arXiv research

Calegari's 4-spheres from fibered knots are proven standard.

New research finds 145 infinite families of CS spheres are standard.

Gluck twisting certain knots results in standard 4-spheres.

Study on Ricci flow on 4-spheres, proving standard sphere convergence.

Standardizes Dunfield-Gong's 4-sphere, solves knot sliceness problem.

Gompf proposed a conjecture on Cappell-Shaneson matrices whose affirmative answer implies that all Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We study Gompf conjecture on Cappell-Shaneson matrices using various algebraic number theoretic techniques. We find a hidden symmetry between…

Smoothly knotted 5RP^2 found in 4-sphere.

We prove that the infinite family of homotopy 4-spheres constructed by Daniel Nash are all diffeomorphic to 4-sphere.

Standard $S^4$ proved to be diffeomorphic to a curious homotopy sphere.

Akbulut has recently shown that an infinite family of Cappell-Shaneson homotopy 4-spheres is diffeomorphic to the standard 4-sphere. In the present paper, a strictly larger family is shown to be standard by a simpler method. This new approach uses no Kirby calculus except through the relatively simple 1979 paper of Akb…

Using techniques from the theory of Kirby calculus we give an explicit construction of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to the standard 4-sphere.

The paper constructs many knotted and linked objects in higher dimensions.

Heegaard diagrams for 5-manifolds help in understanding their structure.

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.

The paper constructs homotopy 4-spheres using pochette surgery.

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

Paper proves every stable 4-sphere has a unique diffeomorphism class.

We discuss the relation between Fintushel-Stern knot surgery operation on 4-manifolds and Scharlemann manifolds, and as a corollary show that they all are standard. Along the way we show that the fishtail can exotically knot in the 4-sphere infinitely many ways.

Study pochette surgery on 4-manifolds, focusing on 4-spheres.

A surface in the 4-sphere is trivially embedded, if it bounds a 3-dimensional handle body in the 4-sphere. For a surface trivially embedded in the 4-sphere, a diffeomorphism over this surface is extensible if and only if this preserves the Rokhlin quadratic form of this embedded surface.

The Price twist creates three 4-manifolds from a 4-sphere.

We show that by performing the Gluck twist along the 2-knot $K^2_{pq}$ derived from two ribbon presentations of the ribbon 1-knot $K(p,q)$ we get the standard 4-sphere $S^4$. In the proof we apply Kirby calculus.

Turing complete flow on 4-sphere preserves volume.

The purpose of this paper is to study geometrically simply-connected homotopy 4-spheres by analyzing $n$-component links with a Dehn surgery realizing $\#^n(S^1\times S^2)$. We call such links $n$R-links. Our main result is that a homotopy 4-sphere that can be built without 1-handles and with only two 2-handles is diff…

New theory proves infinite homology 3-spheres in homology 4-spheres.

Researchers found non-smoothable surfaces in a 4-sphere, solving K3 problems.

New spanning 3-disks found for unlink in 4-sphere.

New homotopy 4-spheres and real projective 4-spaces created.

Examples are given to show that some compact contractible 4-manifolds can be knotted in the 4-sphere. It is then proved that any finitely presented perfect group with a balanced presentation is a knot group for an embedding of some contractible 4-manifold in the 4-sphere.

Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.

New findings about twists in 4-sphere diffeomorphisms.

3-balls in 4-sphere become isotopic in 5-ball.

We present an infinite sequence of smooth embeddings of a connected sum of 6 projective planes in the 4-sphere, which are all ambient homeomorphic, but pairwise ambient non-diffeomorphic. The double covers of the 4-sphere ramified along these surfaces form a family of the exotic $\Bbb CP^2#5\bar{\Bbb CP^2}$ constructed…

We derive the sharp Moser-Trudinger-Onofri inequalities on the standard $n$-sphere and CR $(2n+1)$- sphere as the limit of the sharp fractional Sobolev inequalities for all $n\ge 1$. On the $2$-sphere and $4$-sphere, this was established recently by S.-Y. Chang and F. Wang. Our proof uses an alternative and elementary …

Generators of the 4-sphere's smooth mapping class group via diffeomorphisms of Montesinos twins.

In this paper, Problem 4.17 on R. Kirby's problem list is solved by constructing infinitely many aspherical 4-manifolds that are homology 4-spheres

Researchers create infinite Brunnian links of 3-balls in 4-sphere.

Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.

Infinite Klein bottles with 4-fold meridians found.

In relation to the 4-dimensional smooth Poincaré conjecture we construct a tentative invariant of homotopy 4-spheres using embedded contact homology (ECH) and Seiberg-Witten theory (SWF). But for good reason it is a constant value independent of the sphere, so this null-result demonstrates that one should not try to us…

The only finite nonabelian simple group acting on a homology 3-sphere - necessarily non-freely - is the dodecahedral group $\Bbb A_5 \cong {\rm PSL}(2,5)$ (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group $\Bbb A_5^* \cong {\rm SL}(2,5)$). In the present pa…

This is the first comprehensive introduction to the authors' recent attempts toward a better understanding of the global concepts behind spinor representations of surfaces in 3-space. The important new aspect is a quaternionic-valued function theory, whose "meromorphic functions" are conformal maps into quaternions, wh…

We discuss the constant $σ_{2}$ problem for conic 4-spheres. Based on earlier works of Chang-Han-Yang and Han-Li-Teixeira, we are able to find a necessary condition for the existence problem. In particular, when the condition is sharp, we have the uniqueness result similar to that of Troyanov in dimension 2. It indicat…

We characterize Willmore tori in the 4-sphere with nontrivial normal bundle as Twistor projections of elliptic curves in complex projective space or as inverted minimal tori (with planar ends) in Euclidean 4-space.

Explains a 1978 construction for Yang-Mills instantons.

For a non-orientable closed surface standardly embedded in the 4-sphere, a diffeomorphism over this surface is extendable if and only if this diffeomorphism preserves the Guillou-Marin quadratic form of this embedded surface.

We show that a finite group which admits a faithful, smooth, orientation-preserving action on a homology 4-sphere, and in particular on the 4-sphere, is isomorphic to a subgroup of the orthogonal group SO(5), by explicitly determining the various groups which can occur (up to an indetermination of index two in the case…

New 3D handlebodies in 4-sphere and 5-ball are not isotopic even with same boundary.