arXiv research

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.

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

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.

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

Turing complete flow on 4-sphere preserves volume.

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

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

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…

Gluck twisting certain knots results in standard 4-spheres.

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

Smoothly knotted 5RP^2 found 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.

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

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

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

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

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…

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.

We prove that a constrained Willmore immersion of a 2-torus into the conformal 4-sphere is either of "finite type", that is, has a spectral curve of finite genus, or is of "holomorphic type" which means that it is super conformal or Euclidean minimal with planar ends. This implies that all constrained Willmore tori in …

For every N > 0 there exists a group of deficiency less than -N that arises as the fundamental group of a smooth homology 4-sphere and also as the fundamental group of the complement of a compact contractible submanifold of the 4-sphere. A group is the fundamental group of the complement of a contractible submanifold o…

This is a collection of notes on embedding problems for 3-manifolds. The main question explored is `which 3-manifolds embed smoothly in the 4-sphere?' The terrain of exploration is the Burton/Martelli/Matveev/Petronio census of triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There are 13766 mani…

We prove a convergence theorem on the moduli space of constant $σ_{2}$ metrics for conic 4-spheres. We show that when a numerical condition is convergent to the boundary case, the geometry of conic 4-spheres converges to the boundary case while preserving capacity.

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

Author provides an alternate proof of the free ribbon lemma.

D.Nash defined a family of homotopy 4-spheres in [11]. Proving that his manifolds ${\mathcal S}_{m,n,m',n'}$ are all real $S^4$, we find that they have handle decomposition with no 1-handles, two 2-handles and two 3-handles. The handle structures give new potential counterexamples of Property 2R conjecture.

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 diffeomorphic to $S^4$). It is still unknown if these new examples are in fact exoti…

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

We provide a computer-assisted proof of the holomorphy of the quartic and the octic meromorphic differentials arising in the main Theorem 4.11 of our paper 'The Classification of Branched Willmore spheres in the $3$-Sphere and the $4$-Sphere' (arXiv:1706.01405), using the free mathematical software Sage.