A structure theorem and a splitting theorem for simply-connected smooth 4-manifolds

Stong, Richard

doi:10.4310/mrl.1995.v2.n4.a9

Cited by 7 publications

(11 citation statements)

References 3 publications

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“…Our technique has a similar flavor to an argument of Stong , who proved that if the intersection form

Q_{X}

of a simply connected closed 4‐manifold

X

decomposes as the direct sum of unimodular forms

U_{1} \oplus U_{2}

, then

X

can be smoothly decomposed into two simply connected pieces

X_{1}

and

X_{2}

with

Q_{X_{1}} = U_{1}

and

Q_{X_{2}} = U_{2}

. (Note that this is insufficient for our argument; we need that each piece is a 2‐handlebody.)…”

Section: Exact Calculations For Arbitrarily Large Embedding Numbersmentioning

confidence: 94%

“…In particular this method gives integral homology spheres that bound two negative definite spin 4‐manifolds with different rank, answering a question of Tange [, Question 5.2]. In fact our technique can be generalized using a structure theorem of Stong to show that any simply connected 4‐manifold can be decomposed into a positive definite 4‐manifold and a negative definite 4‐manifold (both simply connected), glued along a rational homology sphere (Theorem ).…”

Section: Introductionmentioning

confidence: 99%

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Embedding 3‐manifolds in spin 4‐manifolds

Aceto

Golla

Larson

2017

Journal of Topology

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An invariant of orientable 3-manifolds is defined by taking the minimum n such that a given 3-manifold embeds in the connected sum of n copies of S 2 × S 2 , and we call this n the embedding number of the 3-manifold. We give some general properties of this invariant, and make calculations for families of lens spaces and Brieskorn spheres. We show how to construct rational and integral homology spheres whose embedding numbers grow arbitrarily large, and which can be calculated exactly if we assume the 11/8-Conjecture. In a different direction we show that any simply connected 4-manifold can be split along a rational homology sphere into a positive definite piece and a negative definite piece.

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“…Our technique has a similar flavor to an argument of Stong , who proved that if the intersection form

Q_{X}

of a simply connected closed 4‐manifold

X

decomposes as the direct sum of unimodular forms

U_{1} \oplus U_{2}

, then

X

can be smoothly decomposed into two simply connected pieces

X_{1}

and

X_{2}

with

Q_{X_{1}} = U_{1}

and

Q_{X_{2}} = U_{2}

. (Note that this is insufficient for our argument; we need that each piece is a 2‐handlebody.)…”

Section: Exact Calculations For Arbitrarily Large Embedding Numbersmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Embedding 3‐manifolds in spin 4‐manifolds

Aceto

Golla

Larson

2017

Journal of Topology

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“…The Involutory Cork Theorem, due to Curtis-Freedman-Hsiang-Stong [10] and Matveyev [21], states that any two (smooth) closed simply-connected 4-manifolds that are homotopy equivalent, and thus homeomorphic, are related by a single cork replacement (see Definition 1.9). It was shown in [10], based on earlier work of Stong [24], that the common complement of the corks can be chosen to be simply-connected. Moreover, as demonstrated in [21], this cork replacement can be accomplished by an involutory cork twist.…”

Section: The Involutory Cork Theoremmentioning

confidence: 99%

“…As all the handles in W lie in U , the complementary cobordism W − U acquires a product structure from the gradient flow of h (with respect to a suitable metric) which gives a diffeomorphism G : X 0 − U 0 → X 1 − U 1 extending f on the boundary ∂X 0 . Since U 0 is of type 1) and π 1 (X 0 − U 0 ) = 1, the handlebody techniques in [24] can be used to encase U 0 (as in Definition 1.3) in a tightly embedded AC cork C 0 in X 0 . In particular this follows from the Encasement Lemma 2.2 below, as noted in Remark 2.3.…”

Section: The Involutory Cork Theoremmentioning

confidence: 99%

Higher order corks

Melvin¹,

Schwartz²

2019

Preprint

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It is shown that any finite list of smooth, closed, simply-connected 4-manifolds that are homeomorphic to a given one X can be obtained by removing a single compact contractible submanifold (or cork) from X, and then regluing it by powers of a boundary diffeomorphism. Furthermore, by allowing the cork to be noncompact, the collection of all the smooth manifolds homeomorphic to X can be obtained in this way. The existence of a universal noncompact cork is also established. † If X C,h and X are diffeomorphic, the embedding C ⊂ X is said to be trivial. The cork (C, h) is trivial if all its embeddings in all 4-manifolds are trivial, or equivalently if h extends to a diffeomorphism of C [4].

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“…Works of Freedman & Taylor, [33], and Stong, [76], show that one can still mimic this decomposition by decomposing along homology 3-spheres into simply connected pieces.…”

Section: Theorem 23 If π Is An Ndl Group Then Top-surgery and The Tmentioning

confidence: 99%