Higher order corks
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Higher order corks Paul Melvin1 · Hannah Schwartz2
Received: 28 July 2020 / Accepted: 30 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract It is shown that any finite list of smooth closed simply-connected 4-manifolds 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. We then use this result to ‘separate’ finite families of corks embedded in a fixed 4-manifold. 0 Introduction A cork is a compact contractible 4-manifold C equipped with a boundary diffeomorphism h : ∂C → ∂C.1 The cork (C, h) is trivial if h extends to a diffeomorphism of C; for example (B 4 , h) is trivial for any h [9]. The cork is finite of order n if h is periodic of order n. Corks of order 2 will be called involutory. If C is embedded in the interior of a 4-manifold X , then the associated cork twist X C,h := (X − int(C)) ∪h C
1 We work throughout in the smooth oriented category, so implicitly assume all manifolds
are smooth and oriented, and all diffeomorphisms are orientation preserving.
B Hannah Schwartz
[email protected]
1
Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010, USA
2
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
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P. Melvin, H. Schwartz
is homeomorphic [11] but not generally diffeomorphic [1] to X . If X C,h and X are diffeomorphic, written X C,h ∼ = X , then C ⊂ X is said to be a trivial embedding of the cork. It can be shown that a cork is nontrivial if and only if it embeds nontrivially in some 4-manifold [3,6]. The first nontrivial (involutory) cork was found by Akbulut [1], showing that cork twists can alter smooth structures on 4-manifolds. It is now known by Curtis–Freedman–Hsiang–Stong [10] and Matveyev [17] that any pair of compact simply-connected 4-manifolds that are h-cobordant rel boundary (so homeomorphic by Freedman [11]) are related by a single involutory cork twist, and that for closed manifolds, the cork may be chosen with simply-connected complement. We call this the “Involutory Cork Theorem” (see Sect. 1.16). Our first main result extends this theorem to arbitrary finite lists of closed simply-connected 4-manifolds (or more generally compact simply-connected 4-manifolds bounded by homology spheres, see Theorem 3.1). This demonstrates the ubiquity of nontrivial higher order corks, meaning corks (C, h) of order n ≥ 3 whose twists X C,h 1 , . . . , X C,h n for suitable embeddings C ⊂ X are smoothly distinct. Such corks were first shown to exist in [6]; see Tange [19] for a weaker existence result. Finite Cork Theorem For any finite list X i (i ∈ Zn ) of closed simplyconnected 4-manifolds all homeomorphic to a given one X = X 0 , there is a cork (C, h) of order n embedded in X with simply-connected complement whose twists X C,h i are diffeomorphic to X i for each i ∈ Zn . Our main application of the Relative Involutory Cork Theorem 1.16, along with the Consolidation Theorem 2.1 (our main techni
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