438 9. BASIC TOPOLOGY OF 3-MANIFOLDSassume that M is irreducible. Since M has a trivial fundamental group, itis geometrically atoroidal. (By definition, if a manifold contains an incom-
pressible torus, its fundamental group must contain Z EEl Z, the fundamental
group of the torus.) Thus, if Conjecture III holds, M admits a metric of
constant positive sectional curvature. Thus M must be 83.2.3. Examples of 3-manifolds and their geometries. There are
eight locally homogeneous Riemannian geometries in dimension 3. Besides
the well-known constant curvature metrics 83 , JR^3 , and 7t^3 , the rest of the
five geometries are given by the standard metrics on 82 xJR, 7t^2 xJR, SL(2, JR),
nil, and sol. (The Ricci fl.ow of homogeneous metrics on 83 is discussed in
Section 5 of Chapter 1 in Volume One.)
Here is a description of the three nonproduct geometries. The geom-
etry SL(2, JR) may be regarded as the universal cover of the Lie group
SL(2, JR) with a metric invariant under left multiplication. The nilpotent 3-
dimensional Lie group nil is the Heisenberg group of strictly upper-triangular
3 x 3 matrices. (The Ricci fl.ow on nil is discussed in Section 7 of Chapter 1
in Volume One.) The geometry sol is that of the 3-dimensional solvable Lie
group defined as the semi-direct product of JR^2 with JR, where the action of
t E JR on JR^2 is given by the matrix ( ~ e~t). (The Ricci fl.ow on sol is
discussed in Section 7 of Chapter 1 in Volume One.)
A closed., orientable manifold with 82 x JR geometry must be either
82 x 81 or JRIP^3 #JRIP^3. It is interesting to note that JRIP^3 #JRJID^3 is the only
connected-sum 3-manifold admitting a locally homogeneous metric. The
product of a closed surface of genus at least two with the circle admits an
7t^2 x 81 metric, i.e., the product of their standard metrics given by the
uniformization theorem. The unit tangent bundle over a surface of negative. Euler characteristic is a 3-manifold with SL(2, JR) geometry. More generally,
any circle bundle over a surface of negative Euler characteristic admits the
SL(2, JR) geometry if the Chern class of the bundle is nonzero. A circle
bundle over the torus with nonzero Chern class is a 3-manifold with nil
geometry. A torus bundle over the circle whose monodromy is a linear map
with distinct real eigenvalues has sol geometry. It can be shown that any
closed 3-manifold with one of these five geometries is finitely covered by one
of the examples just mentioned. For more information, see the excellent
reference [319].
A closed 3-manifold with sol geometry is not a Seifert 3-manifold. It
admits a nontrivial torus decomposition coming from the torus fiber. On
the other hand, any manifold admitting one of the six geometries 83 , JR^3 ,
82 x JR, 7t^2 x JR, SL(2, JR), or nil is a Seifert manifold.
We will say a compact 3-manifold Mis a topological graph manifold
if it admits a torus decomposition such that each complementary piece is a
Seifert manifold. (This is a standard definition in low-dimensional topology.)