Related papers: Homotopy $4$-spheres associated to an infinite ord…

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;…

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…

We construct an infinite order cork (W,f), which means that W is a smooth compact contractible 4-manifold with Stein structure, and f is a self diffeomorphism of the boundary of W, such that the n-fold composition maps f^{n}=f o f o... o f…

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…

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 $C$…

For any positive integer $n$ we give a ${\mathbb Z}^n$-cork with a ${\mathbb Z}^n$-effective embedding in a 4-manifold being homeomorphic to $E(n)$. This means that a cork gives a subset ${\mathbb Z}^n$ in the differential structures on…

We show that the smooth homotopy 4-sphere obtained by Gluck twisting the m-twist n-roll spin of any unknotting number one knot is diffeomorphic to the standard 4-sphere, for any pair of integers (m,n). It follows as a corollary that an…

Every smooth homotopy 4-sphere is diffeomorphic to the 4-sphere.

The goal of this thesis is to prove that $\pi_4(S^3) \simeq \mathbb{Z}/2\mathbb{Z}$ in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory,…

We prove a localization theorem for exotic diffeomorphisms, showing that every diffeomorphism of a compact simply-connected 4-manifold that is isotopic to the identity after stabilizing with one copy of $S^2 \times S^2$, is smoothly…

The author recently proved the existence of an infinite order cork: a compact, contractible submanifold $C$ of a 4-manifold and an infinite order diffeomorphism $f$ of $\partial C$ such that cutting out $C$ and regluing it by distinct…

We use surgery along 2-tori embedded in a union of two copies of a product of punctured 2-tori to produce a new collection of homotopy 4-spheres (4-manifolds homotopy equivalent to $S^4$ and hence homeomorphic to $S^4$ but possibly not…

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…

In this note we describe a family of arguments that link the homotopy-type of a) the diffeomorphism group of the disc $D^n$, b) the space of co-dimension one embedded spheres in a sphere and c) the homotopy-type of the space of co-dimension…

We show that an infinite sequence of homotopy 4-spheres constructed by Cappell-Shaneson are all diffeomorphic to S^4. This generalizes previous results of Akbulut-Kirby and Gompf.

We study the infinite generation in the homotopy groups of the group of diffeomorphisms of $S^1 \times D^{2n-1}$, for $2n \geq 6$, in a range of degrees up to $n-2$. Our analysis relies on understanding the homotopy fibre of a linearisation…

We give a criterion on a group $\pi$ and a homomorphism $w \colon \pi \to C_2$ under which closed $4$-manifolds with fundamental group $\pi$ and orientation character $w$ are classified up to homotopy equivalence by their quadratic…

In this paper we investigate how to simultaneously change homotopy algebras of a certain type and a corresponding infinity morphism between them, and show that this can be done in a homotopically unique way. More precisely, for a reduced…

We provide the first information on diffeotopy groups of exotic smoothings of R^4: For each of uncountably many smoothings, there are uncountably many isotopy classes of self-diffeomorphisms. We realize these by various explicit group…

We prove that there exist infinitely many embedded tori with a common geometric dual in $T^4\#(S^2\times S^2)$ that are homotopic, diffeomorphic, but not isotopic to each other, even after arbitrary many external stabilizations. These…