Chapter 26 - Wall's Finiteness Obstruction

Isotopies of 3-manifolds

An irreducible open 3-manifold W is R 2 -irreducible if it contains no non-trivial planes, i.e. given any proper embedded planein W some component of W −� must have closure an embedded halfspace R 2 ×(0, ∞). In this paper it is shown that if M is a connected, P 2 -irreducible, open 3-manifold such that �1(M) is finitely generated and the universal covering space f M of M is R 2 -irreducible, then either f M is homeomorphic to R 3 or �1(M) is a free product of infinite cyclic groups and fundamental groups of closed, connected surfaces other than S 2 or P 2 . Given any finitely generated group G of this form, uncountably many P 2 -irreducible, open 3-manifolds M are constructed with �1(M) ∼ G such that the universal covering space f M is R 2 -irreducible and not homeomorphic to R 3 ; the f M are pairwise non- homeomorphic. Relations are established between these results and the conjecture that the universal covering space of any irreducible, orientable, closed 3-manifold with infinite fundamental group must be homeomorphic to R 3 .

A question dating back to Waldhausen [10] and discussed in various contexts by Thurston (see [9]) is the problem of the extent to which irreducible 3-manifolds with infinite fundamental group must contain surface groups. To state our results precisely, it is convenient to make the definition that a map i: S 9<M of a closed, orientable connected surface S is essential if it is injective at the level of fundamental groups and the group i*rr1 (S) cannot be conjugated into a subgroup 7rr(coM) of -rr(M), where &oM is a component of OM. This latter condition is equivalent to the statement that the image of the surface S cannot be freely homotoped into OM. One of the main results of this paper is the following:

We study the existence of pluricanonical forms on an algebraic variety of general type with infinite fundamental group. We have a nonvanishing of m th plurigenus of the variety by that of a general fibre of the so-called Shafarevich map of the variety.

Groups associated to a knot in S3 which classify the knot have been constructed by Conway and Gordon [4], Simon [12] and Whitten[16]; see also [1] and [19] for certain special classes of knots. In this note we show that for a knot K in S3 of the groups π1(S3/K)/〈malb〉 classify the knot; here m resp. l denote a meridian resp. longitude of the knot and 〈〉 denotes normal closure. This is based on Thurston's orbifold geometrization theorem [13–15] and the following result (see also [8, 19]):Theorem 1. Let O1 and O2 be good closed irreducible 3-orbifolds which possess a decomposition into geometric pieces (along Euclidean 2-suborbifolds, see [2, 15]) and have infinite (orbifold-) fundamental group. Then O1 and O2 are diffeomorphic (as orbifolds) if and only if π1 O1 and π1 02 are isomorphic.

Let M be an oriented irreducible 3-manifold with infinite fundamental group and empty or toroidal boundary which is not S^1 \times D^2 . Consider any element \phi in the first cohomology of M with integer coefficients. Then one can define the \phi -twisted L^2 -torsion function of the universal covering which is a function from the set of positive real numbers to the set of real numbers. By earlier work of the second author and Schick the evaluation at t=1 determines the volume. In this paper we show that the degree of the L^2 -torsion function, which is a number extracted from its asymptotic behavior at 0 and at \infty , agrees with the Thurston norm of \phi .

We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metrics.

We introduce and study trace fields and invariant trace fields of SL2(C) and PSL2(C) representations of 3-manifold groups. We give conditions on such fields which imply that the underlying 3-manifold is virtually Haken or has virtually positive or 1 first Betti number. In particular, we define the notion of an algebraically trace-proper surface subgroup in a 3-manifold group, and show that any closed orientable irreducible 3-manifold with such a surface subgroup is virtually Haken. We give infinitely many families of closed orientable hyperbolic non-Haken 3-manifolds with algebraically trace-proper surface subgroups. In this paper a 3-manifold is always assumed to be connected and orientable, without loss of generality for the problems to be considered. A 3-manifold is said to be Haken if it is compact, irreducible, and contains a properly embedded incompressible surface. A 3-manifold is said to be virtually Haken if it has a finite cover which is Haken. The well-known virtually Haken conjecture states that every closed and irreducible 3-manifold with infinite fundamental group is virtually Haken. It is also conjectured that every closed Haken 3-manifold has virtually positive first Betti number, that is, the manifold has a finite cover which has positive first Betti number. If M is a closed hyperbolic Haken 3-manifold, then it is further conjectured that M has virtually 1 first Betti number, which means that for any positive integer n, there is a finite cover Mn of M such that the first Betti number of Mn is larger than n. These conjectures are fundamental and difficult issues in 3-manifold topology. In this paper we provide some information about these conjectures by studying traces of representations of 3-manifold groups into SL2ðCÞ or PSL2ðCÞ: For a group G and a representation r : G ! SL2ðCÞ; we define the trace field Kr of r to be the field generated by the traces of all the matrices r (g), g [ G; over the base field Q of rational numbers, that is, Kr ¼ Qðtrðr ðgÞÞ; g [ G Þ:

We give a characterization of k-dimensional (k > 1) Menger manifolds admitting boundaries in the sense of Chapman and Siebenmann.In [5] Chapman and Siebenmann considered the problem of putting a boundary on a Hilbert cube manifold (similar problems in the cases of smooth and piecewise linear manifolds were considered in [2,14]; a parametrical version of the above problem was considered in [13]).It was proved there that if a Q-manifold M satisfies certain minimal necessary homotopy-theoretic conditions (finite type and tameness at oo ), then there are two obstructions o0O(M) and Too(M) to M having a boundary.The first one is an element of the group lim{J?67ri(M -A) : A c M compact}, where %nx is the projective class group functor.If Ooo(M) = 0, then the second obstruction can be defined as an element of the first derived limit of the inverse system lim {W hnx (M -A) : A M compact} , where Whnx is the Whitehead group functor.Further, it was proved in [5] that the different boundaries that can be put on M constitute a whole shape class and that a classification of all possible ways of putting boundaries on M can be done in terms of the group lim {Whnx (M -A) : Ac.M compact}.In the present paper we carry out a similar program for the problem of putting boundaries on pn+x -manifolds, where pn+x denotes the (n + 1 )-dimensional universal Menger compactum (a p"+x -manifold M admits a boundary if there exists a compact p"+x-manifold N such that M = N-Z , where Z is a Z-set in N ; in this case we shall say that N is a compactification of M corresponding to the boundary Z , and conversely, Z is a boundary of M corresponding to the compactification N. We recall also that a closed subset Z of a space X is said to be a Z-set if for each open cover ^ e cov(X) there is a map f:X-*X-Z ^-close to the identity map of X ).Having in mind a deep analogy between the theories of pn+x -manifolds and Q-manifolds [1,[6][7][8][9] it is not surprising that the corresponding results are valid in the case of pn+xmanifolds as well.However, it should be observed that the situation in the last case is much simpler.For example, the analogies of the above-described

Polyhedra for which every homotopy domination over itself is a homotopy equivalence

Covering group theory for topological groups

Projective modules and class groups

The fundamental group at infinity

In this paper, we develop the theory of probabilistic variants of the one-category and diagonal topological complexity, which bound the classical LS-category and topological complexity from below. Unlike any other classical or probabilistic invariants, these invariants are rigid on spaces with finite fundamental group. On Eilenberg–Mac Lane spaces, we identify these new invariants with distributional category and complexity, respectively, and use them to illuminate aspects of the behavior of the latter invariants on aspherical spaces and products of spaces. We also study their properties on covering maps, [Formula: see text]-isomorphisms, [Formula: see text]-spaces, and closed essential manifolds, and consequently, obtain the first examples of closed manifolds beyond the real projective spaces on which the distributional theory disagrees with the classical one.

Topological realizations of chain complexes 1. the general theory