this paper we give examples of closed, orientable, aspherical 3-manifolds on which every action of a finite group is trivial. These 3-manifolds can be fibered over Sl with fiber a closed orientable surface of genus greater than or equal to 3. For such examples, three is the smallest possible genus of the fiber since every 3-manifold fibered over S1 with
... fiber of genus < 2 admits nontrivial involutions . P. E. Conner and F. Raymond  first constructed examples of manifolds exhibiting this total lack of nontrivial periodic homeomorphisms. Their examples were compact nonaspherical 4-manifolds with boundary. We use some of the methods introduced in that paper which provide information as to which finite groups can act effectively on certain manifolds. The starting point is the following theorem of A. Borel. Let G be a group of homeomorphisms of a closed, connected, aspherical manifold M Consider the homomorphism \j/: G -► Out(itx(M)) which sends g e G to the outer automorphism of itx(M) induced by the homeomorphism g. Then if itx(M) has trivial center, \¡/ is a monomorphism. We obtain our results by proving that for certain 3-manifolds this group of outer automorphisms is torsion free. Conner, Raymond, and Weinberger in  construct closed aspherical manifolds (of dimensions 7, 11, 16, 22, 29, and 37) which admit no effective action of any nontrivial group. Also E. Bloomberg in  has constructed nonaspherical, closed, orientable 4-manifolds on which no finite group can operate. The examples presented in this paper are the first 3-manifolds known to admit no periodic maps at all. This result in the PL category was first observed by the second author , whereas now we work in the topological category. We wish to thank Professor Joan Birman for some helpful observations concerning surface maps.