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Vanishing structure set of 3-manifolds

In this short note we update a result proved in [16]. This will complete our program of [12] showing that the structure set vanishes for compact aspherical 3-manifolds.


Introduction
This paper is to note that the program we started in [12] is now complete.
Let us first recall that a compact manifold M with boundary is called topologically rigid if any homotopy equivalence f : (N, ∂N) → (M, ∂M) from another compact manifold with boundary, so that f | ∂N : ∂N → ∂M is a homeomorphism is homotopic to a homeomorphism relative to boundary.
Let M be a compact connected 3-manifold whose fundamental group is torsion free.
We prove the following theorem.
Theorem 1.1. If M is aspherical then M × D n is topologically rigid for n ≥ 2. Here D n denotes the n-dimensional disc.
In [12] and [13] we proved Theorem 1.1 under various conditions. In [12] we proved it for the nonempty boundary case and for the situation when the manifold contains an incompressible square root closed torus. In [13] we assumed the manifold has positive first Betti number. Due to some recent developments in Geometric Topology (see [1], [2], [14], [15] and [16]) we are now able to deduce Theorem 1.1. Also the main ideas from [12] and [13] go behind the proof of this general case.
The first step to prove Theorem 1.1 is to show that the Whitehead group of π 1 (M) is trivial. We deduce the following for this purpose. For terminologies on 3-manifolds used in the proofs see [7] or [12].
Proof of Theorem 1.2. By (Kneser-Milnor) prime decomposition theorem G is isomorphic to the free product of a free group and finitely many groups G 1 , G 2 , . . . , G n where for each i, G i is isomorphic to the fundamental group of an aspherical irreducible 3-manifold M i (see [[13], Lemma 3.1]). Since the Whitehead group of a free product is the direct sum of the Whitehead groups of the individual factors of the free product (see [18]) it is enough to prove that the Whitehead group vanishes for G i . Now by the Geometrization Theorem (conjectured by Thurston and proved by Perelman) M i is either Seifert fibered, Haken or hyperbolic. The hyperbolic case follows from some more general result of Farrell and Jones in [4], for Haken case it follows from Waldhausen's result in [19]. For non-Haken Seifert fibered space the vanishing result is due to Plotnick (see [9]). For the reduced projective class groups K 0 (−) and for the negative K-groups K −i (−) the same sequence of arguments and references work. For details see [3]. In fact, more generally it is shown in [3] that G is K-flat, i.e., W h(G × Z n ) = 0 for all non-negative integer n. This completes the proof of Theorem 1.2.
Below we recall the statement of the Fibered Isomorphism Conjecture of Farrell and Jones. For details about this conjecture see [5]. Here we follow the formulation given in [ [6], Appendix].
Let F be one of the three functors from the category of topological spaces to the category of spectra: (a) the stable topological pseudoisotopy functor P(); (b) the algebraic K-theory functor K(); and (c) the L-theory functor L −∞ (). The L-theory functor also includes an orientation data, that is a homomorphism ω : π 1 (X) → Z 2 . If the topological space is an oriented manifold then this homomorphism is zero.
Let M be a category whoes objects are continuous surjective maps p : E → B between topological spaces E and B. And a morphism between two maps p : E 1 → B 1 and q : E 2 → B 2 is a pair of continuous maps f : E 1 → E 2 , g : B 1 → B 2 such that the following diagram commutes.
There is a functor defined by Quinn in [10] from M to the category of Ω-spectra which associates to the map p a spectrum H(B, F (p)) with the property that H(B, F (p)) = S(E) if B is a single point space. Let Γ be a discrete group and E be a Γ-space which is universal for the class of all virtually cyclic subgroups of Γ and denote E/Γ by B. For definition and properties of universal space see [ [5], Appendix]. Let X be a space on which Γ acts freely and properly discontinuously and p : X × Γ E → E/Γ = B be the map induced by the projection onto the second factor of X × E.
The Fibered Isomorphism Conjecture for Γ states that the map is an (weak) equivalence of spectra. The equality in the above display is induced by the map X × Γ E → X/Γ and using the fact that F is homotopy invariant. If X is simply connected then this is called the Isomorphism Conjecture for Γ. In this paper we consider the case when F () = L −∞ (). We have already mentioned that this L-theory functor contains the orientation data ω : Γ → Z 2 so as to include the case of nonorientable manifolds.
Let us now deduce the following theorem which is an immediate consequence of [ [16], 3(a) of Theorem 2.2] and some recent results from [1] and [2]. Proof. The theorem follows from [ [16], 3(a) of Theorem 2.2] provided we show that the conjecture is true for Γ ≀ H where H is some finite group and Γ belongs to the following classes of groups: 1). Z 2 ⋊ σ Z for all actions σ of Z on Z 2 . 2). Fundamental groups of closed nonpositively curved Riemannian 3-manifolds.
3). Γ ≃ lim i∈I Γ i where {Γ i } is a directed system of groups so that for each i ∈ I the conjecture is true for Γ i ≀ K where K is some finite group.
We now note the following to complete the proof of the Theorem.
(1) follows from [2] where the conjecture is proved for virtually polycyclic groups.
And (3)  Next we recall the definition of structure set and the surgery exact sequence.
Let M be a compact manifold with boundary (may be empty) so that W h(π 1 (M)) = 0. Consider all objects (N, ∂N, f ), where N is a manifold with boundary ∂N and f : N → M is a homotopy equivalence such that f | ∂N : ∂N → ∂M is a homeomorphism. Two such objects (N 1 , ∂N 1 , f 1 ) and (N 2 , ∂N 2 , f 2 ) are equivalent if there is a homeomorphism g : N 1 → N 2 such that the obvious diagram commutes up to homotopy relative to the boundary. The equivalence classes of these objects is the homotopy-topological structure set S(M, ∂M).
Also it is shown in [11] that there is a bijection between S(M × D k , ∂(M × D k )) and S k+dimM (M) provided dimM + k ≥ 5.
The proof of the theorem is now complete since we have already proved the isomorphism H k (M, L 0 ) → L k (G) for all k.