This paper is concerned with fixed-point free
–actions (smooth or locally
linear) on orientable
–manifolds.
We show that the fundamental group plays a predominant role in the equivariant classification of
such
–manifolds.
In particular, it is shown that for any finitely presented group with infinite
center there are at most finitely many distinct smooth (resp. topological)
–manifolds
which support a fixed-point free smooth (resp. locally linear)
–action
and realize the given group as the fundamental group. A similar
statement holds for the number of equivalence classes of fixed-point free
–actions under
some further conditions on the fundamental group. The connection between the classification of
the
–manifolds
and the fundamental group is given by a certain decomposition, called a
fiber-sum decomposition,
of the
–manifolds.
More concretely, each fiber-sum decomposition naturally gives rise to a Z–splitting of
the fundamental group. There are two technical results in this paper which
play a central role in our considerations. One states that the Z–splitting is
a canonical JSJ decomposition of the fundamental group in the sense of
Rips and Sela. Another asserts that if the fundamental group has infinite
center, then the homotopy class of principal orbits of any fixed-point free
–action on the
–manifold must be infinite,
unless the
–manifold
is the mapping torus of a periodic diffeomorphism of some elliptic
–manifold.
Keywords
four-manifolds, circle actions, Rips–Sela theory,
geometrization of $3$–orbifolds
