Related papers: Corks with large shadow-complexity and exotic 4-ma…

We show that there exist infinitely many simply connected compact Stein 4-manifolds with b_2=2 such that they are all homeomorhic but mutually non-diffeomorphic, and they are Stein fillings of the same contact 3-manifold on their…

We use the Heegaard Floer homology cobordism maps to obstruct the existence of a symplectic structure on the Akbulut-Kirby Mazur manifolds whose boundary is a Brieskorn sphere $Y$ among $\Sigma(2,3,13),$ $\Sigma(2,5,7)$ and $\Sigma(3,4,5)$.…

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 proved that the boundary Dehn twist on the Milnor fiber $M_c(2, q, r)$ is an exotic diffeomorphism relative to the boundary if $q, r$ are odd, coprime integers bigger than $3$ and $(q-1)(r-1)/4$ is an odd number. The proof is given by…

We construct a new infinite family of models of exotic 7-spheres. These models are direct generalizations of the Gromoll-Meyer sphere. From their symmetries, geodesics and submanifolds half of them are closer to the standard 7-sphere than…

Let $\mathrm{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\mathrm{pod}_{-4}(n)$ involving the following infinite family of…

A snark -- connected cubic graph with chromatic index $4$ -- is critical if the graph resulting from the removal of any pair of distinct adjacent vertices is $3$-edge-colourable; it is bicritical if the same is true for any pair of distinct…

In this paper, we construct Poincar\'e-Einstein 4-manifolds with various kinds of cusps. In particular, we construct: (1) Infinite families of Einstein metrics on $(0,\infty)\times \mathscr{N}$, where $\mathscr{N}\to T^2$ is a principal…

In this paper we construct a family of simply connected, spin, non-complex, symplectic 4-manifolds which cover all but finitely many allowed lattice points $(\chi, c)$ lying in $0 \leq c \leq 8.76\chi$. Furthermore, as a corollary, we prove…

Kreck and Schafer produced the first examples of stably diffeomorphic closed smooth 4-manifolds which are not homotopy equivalent. They were constructed by applying the doubling construction to 2-complexes over certain finite abelian groups…

We study connected sum at infinity on smooth, open manifolds. This operation requires a choice of proper ray in each manifold summand. In favorable circumstances, the connected sum at infinity operation is independent of ray choices. For…

An odd Seiberg-Witten invariant imposes bounds on the signature of a closed, almost complex 4-manifold with vanishing first Chern class. This applies in particular to symplectic 4-manifolds of Kodaira dimension zero.

In this short note, we present a construction of new symplectic 4-manifolds with non-negative signature using the complex surfaces on Bogomolov-Miyaoka-Yau line $c_1^2 = 9\chi_h$, the fake projective planes and Cartwright-Steger surfaces.…

We study locally flat, compact, oriented surfaces in $4$-manifolds whose exteriors have infinite cyclic fundamental group. We give algebraic topological criteria for two such surfaces, with the same genus $g$, to be related by an ambient…

We study snarks whose edges cannot be covered by fewer than five perfect matchings. Esperet and Mazzuoccolo found an infinite family of such snarks, generalising an example provided by Hagglund. We construct another infinite family, arising…

We construct an infinite family of triples $(G_k,H_k,T_k)$, where $G_k$ are 2-groups of increasing order, $H_k$ are index-2 subgroups of $G_k$, and $T_k$ are pairs of generators of $H_k$. We show that the triples $u_k = (G_k,H_k,T_k)$ are…

We show the homotopy spheres $\Sigma_{n} = -W\smile_{f^{n}}W$, 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…

We study the space of closed anti-invariant forms on an almost complex manifold, possibly non compact. We construct families of (non integrable) almost complex structures on $\R^4$, such that the space of closed $J$-anti-invariant forms is…

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…

We extend a construction of Stipsicz-Szab\'{o} of infinitely many irreducible exotic smooth structures of some closed four-manifolds with even $b_2^+$ and fundamental group $\mathbb{Z}/2\mathbb{Z}$. We use the double node surgery and…