On smoothable surgery for 4–manifolds

Khan, Qayum

doi:10.2140/agt.2007.7.2117

Cited by 2 publications

(5 citation statements)

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“…Remark 1.11. For certain groups π 1 (X) unknown in N DL, such as poly-surface groups, results on exactness at N TOP (X) are found in [24,26,22,9].…”

Section: Homotopy Invariance Of Connected Sums-unstable Versionmentioning

confidence: 99%

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Homotopy invariance of 4-manifold decompositions: Connected sums

Khan

2012

Topology and its Applications

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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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“…Remark 1.11. For certain groups π 1 (X) unknown in N DL, such as poly-surface groups, results on exactness at N TOP (X) are found in [24,26,22,9].…”

Section: Homotopy Invariance Of Connected Sums-unstable Versionmentioning

confidence: 99%

“…Hence # is not always a bijection in the case π 1 (X) = D ∞ ∈ N DL.Remark 1.11. For certain groups π 1 (X) unknown in N DL, such as poly-surface groups, results on exactness at N TOP (X) are found in [24,26,22,9].Remark 1.12. The modular group P SL(2, Z) ∼ = C 2 * C 3 is an example of a free product of N DL groups.…”

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confidence: 99%

Homotopy invariance of 4-manifold decompositions: Connected sums

Khan

2012

Topology and its Applications

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“…For the convenience of the reader, we first recall the relevant hypotheses from the precursor [Kha07,§3]. Let (X, ∂X) be a based, compact, connected TOP 4manifold with fundamental group π and orientation character ω : π → Z × .…”

Section: Exactness At 4-dimensional Normal Invariantsmentioning

confidence: 99%

“…The author is grateful to Jim Davis, who supervised the splitting result of this paper, which was a certain portion of the dissertation [Kha06]. Also, the referee of [Kha07] deserves thanks for suggesting Mostow rigidity for the fibers S. Finally, the author appreciates his discussions with Tom Farrell and Bruce Hughes, from which the relevant techniques for the fibering result were made clear.…”

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confidence: 94%

“…The central theorem of this paper is a generalization of the Cappell-Weinberger theorem [Cap76,Wei87] for splitting compact 5-manifolds along certain incompressible, two-sided 4-submanifolds (Theorem 4.1). Indeed, the development of additional tools for our main splitting theorem motivated the author's initial investigation of 4-manifolds [Kha07].…”

Section: Introductionmentioning

confidence: 99%

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On fibering and splitting of 5-manifolds over the circle

Khan

2008

Topology and its Applications

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Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Most of these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman--Quinn topological surgery. Indeed, our key technique is topological cobordism, which may not be the trace of surgeries.Comment: 22 pages, exposition revised for better self-containmen

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On smoothable surgery for 4–manifolds

Cited by 2 publications

References 30 publications

Homotopy invariance of 4-manifold decompositions: Connected sums

Homotopy invariance of 4-manifold decompositions: Connected sums

On fibering and splitting of 5-manifolds over the circle

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