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4-Dimensional Geometries 129 Part II 4-dimensional Geometries c Geometry & Topology Publications 131 Chapter 7 Geometries and decompositions Every closed connected surface is geometric, i.e., is a quotient of one of the three 2 2 2 model 2-dimensional geometries E , H or S by a free and properly discontinu- ous action of a discrete group of isometries. Every closed irreducible 3-manifold admits a finite decomposition into geometric pieces. (That this should be so was the Geometrization Conjecture of Thurston, which was proven in 2003 by Perelman, through an analysis of the Ricci flow, introduced by Hamilton.) In x1 we shall recall Thurston's definition of geometry, and shall describe briefly the 19 4-dimensional geometries. Our concern in the middle third of this book is not to show how this list arises (as this is properly a question of differen- tial geometry; see [Is55, Fi, Pa96] and [Wl85, Wl86]), but rather to describe the geometries sufficiently well that we may subsequently characterize geomet- ric manifolds up to homotopy equivalence or homeomorphism. In x2 and x3 we relate the notions of \geometry of solvable Lie type" and \infrasolvmani- fold". The limitations of geometry in higher dimensions are illustrated in x4, where it is shown that a closed 4-manifold which admits a finite decomposi- tion into geometric pieces is (essentially) either geometric or aspherical. The geometric viewpoint is nevertheless of considerable interest in connection with complex surfaces [Ue90, Ue91, Wl85, Wl86]. With the exception of the geome- 2 2 2 2 2 2 1 tries S × H , H × H , H × E and SLf × E no closed geometric manifold has a proper geometric decomposition. A number of the geometries support natural Seifert fibrations or compatible complex structures. In x4 we character- ize the groups of aspherical 4-manifolds which are orbifold bundles over flat or hyperbolic 2-orbifolds. We outline what we need about Seifert fibrations and complex surfaces in x5 and x6. Subsequent chapters shall consider in turn geometries whose models are con- tractible (Chapters 8 and 9), geometries with models diffeomorphic to S2 × R2 3 1 (Chapter 10), the geometry S ×E (Chapter 11) and the geometries with com- pact models (Chapter 12). In Chapter 13 we shall consider geometric structures and decompositions of bundle spaces. In the final chapter of the book we shall consider knot manifolds which admit geometries. Geometry & Topology Monographs, Volume 5 (2002) 132 Chapter 7: Geometries and decompositions 7.1 Geometries An n-dimensional geometry X in the sense of Thurston is represented by a pair (X; GX ) where X is a complete 1-connected n-dimensional Riemannian mani- fold and GX is a Lie group which acts effectively, transitively and isometrically on X and which has discrete subgroups Γ which act freely on X so that ΓnX has finite volume. (Such subgroups are called lattices.) Since the stabilizer of a point in X is isomorphic to a closed subgroup of O(n) it is compact, and so ΓnX is compact if and only if ΓnGX is compact. Two such pairs (X; G) and (X0;G0) define the same geometry if there is a diffeomorphism f : X ! X0 which conjugates the action of G onto that of G0 . (Thus the metric is only an adjunct to the definition.) We shall assume that G is maximal among Lie groups acting thus on X , and write Isom(X) = G, and Isomo(X) for the com- ponent of the identity. A closed manifold M is an X-manifold if it is a quotient ΓnX for some lattice in GX . Under an equivalent formulation, M is an X- manifold if it is a quotient ΓnX for some discrete group Γ of isometries acting freely on a 1-connected homogeneous space X = G=K , where G is a connected Lie group and K is a compact subgroup of G such that the intersection of the conjugates of K is trivial, and X has a G-invariant metric. The manifold admits a geometry of type X if it is homeomorphic to such a quotient. If G is solvable we shall say that the geometry is of solvable Lie type. If X = (X; GX ) and Y = (Y; GY ) are two geometries then X ×Y supports a geometry in a nat- ural way; GX×Y = GX × GY if X and Y are irreducible and not isomorphic, but otherwise GX×Y may be larger. The geometries of dimension 1 or 2 are the familiar geometries of constant cur- 1 2 2 2 vature: E , E , H and S . Thurston showed that there are eight maximal 3 3 3 2 1 3 2 1 3 3-dimensional geometries (E , Nil , Sol , SLf , H × E , H , S × E and S .) Manifolds with one of the first five of these geometries are aspherical Seifert 3 fibred 3-manifolds or Sol -manifolds. These are determined among irreducible 3-manifolds by their fundamental groups, which are the PD3 -groups with non- trivial Hirsch-Plotkin radical. A closed 3-manifold M is hyperbolic if and only if it is aspherical and π1(M) has no rank 2 abelian subgroup, while it is an 3 S -manifold if and only if π1(M) is finite, by the work of Perelman [B-P]. 3 (See x11.4 below for more on S -manifolds and their groups.) There are just 2 1 four S × E -manifolds. For a detailed and lucid account of the 3-dimensional geometries see [Sc83']. 4 There are 19 maximal 4-dimensional geometries; one of these (Solm;n ) is in fact 4 a family of closely related geometries, and one (F ) is not realizable by any closed manifold [Fi]. We shall see that the geometry is determined by the fun- Geometry & Topology Monographs, Volume 5 (2002) 7.2 Infranilmanifolds 133 damental group (cf. [Wl86, Ko92]). In addition to the geometries of constant curvature and products of lower dimensional geometries there are seven \new" 4-dimensional geometries. Two of these are modeled on the irreducible Rieman- nian symmetric spaces CP 2 = U(3)=U(2) and H2(C) = SU(2; 1)=S(U(2) × 4 U(1)). The model for the geometry F is the tangent bundle of the hyperbolic plane, which we may identify with R2 ×H2 . Its isometry group is the semidirect 2 ± ± product R ×α SL (2; R), where SL (2; R) = fA 2 GL(2; R) j det A = ±1g, ± 2 and α is the natural action of SL (2; R) on R . The identity component 2 2 2 a b acts on R × H as follows: if u 2 R and A = c d 2 SL(2; R) then az+b 2 2 u(w; z) = (u + w; z) and A(w; z) = (Aw; cz+d ) for all (w; z) 2 R × H . The 1 0 2 4 matrix D = 0 −1 acts via D(w; z) = (Dw; −z¯). All H (C)- and F -manifolds are orientable. The other four new geometries are of solvable Lie type, and shall be described in x2 and x3. In most cases the model X is homeomorphic to R4 , and the corresponding ge- 4 4 3 1 ometric manifolds are aspherical. Six of these geometries (E , Nil , Nil × E , 4 4 4 Solm;n , Sol0 and Sol1 ) are of solvable Lie type; in Chapter 8 we shall show man- ifolds admitting such geometries have Euler characteristic 0 and fundamental group a torsion-free virtually poly-Z group of Hirsch length 4. Such manifolds are determined up to homeomorphism by their fundamental groups, and every such group arises in this way. In Chapter 9 we shall consider closed 4-manifolds 3 1 1 admitting one of the other geometries of aspherical type (H × E , SLf × E , 2 2 2 2 4 2 4 H × E , H × H , H , H (C) and F ). These may be characterised up to s-cobordism by their fundamental group and Euler characteristic. However it is unknown to what extent surgery arguments apply in these cases, and we do not yet have good characterizations of the possible fundamental groups. Al- 4 though no closed 4-manifold admits the geometry F , there are such manifolds with proper geometric decompositions involving this geometry; we shall give examples in Chapter 13. 2 2 2 2 3 1 Three of the remaining geometries (S × E , S × H and S × E ) have models 2 2 3 4 2 2 2 homeomorphic to S ×R or S ×R. The final three (S , CP and S ×S ) have compact models, and there are only eleven such manifolds. We shall discuss these nonaspherical geometries in Chapters 10, 11 and 12. 7.2 Infranilmanifolds The notions of \geometry of solvable Lie type" and \infrasolvmanifold" are closely related. We shall describe briefly the latter class of manifolds, from a rather utilitarian point of view. As we are only interested in closed mani- folds, we shall frame our definitions accordingly. We consider the easier case of Geometry & Topology Monographs, Volume 5 (2002) 134 Chapter 7: Geometries and decompositions infranilmanifolds in this section, and the other infrasolvmanifolds in the next section. A flat n-manifold is a quotient of Rn by a discrete torsion-free subgroup of n n E(n) = Isom(E ) = R oα O(n) (where α is the natural action of O(n) on Rn ). A group π is a flat n-manifold group if it is torsion-free and has a nor- mal subgroup of finite index which is isomorphic to Zn . (These are necessary and sufficient conditions for π to be the fundamental group of a closed flat n-manifold.) The action of π by conjugation on its translation subgroup T (π) (the maximal abelian normal subgroup of π) induces a faithful action of π=T (π) on T (π).
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