Non-existence theorems on infinite order corks

Tange, Motoo

doi:10.48550/arxiv.1609.04344

Cited by 6 publications

(7 citation statements)

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“…More recently, Tange has posted papers extending the methods of this article to exhibit n-fold boundary sums of our Z-corks as Z n -corks [21] and providing constraints on families of manifolds that can be related by Z-corks [20].…”

Section: Introductionmentioning

confidence: 99%

Infinite order corks

Gompf

2017

Geom. Topol.

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Abstract. We construct a compact, contractible 4-manifold C, an infinite order self-diffeomorphism f of its boundary, and a smooth embedding of C into a closed, simply connected 4-manifold X, such that the manifolds obtained by cutting C out of X and regluing it by powers of f are all pairwise nondiffeomorphic. The manifold C can be chosen from among infinitely many homeomorphism types, all obtained by attaching a 2-handle to the meridian of a thickened knot complement.

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Section: Introductionmentioning

confidence: 99%

Infinite order corks

Gompf

2017

Geom. Topol.

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“…Recently higher order corks ( [36], [14]) and surprisingly infinite order corks ( [23], see also [3], [24], [38]) were discovered, but interestingly Tange [37] showed that a natural extension of the above cork theorem does not always hold for an infinite exotic family, by showing a certain finiteness for Ozsváth-Szabó invariants of cork twisted 4-manifolds. More precisely, he gave infinite families of pairwise exotic closed 4-manifolds such that, for any 4-manifold X, any contractible submanifold W , and any self-diffeomorphism f of ∂W , the families cannot be constructed from X by twisting W via powers of f .…”

Section: Introductionmentioning

confidence: 99%

Nonexistence of twists and surgeries generating exotic 4-manifolds

Yasui

2019

Trans. Amer. Math. Soc.

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It is well known that for any exotic pair of simply connected closed oriented 4-manifolds, one is obtained from the other by twisting a compact contractible submanifold via an involution on the boundary. By contrast, here we show that for each positive integer n, there exists a simply connected closed oriented 4-manifold X such that, for any compact (not necessarily connected) codimension zero submanifold W with b 1 (∂W ) < n, the set of all smooth structures on X cannot be generated from X by twisting W and varying the gluing map. As a corollary, we show that there exists no 'universal' compact 4-manifold W such that, for any simply connected closed 4-manifold X, the set of all smooth structures on X is generated from a 4-manifold by twisting a fixed embedded copy of W and varying the gluing map. Moreover, we give similar results for surgeries.

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“…To address infinite lists X i (i ∈ Z) of 4-manifolds homeomorphic to X, such as an enumeration of all the exotic smooth structures on X, it is tempting to search for a single infinite cork (C, h) embedded in X such that X C,h i ∼ = X i for all i (here and below, ∼ = denotes diffeomorphism). Such a cork need not exist, however, as noted by Tange [25]. For example, it follows from the adjunction inequality that knot surgeries on the Kummer surface using any list of knots with unbounded Alexander polynomial degrees cannot be the cork twists associated to a fixed embedding of a single infinite cork (cf.…”

Section: Introductionmentioning

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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Non-existence theorems on infinite order corks

Abstract: Suppose that X, X ′ are simply connected closed exotic 4manifolds. It is well-known that X ′ is obtained by an order 2 cork twist of X. We show that in the case of infinite order cork, this existence theorem does not always hold.

Cited by 6 publications

References 20 publications

Infinite order corks

Infinite order corks

Nonexistence of twists and surgeries generating exotic 4-manifolds

Higher order corks

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