Absolutely exotic 4-manifolds Selman Akbulut1 Daniel Ruberman2 1Department of Mathematics Michigan State University 2Department of Mathematics Brandeis University Hamilton, December 2014 arXiv:1410.1461 Smoothings of manifolds with boundary Smoothings of TOP manifold X ◮ {smooth Y with Y ≈ X} modulo diffeomorphism Smoothings of manifold W with boundary M ◮ How to account for boundary? ≈ ◮ Bordered manifold: (W , j) with j : M −→ ∂W . ◮ Smoothing relative to j is smooth structure with j a diffeo. =∼ ◮ (W , j) =∼ (W ′, j′) if F : W −→ W ′ makes diagram commute. F W / W ′ O O ⊆ ⊆ j j′ ∂Wo M / ∂W ′ Smoothings of manifolds with boundary Alternative view; suppose W smooth, M = ∂W =∼ ◮ τ : M −→ M. ◮ Suppose τ extends to homeomorphism F : W → W . ◮ Pull back to get smooth structure WF on W . ◮ ∼ (W ,τ) = (WF , id). ◮ (W ,τ) =∼ (W , id) if and only if τ extends to diffeomorphism. 7 7 7 Example 1: (Milnor): Exotic sphere Σ = B ∪ϕ B . Induced smooth structure on B7 standard, but (B7, ϕ) exotic. Corks Example 2: (Akbulut) τ 0 0 =∼ Figure 1: The Mazur cork W The diffeomorphism τ : ∂W = M → M gives an exotic smoothing of W rel idM . Main result Question: Is there an absolutely exotic contractible 4-manifold? Answer: Yes! Theorem 1 (Akbulut-Ruberman 2014) There are homeomorphic but not diffeomorphic smooth compact contractible 4-manifolds V and V ′ with diffeomorphic boundaries. In fact, given any relatively exotic 4-manifold (W ,τ) we find a pair V , V ′ of non-diffeomorphic manifolds with the same boundary, both embedded in W and homotopy equivalent to W . Example: Selman’s examples of relatively exotic homotopy S1. Invertible cobordisms Strategy is to kill the symmetry group of the boundary by adding on an invertible homology cobordism. A cobordism X from M to N is invertible if there is a cobordism ′ ′ ∼ X with X ∪N X = M × I (rel idM ). Sometimes X ′ is just X upside down. Relative version: J is doubly slice if there are concordances C from the unknot O to J and C′ going the other way with C ∪ C′ = O× I. The strategy doesn’t quite work, but . Invertible cobordisms Main technical step. Proposition 2 For any M3, there is an invertible homology cobordism X from M to a manifold N such that every diffeomorphism f : N → N extends to X so that it is the identity on M. Extending work of Paoluzzi-Porti (2009) we first show Proposition 3 Any 3-manifold M contains a link ~η with M − ~η hyperbolic and with trivial symmetry group. N = infection along ~η by copies of a doubly slice knot J. Invertible cobordisms Want doubly slice J with special properties ◮ J is hyperbolic. ◮ Symmetry group of J is trivial. Examples: 11n42 (Kinoshita-Terasaka knot), 12n0313, and 12n0430. Figure 2: J = KT knot (the dotted line indicates the ribbon move) Construction of N and X Start with asymmetric ~η ⊂ M as above. Obtain N by gluing copies of P = S3 − nhd(J) to exterior of ~η, matching meridian ↔ longitude. Example: M =+1 surgery on pretzel knot K . 1 K Figure 3: M = K +1 with asymmetric η Proof of Theorem 1 X comes from gluing copies of S3 × I − nhd(C) along ~η × I. ◮ X is an invertible homology cobordism ◮ ∼ π1(X) = π1(M). Let ′ V = W ∪M X and V = W ∪τ X. ′ ∼ Key point: Invertibility of X means (V , idN ) =6 (V , idN ). Suffices to prove assertion about Diff(N), from following. ◮ M − ~η is hyperbolic, with trivial symmetry group. ◮ S3 − J is hyperbolic, with trivial symmetry group. ◮ These imply that (up to isotopy) Diff(N) generated by Dehn twists going across peripheral tori of ~η. Proof of Theorem 1 Asymmetry of J verified (rigorously) by SnapPy with extensions (HIKMOT, Dunfield-Hoffman-Licata). Second follows from JSJ decomposition. But all of those Dehn twists extend over X. On M they’re Dehn 1 2 twists on T = ∂(S × D ) and so are isotopic to idM . So if there’s any diffeomorphism V → V ′ then get one that restricts to idN (which can’t exist). Variations and extensions Construction works for W ≃ S1 (Akbulut anti-corks). Similar technique works to construct V and V ′ with ◮ ∂V =∼ ∂V ′ ◮ V ≃ V ′ but ◮ No relative homotopy equivalence (V , ∂V ) ≃ (V ′, ∂V ′). Simplest version (≃ S2) is derived from Gluck twist on S1 × S2. It has ∂V =∼ 0-surgery on KT knot. Variations and extensions Akbulut announced (but withdrew for now) infinite order cork twist: contractible W (not Mazur anymore) with (W ,τ k ) relatively distinct. Theorem 4 If there are infinitely many relatively distinct (W ,τ k ), then there is a contractible V with infinitely many (absolutely) smooth structures. Idea is to use old theorem of mine constructing invertible homology cobordism of ∂W to hyperbolic N. Symmetry group of N possibly non-trivial, but finiteness (Mostow rigidity) gives theorem. CORK ABSOLUT EXOTIC 2 0 0 0 I pledge to put my slides on my web page: http://people.brandeis.edu/∼ruberman/.