Related papers: Stable equivalence relations on 4-manifolds

We consider stable manifolds of a holomorphic diffeomorphism of a complex manifold. Using a conjugation of the dynamics to a (non-stationary) polynomial normal form, we show that typical stable manifolds are biholomorphic to complex…

We show that a closed four-manifold with $4\frac{1}{2}$-positive curvature operator of the second kind is diffeomorphic to a spherical space form. The curvature assumption is sharp as both $\mathbb{CP}^2$ and $\mathbb{S}^3 \times…

It was shown by Seaman that if a compact, oriented 4-dimensional riemannian manifold (M, g) of positive sectional curvature admits a harmonic 2-form of constant length, its intersection form is definite and such a harmonic form is unique up…

We produce examples of pairwise non-diffeomorphic closed irreducible 4-manifolds with non-trivial free abelian fundamental group of rank less than three and small Euler characteristic. These exotic smooth structures become standard after…

We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply…

We construct infinitely many smooth 4-manifolds which are homotopy equivalent to $S^2$ but do not admit a spine, i.e., a piecewise-linear embedding of $S^2$ which realizes the homotopy equivalence. This is the remaining case in the…

One can define the complexity of a smooth 4-manifold as the minimal sum of the number of disks, strands and crossings in a Kirby diagram. Martelli proved that the number of homeomorphism classes of complexity less than n grows as $n^2$. In…

We prove that homological stability fails for the moduli space of any simply-connected closed smooth 4-manifold in any degree of homology, unlike what happens in all dimensions $\neq 4$. We detect also the homological discrepancy between…

Knots naturally appear in continuous dynamical systems as flow periodic trajectories. However, discrete dynamical systems are also closely connected with the theory of knots and links. For example, for Pixton diffeomorphisms, the…

In this paper, we classify simply connected closed smooth $13$-dimensional manifolds whose cohomology ring is isomorphic to that of $\mb{CP}^3\times S^7$, up to diffeomorphism, homeomorphism, and homotopy equivalence. Furthermore, if such a…

Let $M$ and $N$ be smooth manifolds, with $M$ closed and connected. If the $C^r$--diffeomorphism group of $M$ is elementarily equivalent to the $C^s$--diffeomorphism group of $N$ for some $r,s\in[1,\infty)\cup\{0,\infty\}$, then $r=s$ and…

The homology groups of the automorphism group of a free group are known to stabilize as the number of generators of the free group goes to infinity, and this paper relativizes this result to a family of groups that can be defined in terms…

We prove that for many degrees in a stable range the homotopy groups of the moduli space of metrics of positive scalar curvature on S^n and on other manifolds are non-trivial. This is achieved by further developing and then applying a…

In a small simply-connected closed 4-manifold, we construct infinitely many pairs of exotic codimension-$1$ submanifolds with diffeomorphic complements that remain exotic after any number of stabilizations by $ S^2 \times S^2$. We also give…

We prove a sharp stable $2$-systolic inequality for complex projective space under the scalar curvature lower bound of the normalized Fubini-Study metric. If $M$ is diffeomorphic to $\mathbb{C}\mathrm{P}^n$ and $\mathrm{scal}_g\ge 4n(n+1)$,…

A link in the 3-sphere is homotopically trivial, according to Milnor, if its components bound disjoint maps of disks in the 4-ball. This paper concerns the question of what spaces give rise to the same class of homotopically trivial links…

In this article, we show that, at least for non-simply connected case, there exist an infinite family of nondiffeomorphic symplectic 4-manifolds with the same Seiberg-Witten invariants. The main techniques are knot surgery and a covering…

We give constraints on smooth families of 4-manifolds with boundary using Manolescu's Seiberg-Witten Floer stable homotopy type, provided that the fiberwise restrictions of the families to the boundaries are trivial families of 3-manifolds.…

The problem of splitting a homotopy equivalence along a submanifold is closely related to the surgery exact sequence and to the problem of surgery of manifold pairs. In classical surgery theory there exist two approaches to surgery in the…

Let $F\in\mathrm{Diff}(\mathbb{C}^2,0)$ be a germ of a holomorphic diffeomorphism and let $\Gamma$ be an invariant formal curve of $F$. Assume that the restricted diffeomorphism $F|_{\Gamma}$ is either hyperbolic attracting or rationally…