arXiv subjects
Selman Akbulut
Publications and source records attributed to Selman Akbulut.
On 4-dimensional smooth Poincare conjecture
Here we outline a proof for the 4-dimensional smooth Poincare Conjecture.
Corks and exotic ribbons in $B^{4}$
By using corks we construct diffeomorphic ribbon disks $D\subset B^{4}$, which are non-isotopic rel boundary to each other.
On Shake Slice Knots
Here we discuss $r-$shake slice knots, and their relation to corks, we then prove that $0$-shake slice knots are slice.
Homotopy $4$-spheres associated to an infinite order loose cork
We show the homotopy spheres $Σ_{n} = -W\smile_{f^{n}}W$, formed by doubling the infinite order loose-cork $(W,f)$ by iterates of the cork diffeomorphism $f: \partial W \to \partial W$ is $S^4$. To do this we first show that $Σ_{n} $ are obtained by Gluck twistings of $S^4$; then from this we show how to cancel $3$-handles of $Σ_{n}$ and identify it by $S^{4}$.
Cork twists and automorphisms of $3$-manifolds
Here we study two interesting smooth contractible manifolds, whose boundaries have non-trivial mapping class groups. The first one is a non-Stein contractible manifold, such that every self diffeomorphism of its boundary extends inside; implying that this manifold can not be a loose cork. The second example is a Stein contractible manifold which is a cork, with an interesting cork automorphism $f:\partial W \to \partial W$. By \cite{am} we know that any homotopy $4$-sphere is obtained gluing together two contractible Stein manifolds along their common boundaries by a diffeomorphism. We use the homotopy sphere $Σ= -W\smile_{f}W$ as a test case to investigate if it is $S^4$? We show that $Σ$ is a Gluck twisted $S^4$ twisted along a $2$-knot $S^{2}\hookrightarrow S^4$; by using this we obtain a $3$-handle free handlebody description of $Σ$ and then show $Σ\approx S^4$.
On a homotopy $4$-sphere
We give a brief survey of some facts about homotopy $4$-spheres \cite{a1}, then give a proof that the curious homotopy sphere constructed in \cite{a2} is in fact diffeomorphic to the standard $S^4$, and discuss its relation to infinite order loose corks and anti-corks.
On the Smale Conjecture for Diff$(S^4)$
Recently Watanabe disproved the Smale Conjecture for $S^4$, by showing Diff$(S^{4})\neq SO(5)$. He showed this by proving that their higher homotopy groups are different. Here we prove this more directly by showing $\pi_{0}$Diff$(S^{4})\neq 0$, otherwise a certain loose-cork could not possibly be a loose-cork.
On an infinite order cork automorphisms
Here we give a concrete description of the cork automorphism $f:\partial W\to \partial W$ of the infinite order loose-cork $(W,f)$, defined in \cite{a2}. It is obtained by concatenating the defining ribbon disk of $W$ in $B^4$ by an infinite order isotopy of the boundary knot.
Corks and automorphisms of 3-manifolds
We investigate two specific contractible manifolds (one Stein, and the other non-Stein) whose boundaries have non-trivial mapping class groups. In both cases we show that every diffeomorphism of their boundary extends to a diffeomorphism of the full manifold. In particular, these manifolds cannot be corks. The methods are a mix of 3 and 4-manifold techniques.
Knot concordances in $S^1\times S^2$ and exotic smooth $4$-manifolds
It is known that there is a unique concordance class in the free homotopy class of $S^1\times pt \subset S^1 \times S^2$. The constructive proof of this fact is given by the second author. It turns out that all the concordances in this construction are invertible. The knots $K\subset S^{1}\times S^{2}$ with hyperbolic complements and trivial symmetry group are special interest here, because they can be used to generate absolutely exotic compact 4-manifolds by the recipe given by Akbulut and Ruberman. Here we built absolutely exotic manifold pairs by this construction, and show that this construction keeps the Stein property of the $4$-manifolds we start out with. By using this we establish the existence of an absolutely exotic contractible Stein manifold pair, and absolutely exotic homotopy $S^1\times B^3$ Stein manifold pair.
A simple class of infinitely many absolutely exotic manifolds
We show that the smooth $4$-manifold $M$ obtained by attaching a $2$-handle to $B^4$ along a certain knot $K\subset \partial B^4$ admits infinitely many absolutely exotic copies $M_n$, $n=0,1,2..$, such that each copy $M_n$ is obtained by attaching $2$-handle to a fixed compact smooth contractible manifold $W$ along the iterates $f^{n}(c)$ of a knot $c\subset \partial W$ by a diffeomorphism $f:\partial W \to \partial W$. This generalizes the example in author's 1991 paper, which corresponds to $n=1$ case.
Complex $G_2$-manifolds and Seiberg-Witten Equations
We introduce the notion of complex $G_2$ manifold $M_{\mathbb C}$, and complexification of a $G_2$ manifold $M\subset M_{\mathbb C}$. As an application we show the following: If $(Y,s)$ is a closed oriented $3$-manifold with a $Spin^{c}$ structure, and $(Y,s)\subset (M, φ)$ is an imbedding as an associative submanifold of some $G_2$ manifold (such imbedding always exists), then the isotropic associative deformations of $Y$ in the complexified $G_2$ manifold $M_{\mathbb C}$ is given by Seiberg-Witten equations.
Complex $\text{G}_2$ and Associative Grassmannian
We obtain defining equations of the smooth equivariant compactification of the Grassmannian of the complex associative $3$-planes in $\C^7$, which is the parametrizing variety of all quaternionic subalgebras of the algebra of complex octonions $\OO\cong \C^8$. By studying the torus fixed points, we compute the Poincaré polynomial of the compactification.
Lefschetz Fibrations on Compact Stein Manifolds
Here we prove that up to diffeomorphism every compact Stein manifold W of dimension 2n+2>4 admits a Lefschetz fibration over the two-disk with Stein regular fibers, such that the monodromy of the fibration is a symplectomorphism induced by compositions of right-handed Dehn twists along embedded Lagrangian n-spheres on the generic fiber. This generalizes the Stein surface case of n=1, previously proven by Loi-Piergallini and Akbulut-Ozbagci. More precisely, we show that up to Liouville isomorphism any Weinstein domain W admits a compatible compact convex Lefschetz fibration with Weinstein regular fibers and with the same monodromy description stated above. Moreover, the induced convex open book supports the induced contact structure on the boundary of W.
On Legendrian Embbeddings into Open Book Decompositions
We study Legendrian embeddings of a compact Legendrian submanifold $L$ sitting in a closed contact manifold $(M,ξ)$ whose contact structure is supported by a (contact) open book $\mathcal{OB}$ on $M$. We prove that if $\mathcal{OB}$ has Weinstein pages, then there exist a contact structure $ξ'$ on $M$, isotopic to $ξ$ and supported by $\mathcal{OB}$, and a contactomorphism $f:(M,ξ) \to (M,ξ')$ such that the image $f(L)$ of any such submanifold can be Legendrian isotoped so that it becomes disjoint from the closure of a page of $\mathcal{OB}$.