Homotopy invariance of 4-manifold decompositions: Connected sums

Abstract

Given any homotopy equivalence f : M → X 1 # · · · #X n of closed orientable 4-manifolds, where each fundamental group π 1 (X i ) satisfies Freedman's Null Disc Lemma, we show that M is topologically hcobordant to a connected sum M = M 1 # · · · #M n such that f is h-bordant to some f 1 # · · · #f n with each f i : M i → X i a homotopy equivalence. Moreover, such a replacement M of M is unique up to a connected sum of h-cobordisms. In summary, the existence and uniqueness, up to h-cobordism, of connected sum decompositions of such orientable 4-manifolds M is an invariant of homotopy equivalence.Also we establish that the Borel Conjecture is true in dimension 4, up to s-cobordism, if the fundamental group satisfies the Farrell-Jones Conjecture.arXiv:0907.0308v5 [math.GT] 10 Aug 2012 Homotopy invariance of connected sums-unstable versionOur Main Theorem (Thm. 1.7) is phrased technically in terms of the classes N DL and SES h + , which we define below. The difficulty in the proof is in observing new extensions of the geometric topology developed by S. Cappell [7] and S. Weinberger [44].Definition 1.2 (Freedman). A discrete group G is N DL (or good) if the π 1 -Null Disc Lemma holds for it (see [21] for details). The class N DL is closed under the operations of forming subgroups, extensions, and filtered colimits.This class contains subexponential and exponential growth [19,21,32].Theorem 1.3 (Freedman-Quinn, Freedman-Teichner, Krushkal-Quinn). The class N DL contains all virtually polycyclic groups and all groups of subexponential growth.Example 1.4. Here are some exotic examples in N DL. The semidirect product Z 2 α Z with α = ( 2 1 1 1 ) is polycyclic but has exponential growth. For all integers n = −1, 0, 1, the Baumslag-Solitar group BS(1, n) = Z[1/n] n Z is finitely presented and solvable but not polycyclic. Grigorchuk's infinite 2-group G is finitely generated but not finitely presented and has intermediate growth.Recall that, unless specified in the notation, the structure sets S h TOP and normal invariants N TOP are homeomorphisms on the boundary (that is, rel ∂) [24, §6.2].Definition 1.5. Let Z be a non-empty compact connected topological 4-manifold. Denote the fundamental group π := π 1 (Z) and orientation character ω := w 1 (τ Z ). We declare that Z has class SES h if there exists an exact sequence of based sets:The subclass SES h + includes actions of groups in K-and L-theory (Defn. 2.5). This exact sequence has been proven for the above groups [19, Thm. 11.3A].Theorem 1.6 (Freedman-Quinn). Let X be a compact connected topological manifold of dimension 4. If π 1 (X) has class N DL, then X has class SES h + and satisfies the s-cobordism conjecture (i.e., all s-cobordisms on X are homeomorphic to X × I).Here is the Main Theorem of the paper. The existence and uniqueness question posed in the Title and Abstract, up to h-cobordism, is quantified in # of Part (2). Theorem 1.7. Let X be a compact connected topological manifold of dimension 4.2 Example 1.8. Suppose X is a closed connected topological 4-manifold...

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